Thread: "Introducing "MagicTile""

From: Roice Nelson <roice3@gmail.com>
Date: Sat, 30 Jan 2010 21:04:50 -0600
Subject: Introducing "MagicTile"



--0015175884b6d33edd047e6d21bf
Content-Type: text/plain; charset=ISO-8859-1

Thanks to everyone for the thoughtful feedback on my question this week. I
appreciate it, and it was good to get your perspectives.

I think I'm ready enough to share a first pass of the new Rubik analogue I
started playing with before the MC4D 4.0 fun, which I mentioned the
possibility of heresome
time ago. While you might observe it doesn't quite fall into the
category of hyperpuzzles, it does in at least once sense mentioned below :D
Here is the page with the download, pictures, and a
video.
To describe the analogue idea, I'll just quote the beginning of the
explanation on that page:


> This program aims to support twisty puzzles based on regular polygonal
> tilings having Schlafli
> symbols of the form {p,3}
> for any p>=2. That is, all regular tilings of polygons with two or more
> sides, where three tiles (puzzle faces) meet at a vertex. The Rubik's cube
> is the special case where faces are squares (p=4). The other familiar
> special cases are the Megaminx (p=5) and the Pyraminx (p=3), although you'll
> discover the last takes a slightly different form under this abstraction
> (akin to Jing's Pyraminx ).
> All the other puzzles are new as far as I know, and some may be surprising,
> e.g. the puzzles based on digons
> (p=2).


> Each 2D tiling admits a particular constant curvature (homogenous)
> geometry. The geometry is Spherical for p=2 to p=5, Euclidean (flat) for
> p=6, and Hyperbolic for p>=7. Since you can't "isometrically embed" the
> entire hyperbolic plane in 3-space,
> I have a connection to hyperpuzzling even
> though I'm talking about 2D tilings!

...




I've actually only solved the 3x3x3 on it so far, and I wonder if it may be
more fun to watch than play! I've been calling it MagicTile, though perhaps
there could be something better? As with everything, it is a known work in
progress (the length of the task list has grown to scary proportions). I
have no plans for further development at the moment, though I'll happily fix
any glaring bugs.

Enjoy!
Roice

P.S. This is the only "twisty puzzle" group I'm active in, so if any of you
are also members of other groups and think they would be interested to hear
about these new puzzles, I'll appreciate the exposure :)

--0015175884b6d33edd047e6d21bf
Content-Type: text/html; charset=ISO-8859-1
Content-Transfer-Encoding: quoted-printable

Thanks to everyone for the thoughtful feedback on my question this week. =
=A0I appreciate it, and it was good to get your perspectives.


I think I'm ready enough to share a first pass of the new Rubi=
k analogue I started playing with before the MC4D 4.0 fun, which I =3D"http://games.groups.yahoo.com/group/4D_Cubing/message/541" target=3D"_b=
lank">mentioned the possibility of here
some time ago.=A0 While you mig=
ht observe it doesn't quite fall into the category of hyperpuzzles, it =
does in at least once sense mentioned below=A0:D =A0Here is the http://www.gravitation3d.com/magictile" target=3D"_blank">page with the dow=
nload, pictures, and a video
.=A0 To describe the analogue idea, I'l=
l just quote the beginning of the explanation on that page:



rgin-right:0px;margin-bottom:0px;margin-left:0.8ex;border-left-width:1px;bo=
rder-left-color:rgb(204, 204, 204);border-left-style:solid;padding-left:1ex=
">


margin-right:0px;margin-bottom:0px;margin-left:0.8ex;border-left-width:1px;=
border-left-color:rgb(204, 204, 204);border-left-style:solid;padding-left:1=
ex">

This program aims to support twisty puzzles based on=A0.wikipedia.org/wiki/Regular_tessellation" target=3D"_blank">#000000">regular polygonal tilingsan>=A0having mbol" target=3D"_blank">ion:none">Schlafli symbols=A0of the form {p,3} for any p&=
gt;=3D2. That is, all regular tilings of polygons with two or more sides, w=
here three tiles (puzzle faces) meet at a vertex. The Rubik's cube is t=
he special case where faces are squares (p=3D4). The other familiar special=
cases are the Megaminx (p=3D5) and the Pyraminx (p=3D3), although you'=
ll discover the last takes a slightly different form under this abstraction=
(akin to=A0=3D"_blank">Ji=
ng's Pyraminx
). All the other puzzles are new as far =
as I know, and some may be surprising, e.g. the puzzles based on "http://en.wikipedia.org/wiki/Digon" target=3D"_blank">00">digons=A0(p=3D2)=
.


margin-bottom:0px;margin-left:0.8ex;border-left-width:1px;border-left-color=
:rgb(204, 204, 204);border-left-style:solid;padding-left:1ex">

margin-right:0px;margin-bottom:0px;margin-left:0.8ex;border-left-width:1px;=
border-left-color:rgb(204, 204, 204);border-left-style:solid;padding-left:1=
ex">

Each 2D tiling admits a particular constant curvature (homogenous) geometry=
. The geometry is Spherical for p=3D2 to p=3D5, Euclidean (flat) for p=3D6,=
and Hyperbolic for p>=3D7. Since=A0du/~dwh/papers/crochet/crochet.html" target=3D"_blank">00">you can't "isometrically =
embed" the entire hyperbolic plane in 3-space
, I hav=
e a connection to=A0g/" target=3D"_blank">n:none">hyperpuzzling=A0even though I'm talking about=
2D tilings!=A0


margin-bottom:0px;margin-left:0.8ex;border-left-width:1px;border-left-color=
:rgb(204, 204, 204);border-left-style:solid;padding-left:1ex">
...
argin-right:0px;margin-bottom:0px;margin-left:0.8ex;border-left-width:1px;b=
order-left-color:rgb(204, 204, 204);border-left-style:solid;padding-left:1e=
x">

=A0

I've actually only solved the 3x3x3 on i=
t so far, and I wonder if it may be more fun to watch than play! =A0I'v=
e been calling it MagicTile, though perhaps there could be something better=
? =A0As with everything, it is a known work in progress (the length of the =
task list has grown to scary proportions). =A0I have no plans for further d=
evelopment at the moment, though I'll happily fix any glaring bugs.v>



Enjoy!
Roice

P.S. This is the only "twisty puzzle" =
group I'm active in, so if any of you are also members of other groups =
and think they would be interested to hear about these new puzzles,=A0I'=
;ll appreciate the exposure :)




--0015175884b6d33edd047e6d21bf--




From: Matthew Galla <mgalla@trinity.edu>
Date: Sun, 31 Jan 2010 03:37:05 -0600
Subject: Re: [MC4D] Introducing "MagicTile"



--001485f78c6894ba00047e729c71
Content-Type: text/plain; charset=windows-1252
Content-Transfer-Encoding: quoted-printable

Very nice, Roice.

Although the options are clearly limited, this program is a work of art. Th=
e
hyperbolic face patterns combined with the scrambled colors are absolutely
beautiful.

I played around with some puzzles I already understood, and it takes a whil=
e
to get used to the little quirks in your program, but well worth it! I did
notice that the outermost face for the cube and megaminx series is
controlled opposite to my intuition. If I am thinking in terms of macros an=
d
try to apply a macro I know works near the center of the puzzle on the
outermost face, I find that I must invert every move on the outermost face.
A closer look reveals that this is because the outermost face is inverted.
Now this is just an idea, but have you considered inverting the movement fo=
r
just the outermost face? Although it my confuse some things visually, I
think it may be an overall improvement solving-wise.

Also, most of your 2-layer puzzles are currently not working (which I'm sur=
e
you already know). Are you looking into correcting this function of the
program? If so, can we expect puzzles with an even number of layers >2? For
puzzles with even layers (excluding cube) the visual pieces will have to
pass under/over/through each other. This is an inevitable behavior if you
restrict the exterior shape of a puzzle (which your program does because it
forces it to be drawn on a hyperplane). However, as you demonstrated with
the two-layered megaminx (impossiball) this is clearly do-able.

I am also looking forward to an updatewhere we can reorient some of these
puzzles! This is allowable on the cubical and dodecahedral puzzles by
holding down every layer number, but I would love to watch some these
hyperplane tesselations shift!

My favorite thing about your program, however, is the identical puzzles wit=
h
different sticker patterns. I am very interested to know how you came up
with the different patterns of colors on say, {6,3}, as well as the other
puzzles with multiple color-pattern options.

All in all, an excellent program that opens up a world of puzzles I had
never considered before! Although, I should say that none of the puzzles in
your program are very hard ;)

Thank you for once again expanding the limits on twisty puzzles!
Matt Galla
PS How many moves counts as an official scramble so I can start submitting
my solves? :)
On Sat, Jan 30, 2010 at 9:04 PM, Roice Nelson wrote:

>
>
> Thanks to everyone for the thoughtful feedback on my question this week. =
I
> appreciate it, and it was good to get your perspectives.
>
> I think I'm ready enough to share a first pass of the new Rubik analogue =
I
> started playing with before the MC4D 4.0 fun, which I mentioned the
> possibility of here/541>some time ago. While you might observe it doesn't quite fall into the
> category of hyperpuzzles, it does in at least once sense mentioned below =
:D
> Here is the page with the download, pictures, and a videovitation3d.com/magictile>.
> To describe the analogue idea, I'll just quote the beginning of the
> explanation on that page:
>
>
>> This program aims to support twisty puzzles based on regular polygonal
>> tilings having Schla=
fli
>> symbols of the form {p,3}
>> for any p>=3D2. That is, all regular tilings of polygons with two or mor=
e
>> sides, where three tiles (puzzle faces) meet at a vertex. The Rubik's cu=
be
>> is the special case where faces are squares (p=3D4). The other familiar
>> special cases are the Megaminx (p=3D5) and the Pyraminx (p=3D3), althoug=
h you'll
>> discover the last takes a slightly different form under this abstraction
>> (akin to Jing's Pyraminx )=
.
>> All the other puzzles are new as far as I know, and some may be surprisi=
ng,
>> e.g. the puzzles based on digons
>> (p=3D2).
>
>
>> Each 2D tiling admits a particular constant curvature (homogenous)
>> geometry. The geometry is Spherical for p=3D2 to p=3D5, Euclidean (flat)=
for
>> p=3D6, and Hyperbolic for p>=3D7. Since you can't "isometrically embed" =
the
>> entire hyperbolic plane in 3-spacers/crochet/crochet.html>,
>> I have a connection to hyperpuzzling/4D_Cubing/> even
>> though I'm talking about 2D tilings!
>
> ...
>
>
>
>
> I've actually only solved the 3x3x3 on it so far, and I wonder if it may =
be
> more fun to watch than play! I've been calling it MagicTile, though perh=
aps
> there could be something better? As with everything, it is a known work =
in
> progress (the length of the task list has grown to scary proportions). I
> have no plans for further development at the moment, though I'll happily =
fix
> any glaring bugs.
>
> Enjoy!
> Roice
>
> P.S. This is the only "twisty puzzle" group I'm active in, so if any of y=
ou
> are also members of other groups and think they would be interested to he=
ar
> about these new puzzles, I'll appreciate the exposure :)
>
>=20=20
>

--001485f78c6894ba00047e729c71
Content-Type: text/html; charset=windows-1252
Content-Transfer-Encoding: quoted-printable

Very nice, Roice.

=A0

Although the options are clearly limited, this program is a work of ar=
t. The hyperbolic face patterns combined with the scrambled colors are abso=
lutely beautiful.

=A0

I played around with some puzzles I already understood, and it takes a=
while to get used to the little quirks in your program, but well worth it!=
I did notice that the outermost face for=A0the cube and megaminx series is=
controlled opposite to my intuition. If I am thinking in terms of macros a=
nd try to apply a macro I know works near the center of the puzzle on the o=
utermost face, I find that I must invert every move on the outermost face. =
A closer look reveals that this is because the outermost face is inverted. =
Now this is just an idea, but have you considered inverting the movement fo=
r just the outermost face? Although it my confuse some things visually, I t=
hink it may be an overall improvement solving-wise.


=A0

Also, most of your 2-layer puzzles are currently not working (which I&=
#39;m sure you already know). Are you looking into correcting this function=
of the program? If so, can we expect puzzles with an even number of layers=
>2? For puzzles with even layers (excluding cube) the visual pieces wil=
l have to pass under/over/through each other. This is an inevitable behavio=
r if you restrict the exterior shape of a puzzle (which your program does b=
ecause it forces it to be drawn on a hyperplane). However, as you demonstra=
ted with the two-layered megaminx (impossiball) this is clearly do-able.iv>

=A0

I am also looking forward to an updatewhere we can reorient some of th=
ese puzzles! This is allowable on the cubical and dodecahedral puzzles by h=
olding down every layer number, but I would love to watch some these hyperp=
lane tesselations shift!


=A0

My favorite thing about your program, however, is the identical puzzle=
s with different sticker patterns. I am very interested to know how you cam=
e up with the different patterns of colors on say, {6,3}, as well as the ot=
her puzzles with multiple color-pattern options.


=A0

All in all, an excellent program that opens up a world of puzzles I ha=
d never considered before! Although, I should say that none of the puzzles =
in your program are very hard ;)

=A0

Thank you for once again expanding the limits on twisty puzzles!

Matt Galla

PS How many moves counts as an official scramble so I can start submit=
ting my solves? :)

On Sat, Jan 30, 2010 at 9:04 PM, Roice Nelson pan dir=3D"ltr"><roice3@gmail.com>> wrote:

; PADDING-LEFT: 1ex" class=3D"gmail_quote">
=A0=20



Thanks to everyone for the thoughtful feedback on my question this week.=
=A0I appreciate it, and it was good to get your perspectives.=20


I think I'm ready enough to share a first pass of the new Rubi=
k analogue I started playing with before the MC4D 4.0 fun, which I
=3D"http://games.groups.yahoo.com/group/4D_Cubing/message/541" target=3D"_b=
lank">mentioned the possibility of here
some time ago.=A0 While you mig=
ht observe it doesn't quite fall into the category of hyperpuzzles, it =
does in at least once sense mentioned below=A0:D =A0Here is the http://www.gravitation3d.com/magictile" target=3D"_blank">page with the dow=
nload, pictures, and a video
.=A0 To describe the analogue idea, I'l=
l just quote the beginning of the explanation on that page:




l_quote">

l_quote">This program aims to support twisty puzzles based on=A0http://en.wikipedia.org/wiki/Regular_tessellation" target=3D"_blank">color=3D"#000000">regular polygonal t=
ilings
=A0having chlafli_symbol" target=3D"_blank">XT-DECORATION: none">Schlafli symbols=A0of the form {p,3}=
for any p>=3D2. That is, all regular tilings of polygons with two or mo=
re sides, where three tiles (puzzle faces) meet at a vertex. The Rubik'=
s cube is the special case where faces are squares (p=3D4). The other famil=
iar special cases are the Megaminx (p=3D5) and the Pyraminx (p=3D3), althou=
gh you'll discover the last takes a slightly different form under this =
abstraction (akin to=A0W2c" target=3D"_blank">ON: none">Jing's Pyraminx). All the other puzzles are=
new as far as I know, and some may be surprising, e.g. the puzzles based o=
n lor=3D"#000000">digons<=
/a>=A0(p=3D2).


l_quote">

l_quote">Each 2D tiling admits a particular constant curvature (homogenous)=
geometry. The geometry is Spherical for p=3D2 to p=3D5, Euclidean (flat) f=
or p=3D6, and Hyperbolic for p>=3D7. Since=A0
cornell.edu/~dwh/papers/crochet/crochet.html" target=3D"_blank">=3D"#000000">you can't "isom=
etrically embed" the entire hyperbolic plane in 3-space
<=
/a>, I have a connection to=A0
p/4D_Cubing/" target=3D"_blank">-DECORATION: none">hyperpuzzling=A0even though I'm ta=
lking about 2D tilings!=A0


l_quote">...

l_quote">=A0


I've actually only solved the 3x3x3 on it so far, and I wonder=
if it may be more fun to watch than play! =A0I've been calling it Magi=
cTile, though perhaps there could be something better? =A0As with everythin=
g, it is a known work in progress (the length of the task list has grown to=
scary proportions). =A0I have no plans for further development at the mome=
nt, though I'll happily fix any glaring bugs.



Enjoy!
Roice

P.S. This is the only "twisty puzzle&q=
uot; group I'm active in, so if any of you are also members of other gr=
oups and think they would be interested to hear about these new puzzles,=A0=
I'll appreciate the exposure :)





=



--001485f78c6894ba00047e729c71--




From: Roice Nelson <roice3@gmail.com>
Date: Sun, 31 Jan 2010 15:26:39 -0600
Subject: Re: [MC4D] Introducing "MagicTile"



--0015175884b636ddb5047e7c86b8
Content-Type: text/plain; charset=ISO-8859-1

Thanks Matt!

I agree with your thoughts about inverting the outermost face, and have
added it to "the scary list". The same issue comes up in the 4D puzzles (at
least in my implementation of Magic120Cell since I allow showing cells
mirrored by the projection). I had originally reversed the twisting of
those faces, but ended up reverting that when I discovered some bug entropy
related to it. Also, I worried it could lead to confusion, like "hey,
that's not counterclockwise!". On the other hand, it makes the projection
effects explicit. It might actually be nice to have an option for this,
which I think I'll do when I get to it.

The two layer puzzles are an interesting topic for sure. I put them in the
list even though they don't currently do anything (except in the Megaminx
family), because they look pretty. I was on the fence about enabling the
Megaminx behavior as is, but did so solely because of the Impossiball. It
is the only puzzle in the list right now which overlaps material when
twisting. I actually feel there is a better length-2 analogy for the
Megaminx than the Impossiball, which is twisting a slicing circle that is a
"great circle" (cuts the unprojected puzzle sphere in half). The order of
this twist is 2 instead of 5 and swaps opposite faces. It also twists
without overlapping any material, which is why I prefer it. The reason I
didn't include this as the length-2 version is that this twist is
edge-centered rather than face-centered, and I had made the (arbitrary)
decision to restrict to the latter in the first version. I have to keep
myself sane somehow :)

The situation is similar in other puzzles. Check out the length-2 digonal
and trigonal puzzles and note that the nicest twists there (which don't
overlap material) are vertex-centered. To answer your question about
supporting all even length puzzles vs. just length-2, absolutely. The guts
are capable, I just didn't expose those as menu options until the behavior
is better worked out. And I welcome further discussion of the specification
of these puzzles! In particular, what would be a good way to specify
edge/vertex twists?

When you get to the infinite tilings, thinking about even-layers becomes
even stranger, and there is analogizing work to be done :) In the hexagonal
case, I think the right twist of the length-2 puzzle would be a translation
of half of the puzzle! (do you agree?) Again, the reason I favor this is
because no material overlaps. The problem I ran into was how to specify
such twists elegantly. Try thinking about it and backing yourself into some
corners :) One thing I can say is that a { clicked cell + direction }
simply isn't enough information to specify it. The same is true in the
hyperbolic cases.

Related to the last paragraph, it is interesting that outer twists for the
infinite tilings don't make sense regardless of whether the puzzle is even
or odd. What would a slice-2 twist on a length-3 hexagonal puzzle do? The
topology restricts the movement, which (I think) makes sense to me if I
picture the hexagonal puzzle on a torus instead of unrolled as in MagicTile.
Anyway, I'm perhaps getting a little off topic, but the reason to mention
this is that we won't be able to specify reorientations for the infinite
tilings as a twist "with all slices down". However, I'd love to see
reorientations (both view rotations and panning) done some other way in both
the spherical/hyperbolic cases. To see a really nice example of panning
hyperbolic space, check out Don's hyperbolic tessellation
applet!
The are some big challenges to get this working in MagicTile, and
performance is one of the largest, since so much needs to be drawn.

In regards to your question about the tiling patterns, there is a simple
procedure I used to get the current list, and it involved specifying two
numbers. First was the number of reflections from a "fundamental" cell to
an "orbit" cell (I actually called them masters/slaves in the code). Second
was which polygon segment to do this reflection across. So for example,
take a look at the 6-colored octagonal puzzle. This would be 2,4
(equivalently 4,2). The white center cell is reflected twice across the 4th
segment of each adjacent cell, and recursively thereafter. I wish I
understood this all better, as I actually just had the program run through a
loop to see which of the configurations "converged". Some end up fitting
together and some don't. Math Magic! Hope this helped clarify.
Tilings.org has some papers with lists that would probably help expose the
magic more. Btw, the patterns that work in the hexagonal case end up
producing puzzles where the number of colors are perfect squares, go figure!
And I wasn't able to find working patterns for polygons with 11 and 13
sides, though I wonder if they exist and I wasn't recursing deeply enough.

So there is plenty more discussion that could happen, but I don't want to
overload it right off the bat. One last thing though related to your final
comment. I put a feature in the program just for you Matt! Under "Options
-> Edit Settings...", play with the "Slicing Circles Expansion Factor".
This is analogous to "deepening the cuts" on the original puzzles, which I
know you've wanted in the 4D puzzles. It is a fully experimental setting
and I know cases where it doesn't work well, but often it does. Try 1.4 on
a Megaminx for a more difficult puzzle!

Also, if anyone feels some of this discussion shouldn't be on the
hypercubing mailing list, let me know. I'm a little worried some might feel
the hyperpuzzling connection a little too tenuous.

All the best,
Roice


On Sun, Jan 31, 2010 at 3:37 AM, Matthew Galla wrote:

>
>
> Very nice, Roice.
>
> Although the options are clearly limited, this program is a work of art.
> The hyperbolic face patterns combined with the scrambled colors are
> absolutely beautiful.
>
> I played around with some puzzles I already understood, and it takes a
> while to get used to the little quirks in your program, but well worth it! I
> did notice that the outermost face for the cube and megaminx series is
> controlled opposite to my intuition. If I am thinking in terms of macros and
> try to apply a macro I know works near the center of the puzzle on the
> outermost face, I find that I must invert every move on the outermost face.
> A closer look reveals that this is because the outermost face is inverted.
> Now this is just an idea, but have you considered inverting the movement for
> just the outermost face? Although it my confuse some things visually, I
> think it may be an overall improvement solving-wise.
>
> Also, most of your 2-layer puzzles are currently not working (which I'm
> sure you already know). Are you looking into correcting this function of the
> program? If so, can we expect puzzles with an even number of layers >2? For
> puzzles with even layers (excluding cube) the visual pieces will have to
> pass under/over/through each other. This is an inevitable behavior if you
> restrict the exterior shape of a puzzle (which your program does because it
> forces it to be drawn on a hyperplane). However, as you demonstrated with
> the two-layered megaminx (impossiball) this is clearly do-able.
>
> I am also looking forward to an updatewhere we can reorient some of these
> puzzles! This is allowable on the cubical and dodecahedral puzzles by
> holding down every layer number, but I would love to watch some these
> hyperplane tesselations shift!
>
> My favorite thing about your program, however, is the identical puzzles
> with different sticker patterns. I am very interested to know how you came
> up with the different patterns of colors on say, {6,3}, as well as the other
> puzzles with multiple color-pattern options.
>
> All in all, an excellent program that opens up a world of puzzles I had
> never considered before! Although, I should say that none of the puzzles in
> your program are very hard ;)
>
> Thank you for once again expanding the limits on twisty puzzles!
> Matt Galla
> PS How many moves counts as an official scramble so I can start submitting
> my solves? :)
> On Sat, Jan 30, 2010 at 9:04 PM, Roice Nelson wrote:
>
>>
>>
>> Thanks to everyone for the thoughtful feedback on my question this week.
>> I appreciate it, and it was good to get your perspectives.
>>
>> I think I'm ready enough to share a first pass of the new Rubik analogue I
>> started playing with before the MC4D 4.0 fun, which I mentioned the
>> possibility of heresome time ago. While you might observe it doesn't quite fall into the
>> category of hyperpuzzles, it does in at least once sense mentioned below :D
>> Here is the page with the download, pictures, and a video.
>> To describe the analogue idea, I'll just quote the beginning of the
>> explanation on that page:
>>
>>
>>> This program aims to support twisty puzzles based on regular polygonal
>>> tilings having Schlafli
>>> symbols of the form {p,3}
>>> for any p>=2. That is, all regular tilings of polygons with two or more
>>> sides, where three tiles (puzzle faces) meet at a vertex. The Rubik's cube
>>> is the special case where faces are squares (p=4). The other familiar
>>> special cases are the Megaminx (p=5) and the Pyraminx (p=3), although you'll
>>> discover the last takes a slightly different form under this abstraction
>>> (akin to Jing's Pyraminx ).
>>> All the other puzzles are new as far as I know, and some may be surprising,
>>> e.g. the puzzles based on digons
>>> (p=2).
>>
>>
>>> Each 2D tiling admits a particular constant curvature (homogenous)
>>> geometry. The geometry is Spherical for p=2 to p=5, Euclidean (flat) for
>>> p=6, and Hyperbolic for p>=7. Since you can't "isometrically embed" the
>>> entire hyperbolic plane in 3-space,
>>> I have a connection to hyperpuzzling even
>>> though I'm talking about 2D tilings!
>>
>> ...
>>
>>
>>
>>
>> I've actually only solved the 3x3x3 on it so far, and I wonder if it may
>> be more fun to watch than play! I've been calling it MagicTile, though
>> perhaps there could be something better? As with everything, it is a known
>> work in progress (the length of the task list has grown to scary
>> proportions). I have no plans for further development at the moment, though
>> I'll happily fix any glaring bugs.
>>
>> Enjoy!
>> Roice
>>
>> P.S. This is the only "twisty puzzle" group I'm active in, so if any of
>> you are also members of other groups and think they would be interested to
>> hear about these new puzzles, I'll appreciate the exposure :)
>>
>>
>
>
>

--0015175884b636ddb5047e7c86b8
Content-Type: text/html; charset=ISO-8859-1
Content-Transfer-Encoding: quoted-printable

Thanks Matt!


I agree with your thoughts about inverting =
the outermost face, and have added it to "the scary list". =A0The=
same issue comes up in the 4D puzzles (at least in my implementation of Ma=
gic120Cell since I allow showing cells mirrored by the projection). =A0I ha=
d originally reversed the twisting of those faces, but ended up reverting t=
hat when I discovered some bug entropy related to it. =A0Also, I worried it=
could lead to confusion, like "hey, that's not counterclockwise!&=
quot;. =A0On the other hand, it makes the projection effects explicit. =A0I=
t might actually be nice to have an option for this, which I think I'll=
do when I get to it.


The two layer puzzles are an interesting topic for sure=
. =A0I put them in the list even though they don't currently do anythin=
g (except in the Megaminx family), because they look pretty. =A0I was on th=
e fence about enabling the Megaminx behavior as is, but did so solely becau=
se of the Impossiball. =A0It is the only puzzle in the list right now which=
overlaps material when twisting. =A0I actually feel there is a better leng=
th-2 analogy for the Megaminx than the Impossiball, which is twisting a sli=
cing circle that is a "great circle" (cuts the unprojected puzzle=
sphere in half). =A0The order of this twist is 2 instead of 5 and swaps op=
posite faces. =A0It also twists without overlapping any material, which is =
why I prefer it. =A0The reason I didn't include this as the length-2 ve=
rsion is that this twist is edge-centered rather than face-centered, and I =
had made the (arbitrary) decision to restrict to the latter in the first ve=
rsion. =A0I have to keep myself sane somehow :)


The situation is similar in other puzzles. =A0Check out the =
length-2 digonal and trigonal puzzles and note that the nicest twists there=
(which don't overlap material) are vertex-centered. =A0To answer your =
question about supporting all even length puzzles vs. just length-2, absolu=
tely. =A0The guts are capable, I just didn't expose those as menu optio=
ns until the behavior is better worked out. =A0And I welcome further discus=
sion of the specification of these puzzles! =A0In particular, what would be=
a good way to specify edge/vertex twists?


When you get to the infinite tilings, thinking about ev=
en-layers becomes even stranger, and there is analogizing work to be done :=
) =A0In the hexagonal case, I think the right twist of the length-2 puzzle =
would be a translation of half of the puzzle! =A0(do you agree?) =A0Again, =
the reason I favor this is because no material overlaps. =A0The problem I r=
an into was how to specify such twists elegantly. =A0Try thinking about it =
and backing yourself into some corners :) =A0One thing I can say is that a =
{ clicked cell + direction } simply isn't enough information to specify=
it. =A0The same is true in the hyperbolic cases.


Related to the last paragraph, it is interesting that o=
uter twists for the infinite tilings don't make sense regardless of whe=
ther the puzzle is even or odd. =A0What would a slice-2 twist on a length-3=
hexagonal puzzle do? =A0The topology restricts the movement, which (I thin=
k) makes sense to me if I picture the hexagonal puzzle on a torus instead o=
f unrolled as in MagicTile. =A0Anyway, I'm perhaps getting a little off=
topic, but the reason to mention this is that we won't be able to spec=
ify reorientations for the infinite tilings as a twist "with all slice=
s down". =A0However, I'd love to see reorientations (both view rot=
ations and panning) done some other way in both the spherical/hyperbolic ca=
ses. =A0To see a really nice example of panning hyperbolic space, check out=
Don's hyperb=
olic tessellation applet
! =A0The are some big challenges to get this wo=
rking in MagicTile, and performance is one of the largest, since so much ne=
eds to be drawn.


In regards to your question about the tiling patterns, =
there is a simple procedure I used to get the current list, and it involved=
specifying two numbers. =A0First was the number of reflections from a &quo=
t;fundamental" cell to an "orbit" cell (I actually called th=
em masters/slaves in the code). =A0Second was which polygon segment to do t=
his reflection across. =A0So for example, take a look at the 6-colored octa=
gonal puzzle. =A0This would be 2,4 (equivalently 4,2). =A0The white center =
cell is reflected twice across the 4th segment of each adjacent cell, and r=
ecursively thereafter. =A0I wish I understood this all better, as I actuall=
y just had the program run through a loop to see which of the configuration=
s "converged". =A0Some end up fitting together and some don't=
. =A0Math Magic! =A0Hope this helped clarify. =A0Tilings.org has some paper=
s with lists that would probably help expose the magic more. =A0Btw, the pa=
tterns that work in the hexagonal case end up producing puzzles where the n=
umber of colors are perfect squares, go figure! =A0And I wasn't able to=
find working patterns for polygons with 11 and 13 sides, though I wonder i=
f they exist and I wasn't recursing deeply enough.


So there is plenty more discussion that could happen, b=
ut I don't want to overload it right off the bat. =A0One last thing tho=
ugh related to your final comment. =A0I put a feature in the program just f=
or you Matt! =A0Under "Options -> Edit Settings...", play with=
the "Slicing Circles Expansion Factor". =A0This is analogous to =
"deepening the cuts" on the original puzzles, which I know you=
9;ve wanted in the 4D puzzles. =A0It is a fully experimental setting and I =
know cases where it doesn't work well, but often it does. =A0Try 1.4 on=
a Megaminx for a more difficult puzzle!


Also, if anyone feels some of this discussion shouldn&#=
39;t be on the hypercubing mailing list, let me know. =A0I'm a little w=
orried some might feel the hyperpuzzling connection a little too tenuous.div>

All the best,
Roice

=

On Sun, Jan 31, 2010 at 3:37 AM, Matthew Gal=
la <mgalla@trini=
ty.edu
>
wrote:

x #ccc solid;padding-left:1ex;">






=20=20=20=20=20=20=20=20
















Very nice, Roice.

=A0

Although the options are clearly limited, this program is a work of ar=
t. The hyperbolic face patterns combined with the scrambled colors are abso=
lutely beautiful.

=A0

I played around with some puzzles I already understood, and it takes a=
while to get used to the little quirks in your program, but well worth it!=
I did notice that the outermost face for=A0the cube and megaminx series is=
controlled opposite to my intuition. If I am thinking in terms of macros a=
nd try to apply a macro I know works near the center of the puzzle on the o=
utermost face, I find that I must invert every move on the outermost face. =
A closer look reveals that this is because the outermost face is inverted. =
Now this is just an idea, but have you considered inverting the movement fo=
r just the outermost face? Although it my confuse some things visually, I t=
hink it may be an overall improvement solving-wise.



=A0

Also, most of your 2-layer puzzles are currently not working (which I&=
#39;m sure you already know). Are you looking into correcting this function=
of the program? If so, can we expect puzzles with an even number of layers=
>2? For puzzles with even layers (excluding cube) the visual pieces wil=
l have to pass under/over/through each other. This is an inevitable behavio=
r if you restrict the exterior shape of a puzzle (which your program does b=
ecause it forces it to be drawn on a hyperplane). However, as you demonstra=
ted with the two-layered megaminx (impossiball) this is clearly do-able.iv>


=A0

I am also looking forward to an updatewhere we can reorient some of th=
ese puzzles! This is allowable on the cubical and dodecahedral puzzles by h=
olding down every layer number, but I would love to watch some these hyperp=
lane tesselations shift!



=A0

My favorite thing about your program, however, is the identical puzzle=
s with different sticker patterns. I am very interested to know how you cam=
e up with the different patterns of colors on say, {6,3}, as well as the ot=
her puzzles with multiple color-pattern options.



=A0

All in all, an excellent program that opens up a world of puzzles I ha=
d never considered before! Although, I should say that none of the puzzles =
in your program are very hard ;)

=A0

Thank you for once again expanding the limits on twisty puzzles!

Matt Galla

PS How many moves counts as an official scramble so I can start submit=
ting my solves? :)

On Sat, Jan 30, 2010 at 9:04 PM, Roice Nelson pan dir=3D"ltr"><r=
oice3@gmail.com
> wrote:

dding-left:1ex" class=3D"gmail_quote">
=A0=20



Thanks to everyone for the thoughtful feedback on my question this week.=
=A0I appreciate it, and it was good to get your perspectives.=20


I think I'm ready enough to share a first pass of the new =
Rubik analogue I started playing with before the MC4D 4.0 fun, which I ref=3D"http://games.groups.yahoo.com/group/4D_Cubing/message/541" target=3D=
"_blank">mentioned the possibility of here
some time ago.=A0 While you =
might observe it doesn't quite fall into the category of hyperpuzzles, =
it does in at least once sense mentioned below=A0:D =A0Here is the =3D"http://www.gravitation3d.com/magictile" target=3D"_blank">page with the=
download, pictures, and a video
.=A0 To describe the analogue idea, I&#=
39;ll just quote the beginning of the explanation on that page:





_quote">

_quote">This program aims to support twisty puzzles based on=A0ttp://en.wikipedia.org/wiki/Regular_tessellation" target=3D"_blank">olor=3D"#000000">regular polygonal til=
ings
=A0having lafli_symbol" target=3D"_blank">-decoration:none">Schlafli symbols=A0of the form {p,3} fo=
r any p>=3D2. That is, all regular tilings of polygons with two or more =
sides, where three tiles (puzzle faces) meet at a vertex. The Rubik's c=
ube is the special case where faces are squares (p=3D4). The other familiar=
special cases are the Megaminx (p=3D5) and the Pyraminx (p=3D3), although =
you'll discover the last takes a slightly different form under this abs=
traction (akin to=A0" target=3D"_blank">none">Jing's Pyraminx). All the other puzzles are new=
as far as I know, and some may be surprising, e.g. the puzzles based on href=3D"http://en.wikipedia.org/wiki/Digon" target=3D"_blank">=3D"#000000">digons=
=A0(p=3D2).



_quote">

_quote">Each 2D tiling admits a particular constant curvature (homogenous) =
geometry. The geometry is Spherical for p=3D2 to p=3D5, Euclidean (flat) fo=
r p=3D6, and Hyperbolic for p>=3D7. Since=A0ornell.edu/~dwh/papers/crochet/crochet.html" target=3D"_blank">=3D"#000000">you can't "isome=
trically embed" the entire hyperbolic plane in 3-space
a>, I have a connection to=A0
/4D_Cubing/" target=3D"_blank">decoration:none">hyperpuzzling=A0even though I'm talk=
ing about 2D tilings!=A0



_quote">...

_quote">=A0


I've actually only solved the 3x3x3 on it so far, and I wonder=
if it may be more fun to watch than play! =A0I've been calling it Magi=
cTile, though perhaps there could be something better? =A0As with everythin=
g, it is a known work in progress (the length of the task list has grown to=
scary proportions). =A0I have no plans for further development at the mome=
nt, though I'll happily fix any glaring bugs.




Enjoy!
Roice

P.S. This is the only "twisty puzzle&q=
uot; group I'm active in, so if any of you are also members of other gr=
oups and think they would be interested to hear about these new puzzles,=A0=
I'll appreciate the exposure :)






te>











=20=20=20=20
=20=20=20=20













--0015175884b636ddb5047e7c86b8--




From: Roice Nelson <roice3@gmail.com>
Date: Sun, 31 Jan 2010 17:52:47 -0600
Subject: Re: [MC4D] Introducing "MagicTile"



--0015175884b6da4e18047e7e90ce
Content-Type: text/plain; charset=ISO-8859-1

Greetings again,

I wanted to make a minor clarification. I didn't describe my desired
order-2 twist for a length-2 Megaminx well, and incorrectly wrote that this
twist would swap opposite faces. It would swap half the material for two
pairs of opposite faces. It would also completely swap a further two pairs
of faces (one pair is of adjacent faces, one is not, but neither are
opposite). Grab a Megaminx and picture an entire half of the puzzle being
rotated 180 degrees to see the twist. Hopefully the goal was clear enough
though... a length-2 Megaminx that wouldn't suffer from the weirdness of the
Impossiball. It seems a worthy aesthetic goal for puzzles to have no
overlapping material when twisting, but I'm always curious of other opinions
on things like that.

Cheers,
Roice


On Sun, Jan 31, 2010 at 3:26 PM, Roice Nelson wrote:

> Thanks Matt!
>
> I agree with your thoughts about inverting the outermost face, and have
> added it to "the scary list". The same issue comes up in the 4D puzzles (at
> least in my implementation of Magic120Cell since I allow showing cells
> mirrored by the projection). I had originally reversed the twisting of
> those faces, but ended up reverting that when I discovered some bug entropy
> related to it. Also, I worried it could lead to confusion, like "hey,
> that's not counterclockwise!". On the other hand, it makes the projection
> effects explicit. It might actually be nice to have an option for this,
> which I think I'll do when I get to it.
>
> The two layer puzzles are an interesting topic for sure. I put them in the
> list even though they don't currently do anything (except in the Megaminx
> family), because they look pretty. I was on the fence about enabling the
> Megaminx behavior as is, but did so solely because of the Impossiball. It
> is the only puzzle in the list right now which overlaps material when
> twisting. I actually feel there is a better length-2 analogy for the
> Megaminx than the Impossiball, which is twisting a slicing circle that is a
> "great circle" (cuts the unprojected puzzle sphere in half). The order of
> this twist is 2 instead of 5 and swaps opposite faces. It also twists
> without overlapping any material, which is why I prefer it. The reason I
> didn't include this as the length-2 version is that this twist is
> edge-centered rather than face-centered, and I had made the (arbitrary)
> decision to restrict to the latter in the first version. I have to keep
> myself sane somehow :)
>
> The situation is similar in other puzzles. Check out the length-2 digonal
> and trigonal puzzles and note that the nicest twists there (which don't
> overlap material) are vertex-centered. To answer your question about
> supporting all even length puzzles vs. just length-2, absolutely. The guts
> are capable, I just didn't expose those as menu options until the behavior
> is better worked out. And I welcome further discussion of the specification
> of these puzzles! In particular, what would be a good way to specify
> edge/vertex twists?
>
> When you get to the infinite tilings, thinking about even-layers becomes
> even stranger, and there is analogizing work to be done :) In the hexagonal
> case, I think the right twist of the length-2 puzzle would be a translation
> of half of the puzzle! (do you agree?) Again, the reason I favor this is
> because no material overlaps. The problem I ran into was how to specify
> such twists elegantly. Try thinking about it and backing yourself into some
> corners :) One thing I can say is that a { clicked cell + direction }
> simply isn't enough information to specify it. The same is true in the
> hyperbolic cases.
>
> Related to the last paragraph, it is interesting that outer twists for the
> infinite tilings don't make sense regardless of whether the puzzle is even
> or odd. What would a slice-2 twist on a length-3 hexagonal puzzle do? The
> topology restricts the movement, which (I think) makes sense to me if I
> picture the hexagonal puzzle on a torus instead of unrolled as in MagicTile.
> Anyway, I'm perhaps getting a little off topic, but the reason to mention
> this is that we won't be able to specify reorientations for the infinite
> tilings as a twist "with all slices down". However, I'd love to see
> reorientations (both view rotations and panning) done some other way in both
> the spherical/hyperbolic cases. To see a really nice example of panning
> hyperbolic space, check out Don's hyperbolic tessellation applet!
> The are some big challenges to get this working in MagicTile, and
> performance is one of the largest, since so much needs to be drawn.
>
> In regards to your question about the tiling patterns, there is a simple
> procedure I used to get the current list, and it involved specifying two
> numbers. First was the number of reflections from a "fundamental" cell to
> an "orbit" cell (I actually called them masters/slaves in the code). Second
> was which polygon segment to do this reflection across. So for example,
> take a look at the 6-colored octagonal puzzle. This would be 2,4
> (equivalently 4,2). The white center cell is reflected twice across the 4th
> segment of each adjacent cell, and recursively thereafter. I wish I
> understood this all better, as I actually just had the program run through a
> loop to see which of the configurations "converged". Some end up fitting
> together and some don't. Math Magic! Hope this helped clarify.
> Tilings.org has some papers with lists that would probably help expose the
> magic more. Btw, the patterns that work in the hexagonal case end up
> producing puzzles where the number of colors are perfect squares, go figure!
> And I wasn't able to find working patterns for polygons with 11 and 13
> sides, though I wonder if they exist and I wasn't recursing deeply enough.
>
> So there is plenty more discussion that could happen, but I don't want to
> overload it right off the bat. One last thing though related to your final
> comment. I put a feature in the program just for you Matt! Under "Options
> -> Edit Settings...", play with the "Slicing Circles Expansion Factor".
> This is analogous to "deepening the cuts" on the original puzzles, which I
> know you've wanted in the 4D puzzles. It is a fully experimental setting
> and I know cases where it doesn't work well, but often it does. Try 1.4 on
> a Megaminx for a more difficult puzzle!
>
> Also, if anyone feels some of this discussion shouldn't be on the
> hypercubing mailing list, let me know. I'm a little worried some might feel
> the hyperpuzzling connection a little too tenuous.
>
> All the best,
> Roice
>
>
> On Sun, Jan 31, 2010 at 3:37 AM, Matthew Galla wrote:
>
>>
>>
>> Very nice, Roice.
>>
>> Although the options are clearly limited, this program is a work of art.
>> The hyperbolic face patterns combined with the scrambled colors are
>> absolutely beautiful.
>>
>> I played around with some puzzles I already understood, and it takes a
>> while to get used to the little quirks in your program, but well worth it! I
>> did notice that the outermost face for the cube and megaminx series is
>> controlled opposite to my intuition. If I am thinking in terms of macros and
>> try to apply a macro I know works near the center of the puzzle on the
>> outermost face, I find that I must invert every move on the outermost face.
>> A closer look reveals that this is because the outermost face is inverted.
>> Now this is just an idea, but have you considered inverting the movement for
>> just the outermost face? Although it my confuse some things visually, I
>> think it may be an overall improvement solving-wise.
>>
>> Also, most of your 2-layer puzzles are currently not working (which I'm
>> sure you already know). Are you looking into correcting this function of the
>> program? If so, can we expect puzzles with an even number of layers >2? For
>> puzzles with even layers (excluding cube) the visual pieces will have to
>> pass under/over/through each other. This is an inevitable behavior if you
>> restrict the exterior shape of a puzzle (which your program does because it
>> forces it to be drawn on a hyperplane). However, as you demonstrated with
>> the two-layered megaminx (impossiball) this is clearly do-able.
>>
>> I am also looking forward to an updatewhere we can reorient some of these
>> puzzles! This is allowable on the cubical and dodecahedral puzzles by
>> holding down every layer number, but I would love to watch some these
>> hyperplane tesselations shift!
>>
>> My favorite thing about your program, however, is the identical puzzles
>> with different sticker patterns. I am very interested to know how you came
>> up with the different patterns of colors on say, {6,3}, as well as the other
>> puzzles with multiple color-pattern options.
>>
>> All in all, an excellent program that opens up a world of puzzles I had
>> never considered before! Although, I should say that none of the puzzles in
>> your program are very hard ;)
>>
>> Thank you for once again expanding the limits on twisty puzzles!
>> Matt Galla
>> PS How many moves counts as an official scramble so I can start submitting
>> my solves? :)
>> On Sat, Jan 30, 2010 at 9:04 PM, Roice Nelson wrote:
>>
>>>
>>>
>>> Thanks to everyone for the thoughtful feedback on my question this week.
>>> I appreciate it, and it was good to get your perspectives.
>>>
>>> I think I'm ready enough to share a first pass of the new Rubik analogue
>>> I started playing with before the MC4D 4.0 fun, which I mentioned the
>>> possibility of heresome time ago. While you might observe it doesn't quite fall into the
>>> category of hyperpuzzles, it does in at least once sense mentioned below :D
>>> Here is the page with the download, pictures, and a video.
>>> To describe the analogue idea, I'll just quote the beginning of the
>>> explanation on that page:
>>>
>>>
>>>> This program aims to support twisty puzzles based on regular polygonal
>>>> tilings having Schlafli
>>>> symbols of the form
>>>> {p,3} for any p>=2. That is, all regular tilings of polygons with two or
>>>> more sides, where three tiles (puzzle faces) meet at a vertex. The Rubik's
>>>> cube is the special case where faces are squares (p=4). The other familiar
>>>> special cases are the Megaminx (p=5) and the Pyraminx (p=3), although you'll
>>>> discover the last takes a slightly different form under this abstraction
>>>> (akin to Jing's Pyraminx ).
>>>> All the other puzzles are new as far as I know, and some may be surprising,
>>>> e.g. the puzzles based on digons
>>>> (p=2).
>>>
>>>
>>>> Each 2D tiling admits a particular constant curvature (homogenous)
>>>> geometry. The geometry is Spherical for p=2 to p=5, Euclidean (flat) for
>>>> p=6, and Hyperbolic for p>=7. Since you can't "isometrically embed" the
>>>> entire hyperbolic plane in 3-space,
>>>> I have a connection to hyperpuzzling even
>>>> though I'm talking about 2D tilings!
>>>
>>> ...
>>>
>>>
>>>
>>>
>>> I've actually only solved the 3x3x3 on it so far, and I wonder if it may
>>> be more fun to watch than play! I've been calling it MagicTile, though
>>> perhaps there could be something better? As with everything, it is a known
>>> work in progress (the length of the task list has grown to scary
>>> proportions). I have no plans for further development at the moment, though
>>> I'll happily fix any glaring bugs.
>>>
>>> Enjoy!
>>> Roice
>>>
>>> P.S. This is the only "twisty puzzle" group I'm active in, so if any of
>>> you are also members of other groups and think they would be interested to
>>> hear about these new puzzles, I'll appreciate the exposure :)
>>>
>>>
>>
>>
>>
>
>
>

--0015175884b6da4e18047e7e90ce
Content-Type: text/html; charset=ISO-8859-1
Content-Transfer-Encoding: quoted-printable

Greetings again,


I wanted to make a minor clarification.=
=A0I didn't describe my desired order-2 twist for a length-2 Megaminx =
well, and incorrectly wrote that this twist would swap opposite faces. =A0I=
t would swap half the material for two pairs of opposite faces. =A0It would=
also completely swap a further two pairs of faces (one pair is of adjacent=
faces, one is not, but neither are opposite). =A0Grab a Megaminx and pictu=
re an entire half of the puzzle being rotated 180 degrees to see the twist.=
=A0Hopefully the goal was clear enough though... a length-2 Megaminx that =
wouldn't suffer from the weirdness of the Impossiball. =A0It seems a wo=
rthy aesthetic goal for puzzles to have no overlapping material when twisti=
ng, but I'm always curious of other opinions on things like that.


Cheers,
Roice


"gmail_quote">On Sun, Jan 31, 2010 at 3:26 PM, Roice Nelson r"><roice3@gmail.com> =
wrote:

x #ccc solid;padding-left:1ex;">Thanks Matt!

I agree wit=
h your thoughts about inverting the outermost face, and have added it to &q=
uot;the scary list". =A0The same issue comes up in the 4D puzzles (at =
least in my implementation of Magic120Cell since I allow showing cells mirr=
ored by the projection). =A0I had originally reversed the twisting of those=
faces, but ended up reverting that when I discovered some bug entropy rela=
ted to it. =A0Also, I worried it could lead to confusion, like "hey, t=
hat's not counterclockwise!". =A0On the other hand, it makes the p=
rojection effects explicit. =A0It might actually be nice to have an option =
for this, which I think I'll do when I get to it.



The two layer puzzles are an interesting topic for sure=
. =A0I put them in the list even though they don't currently do anythin=
g (except in the Megaminx family), because they look pretty. =A0I was on th=
e fence about enabling the Megaminx behavior as is, but did so solely becau=
se of the Impossiball. =A0It is the only puzzle in the list right now which=
overlaps material when twisting. =A0I actually feel there is a better leng=
th-2 analogy for the Megaminx than the Impossiball, which is twisting a sli=
cing circle that is a "great circle" (cuts the unprojected puzzle=
sphere in half). =A0The order of this twist is 2 instead of 5 and swaps op=
posite faces. =A0It also twists without overlapping any material, which is =
why I prefer it. =A0The reason I didn't include this as the length-2 ve=
rsion is that this twist is edge-centered rather than face-centered, and I =
had made the (arbitrary) decision to restrict to the latter in the first ve=
rsion. =A0I have to keep myself sane somehow :)



The situation is similar in other puzzles. =A0Check out the =
length-2 digonal and trigonal puzzles and note that the nicest twists there=
(which don't overlap material) are vertex-centered. =A0To answer your =
question about supporting all even length puzzles vs. just length-2, absolu=
tely. =A0The guts are capable, I just didn't expose those as menu optio=
ns until the behavior is better worked out. =A0And I welcome further discus=
sion of the specification of these puzzles! =A0In particular, what would be=
a good way to specify edge/vertex twists?



When you get to the infinite tilings, thinking about ev=
en-layers becomes even stranger, and there is analogizing work to be done :=
) =A0In the hexagonal case, I think the right twist of the length-2 puzzle =
would be a translation of half of the puzzle! =A0(do you agree?) =A0Again, =
the reason I favor this is because no material overlaps. =A0The problem I r=
an into was how to specify such twists elegantly. =A0Try thinking about it =
and backing yourself into some corners :) =A0One thing I can say is that a =
{ clicked cell + direction } simply isn't enough information to specify=
it. =A0The same is true in the hyperbolic cases.



Related to the last paragraph, it is interesting that o=
uter twists for the infinite tilings don't make sense regardless of whe=
ther the puzzle is even or odd. =A0What would a slice-2 twist on a length-3=
hexagonal puzzle do? =A0The topology restricts the movement, which (I thin=
k) makes sense to me if I picture the hexagonal puzzle on a torus instead o=
f unrolled as in MagicTile. =A0Anyway, I'm perhaps getting a little off=
topic, but the reason to mention this is that we won't be able to spec=
ify reorientations for the infinite tilings as a twist "with all slice=
s down". =A0However, I'd love to see reorientations (both view rot=
ations and panning) done some other way in both the spherical/hyperbolic ca=
ses. =A0To see a really nice example of panning hyperbolic space, check out=
Don's =3D"_blank">hyperbolic tessellation applet! =A0The are some big challen=
ges to get this working in MagicTile, and performance is one of the largest=
, since so much needs to be drawn.



In regards to your question about the tiling patterns, =
there is a simple procedure I used to get the current list, and it involved=
specifying two numbers. =A0First was the number of reflections from a &quo=
t;fundamental" cell to an "orbit" cell (I actually called th=
em masters/slaves in the code). =A0Second was which polygon segment to do t=
his reflection across. =A0So for example, take a look at the 6-colored octa=
gonal puzzle. =A0This would be 2,4 (equivalently 4,2). =A0The white center =
cell is reflected twice across the 4th segment of each adjacent cell, and r=
ecursively thereafter. =A0I wish I understood this all better, as I actuall=
y just had the program run through a loop to see which of the configuration=
s "converged". =A0Some end up fitting together and some don't=
. =A0Math Magic! =A0Hope this helped clarify. =A0Tilings.org has some paper=
s with lists that would probably help expose the magic more. =A0Btw, the pa=
tterns that work in the hexagonal case end up producing puzzles where the n=
umber of colors are perfect squares, go figure! =A0And I wasn't able to=
find working patterns for polygons with 11 and 13 sides, though I wonder i=
f they exist and I wasn't recursing deeply enough.



So there is plenty more discussion that could happen, b=
ut I don't want to overload it right off the bat. =A0One last thing tho=
ugh related to your final comment. =A0I put a feature in the program just f=
or you Matt! =A0Under "Options -> Edit Settings...", play with=
the "Slicing Circles Expansion Factor". =A0This is analogous to =
"deepening the cuts" on the original puzzles, which I know you=
9;ve wanted in the 4D puzzles. =A0It is a fully experimental setting and I =
know cases where it doesn't work well, but often it does. =A0Try 1.4 on=
a Megaminx for a more difficult puzzle!



Also, if anyone feels some of this discussion shouldn&#=
39;t be on the hypercubing mailing list, let me know. =A0I'm a little w=
orried some might feel the hyperpuzzling connection a little too tenuous.div>


All the best,
Roice
class=3D"h5">


On Sun, Ja=
n 31, 2010 at 3:37 AM, Matthew Galla <o:mgalla@trinity.edu" target=3D"_blank">mgalla@trinity.edu> w=
rote:


x #ccc solid;padding-left:1ex">






=20=20=20=20=20=20=20=20
















Very nice, Roice.

=A0

Although the options are clearly limited, this program is a work of ar=
t. The hyperbolic face patterns combined with the scrambled colors are abso=
lutely beautiful.

=A0

I played around with some puzzles I already understood, and it takes a=
while to get used to the little quirks in your program, but well worth it!=
I did notice that the outermost face for=A0the cube and megaminx series is=
controlled opposite to my intuition. If I am thinking in terms of macros a=
nd try to apply a macro I know works near the center of the puzzle on the o=
utermost face, I find that I must invert every move on the outermost face. =
A closer look reveals that this is because the outermost face is inverted. =
Now this is just an idea, but have you considered inverting the movement fo=
r just the outermost face? Although it my confuse some things visually, I t=
hink it may be an overall improvement solving-wise.




=A0

Also, most of your 2-layer puzzles are currently not working (which I&=
#39;m sure you already know). Are you looking into correcting this function=
of the program? If so, can we expect puzzles with an even number of layers=
>2? For puzzles with even layers (excluding cube) the visual pieces wil=
l have to pass under/over/through each other. This is an inevitable behavio=
r if you restrict the exterior shape of a puzzle (which your program does b=
ecause it forces it to be drawn on a hyperplane). However, as you demonstra=
ted with the two-layered megaminx (impossiball) this is clearly do-able.iv>



=A0

I am also looking forward to an updatewhere we can reorient some of th=
ese puzzles! This is allowable on the cubical and dodecahedral puzzles by h=
olding down every layer number, but I would love to watch some these hyperp=
lane tesselations shift!




=A0

My favorite thing about your program, however, is the identical puzzle=
s with different sticker patterns. I am very interested to know how you cam=
e up with the different patterns of colors on say, {6,3}, as well as the ot=
her puzzles with multiple color-pattern options.




=A0

All in all, an excellent program that opens up a world of puzzles I ha=
d never considered before! Although, I should say that none of the puzzles =
in your program are very hard ;)

=A0

Thank you for once again expanding the limits on twisty puzzles!

Matt Galla

PS How many moves counts as an official scramble so I can start submit=
ting my solves? :)

On Sat, Jan 30, 2010 at 9:04 PM, Roice Nelson pan dir=3D"ltr"><r=
oice3@gmail.com
> wrote:

dding-left:1ex" class=3D"gmail_quote">
=A0=20



Thanks to everyone for the thoughtful feedback on my question this week.=
=A0I appreciate it, and it was good to get your perspectives.=20


I think I'm ready enough to share a first pass of the new =
Rubik analogue I started playing with before the MC4D 4.0 fun, which I ref=3D"http://games.groups.yahoo.com/group/4D_Cubing/message/541" target=3D=
"_blank">mentioned the possibility of here
some time ago.=A0 While you =
might observe it doesn't quite fall into the category of hyperpuzzles, =
it does in at least once sense mentioned below=A0:D =A0Here is the =3D"http://www.gravitation3d.com/magictile" target=3D"_blank">page with the=
download, pictures, and a video
.=A0 To describe the analogue idea, I&#=
39;ll just quote the beginning of the explanation on that page:






_quote">

_quote">This program aims to support twisty puzzles based on=A0ttp://en.wikipedia.org/wiki/Regular_tessellation" target=3D"_blank">olor=3D"#000000">regular polygonal til=
ings
=A0having lafli_symbol" target=3D"_blank">-decoration:none">Schlafli symbols=A0of the form {p,3} fo=
r any p>=3D2. That is, all regular tilings of polygons with two or more =
sides, where three tiles (puzzle faces) meet at a vertex. The Rubik's c=
ube is the special case where faces are squares (p=3D4). The other familiar=
special cases are the Megaminx (p=3D5) and the Pyraminx (p=3D3), although =
you'll discover the last takes a slightly different form under this abs=
traction (akin to=A0" target=3D"_blank">none">Jing's Pyraminx). All the other puzzles are new=
as far as I know, and some may be surprising, e.g. the puzzles based on href=3D"http://en.wikipedia.org/wiki/Digon" target=3D"_blank">=3D"#000000">digons=
=A0(p=3D2).




_quote">

_quote">Each 2D tiling admits a particular constant curvature (homogenous) =
geometry. The geometry is Spherical for p=3D2 to p=3D5, Euclidean (flat) fo=
r p=3D6, and Hyperbolic for p>=3D7. Since=A0ornell.edu/~dwh/papers/crochet/crochet.html" target=3D"_blank">=3D"#000000">you can't "isome=
trically embed" the entire hyperbolic plane in 3-space
a>, I have a connection to=A0
/4D_Cubing/" target=3D"_blank">decoration:none">hyperpuzzling=A0even though I'm talk=
ing about 2D tilings!=A0




_quote">...

_quote">=A0


I've actually only solved the 3x3x3 on it so far, and I wonder=
if it may be more fun to watch than play! =A0I've been calling it Magi=
cTile, though perhaps there could be something better? =A0As with everythin=
g, it is a known work in progress (the length of the task list has grown to=
scary proportions). =A0I have no plans for further development at the mome=
nt, though I'll happily fix any glaring bugs.





Enjoy!
Roice

P.S. This is the only "twisty puzzle&q=
uot; group I'm active in, so if any of you are also members of other gr=
oups and think they would be interested to hear about these new puzzles,=A0=
I'll appreciate the exposure :)







te>











=20=20=20=20
=20=20=20=20















--0015175884b6da4e18047e7e90ce--




From: Roice Nelson <roice3@gmail.com>
Date: Mon, 1 Feb 2010 00:36:32 -0600
Subject: Re: [MC4D] Introducing "MagicTile"



--000325559052c5b0a9047e843458
Content-Type: text/plain; charset=ISO-8859-1

This is embarrassing, but I was just playing with MagicTile and need to make
another correction to something I claimed - I probably should let my emails
sit a while before sending them! My fingers are crossed that this is the
last spam I feel compelled to make on this.

I was off track about vertex-centered twists on the trigonal puzzle being
the right ones for a length-2 version. It actually appears there is no good
(non-trivial and non-overlapping) twist for a length-2 trigonal puzzle. The
twists based on great circles that would slice faces in half would have
order 1, meaning they would have to rotate 360 degrees before the puzzle
could fit back together (sort of a trivial twist - there were some of these
in MC4D too, which we ended up disallowing). I think the trivial order is
related to the fact that the center of these twists don't correspond to a
face center, edge, or vertex.

To at least try to make a useful observation, for puzzles with Schlafli
symbol { p, q }, face-centered twists will have order p and vertex-centered
twists will have order q. Edge-centered twists will have order 2, since two
faces meet at each edge...

Have a nice week all,
Roice


On Sun, Jan 31, 2010 at 5:52 PM, Roice Nelson wrote:

> Greetings again,
>
> I wanted to make a minor clarification. I didn't describe my desired
> order-2 twist for a length-2 Megaminx well, and incorrectly wrote that this
> twist would swap opposite faces. It would swap half the material for two
> pairs of opposite faces. It would also completely swap a further two pairs
> of faces (one pair is of adjacent faces, one is not, but neither are
> opposite). Grab a Megaminx and picture an entire half of the puzzle being
> rotated 180 degrees to see the twist. Hopefully the goal was clear enough
> though... a length-2 Megaminx that wouldn't suffer from the weirdness of the
> Impossiball. It seems a worthy aesthetic goal for puzzles to have no
> overlapping material when twisting, but I'm always curious of other opinions
> on things like that.
>
> Cheers,
> Roice
>
>
> On Sun, Jan 31, 2010 at 3:26 PM, Roice Nelson wrote:
>
>> Thanks Matt!
>>
>> I agree with your thoughts about inverting the outermost face, and have
>> added it to "the scary list". The same issue comes up in the 4D puzzles (at
>> least in my implementation of Magic120Cell since I allow showing cells
>> mirrored by the projection). I had originally reversed the twisting of
>> those faces, but ended up reverting that when I discovered some bug entropy
>> related to it. Also, I worried it could lead to confusion, like "hey,
>> that's not counterclockwise!". On the other hand, it makes the projection
>> effects explicit. It might actually be nice to have an option for this,
>> which I think I'll do when I get to it.
>>
>> The two layer puzzles are an interesting topic for sure. I put them in
>> the list even though they don't currently do anything (except in the
>> Megaminx family), because they look pretty. I was on the fence about
>> enabling the Megaminx behavior as is, but did so solely because of the
>> Impossiball. It is the only puzzle in the list right now which overlaps
>> material when twisting. I actually feel there is a better length-2 analogy
>> for the Megaminx than the Impossiball, which is twisting a slicing circle
>> that is a "great circle" (cuts the unprojected puzzle sphere in half). The
>> order of this twist is 2 instead of 5 and swaps opposite faces. It also
>> twists without overlapping any material, which is why I prefer it. The
>> reason I didn't include this as the length-2 version is that this twist is
>> edge-centered rather than face-centered, and I had made the (arbitrary)
>> decision to restrict to the latter in the first version. I have to keep
>> myself sane somehow :)
>>
>> The situation is similar in other puzzles. Check out the length-2 digonal
>> and trigonal puzzles and note that the nicest twists there (which don't
>> overlap material) are vertex-centered. To answer your question about
>> supporting all even length puzzles vs. just length-2, absolutely. The guts
>> are capable, I just didn't expose those as menu options until the behavior
>> is better worked out. And I welcome further discussion of the specification
>> of these puzzles! In particular, what would be a good way to specify
>> edge/vertex twists?
>>
>> When you get to the infinite tilings, thinking about even-layers becomes
>> even stranger, and there is analogizing work to be done :) In the hexagonal
>> case, I think the right twist of the length-2 puzzle would be a translation
>> of half of the puzzle! (do you agree?) Again, the reason I favor this is
>> because no material overlaps. The problem I ran into was how to specify
>> such twists elegantly. Try thinking about it and backing yourself into some
>> corners :) One thing I can say is that a { clicked cell + direction }
>> simply isn't enough information to specify it. The same is true in the
>> hyperbolic cases.
>>
>> Related to the last paragraph, it is interesting that outer twists for the
>> infinite tilings don't make sense regardless of whether the puzzle is even
>> or odd. What would a slice-2 twist on a length-3 hexagonal puzzle do? The
>> topology restricts the movement, which (I think) makes sense to me if I
>> picture the hexagonal puzzle on a torus instead of unrolled as in MagicTile.
>> Anyway, I'm perhaps getting a little off topic, but the reason to mention
>> this is that we won't be able to specify reorientations for the infinite
>> tilings as a twist "with all slices down". However, I'd love to see
>> reorientations (both view rotations and panning) done some other way in both
>> the spherical/hyperbolic cases. To see a really nice example of panning
>> hyperbolic space, check out Don's hyperbolic tessellation applet!
>> The are some big challenges to get this working in MagicTile, and
>> performance is one of the largest, since so much needs to be drawn.
>>
>> In regards to your question about the tiling patterns, there is a simple
>> procedure I used to get the current list, and it involved specifying two
>> numbers. First was the number of reflections from a "fundamental" cell to
>> an "orbit" cell (I actually called them masters/slaves in the code). Second
>> was which polygon segment to do this reflection across. So for example,
>> take a look at the 6-colored octagonal puzzle. This would be 2,4
>> (equivalently 4,2). The white center cell is reflected twice across the 4th
>> segment of each adjacent cell, and recursively thereafter. I wish I
>> understood this all better, as I actually just had the program run through a
>> loop to see which of the configurations "converged". Some end up fitting
>> together and some don't. Math Magic! Hope this helped clarify.
>> Tilings.org has some papers with lists that would probably help expose the
>> magic more. Btw, the patterns that work in the hexagonal case end up
>> producing puzzles where the number of colors are perfect squares, go figure!
>> And I wasn't able to find working patterns for polygons with 11 and 13
>> sides, though I wonder if they exist and I wasn't recursing deeply enough.
>>
>> So there is plenty more discussion that could happen, but I don't want to
>> overload it right off the bat. One last thing though related to your final
>> comment. I put a feature in the program just for you Matt! Under "Options
>> -> Edit Settings...", play with the "Slicing Circles Expansion Factor".
>> This is analogous to "deepening the cuts" on the original puzzles, which I
>> know you've wanted in the 4D puzzles. It is a fully experimental setting
>> and I know cases where it doesn't work well, but often it does. Try 1.4 on
>> a Megaminx for a more difficult puzzle!
>>
>> Also, if anyone feels some of this discussion shouldn't be on the
>> hypercubing mailing list, let me know. I'm a little worried some might feel
>> the hyperpuzzling connection a little too tenuous.
>>
>> All the best,
>> Roice
>>
>>
>> On Sun, Jan 31, 2010 at 3:37 AM, Matthew Galla wrote:
>>
>>>
>>>
>>> Very nice, Roice.
>>>
>>> Although the options are clearly limited, this program is a work of art.
>>> The hyperbolic face patterns combined with the scrambled colors are
>>> absolutely beautiful.
>>>
>>> I played around with some puzzles I already understood, and it takes a
>>> while to get used to the little quirks in your program, but well worth it! I
>>> did notice that the outermost face for the cube and megaminx series is
>>> controlled opposite to my intuition. If I am thinking in terms of macros and
>>> try to apply a macro I know works near the center of the puzzle on the
>>> outermost face, I find that I must invert every move on the outermost face.
>>> A closer look reveals that this is because the outermost face is inverted.
>>> Now this is just an idea, but have you considered inverting the movement for
>>> just the outermost face? Although it my confuse some things visually, I
>>> think it may be an overall improvement solving-wise.
>>>
>>> Also, most of your 2-layer puzzles are currently not working (which I'm
>>> sure you already know). Are you looking into correcting this function of the
>>> program? If so, can we expect puzzles with an even number of layers >2? For
>>> puzzles with even layers (excluding cube) the visual pieces will have to
>>> pass under/over/through each other. This is an inevitable behavior if you
>>> restrict the exterior shape of a puzzle (which your program does because it
>>> forces it to be drawn on a hyperplane). However, as you demonstrated with
>>> the two-layered megaminx (impossiball) this is clearly do-able.
>>>
>>> I am also looking forward to an updatewhere we can reorient some of these
>>> puzzles! This is allowable on the cubical and dodecahedral puzzles by
>>> holding down every layer number, but I would love to watch some these
>>> hyperplane tesselations shift!
>>>
>>> My favorite thing about your program, however, is the identical puzzles
>>> with different sticker patterns. I am very interested to know how you came
>>> up with the different patterns of colors on say, {6,3}, as well as the other
>>> puzzles with multiple color-pattern options.
>>>
>>> All in all, an excellent program that opens up a world of puzzles I had
>>> never considered before! Although, I should say that none of the puzzles in
>>> your program are very hard ;)
>>>
>>> Thank you for once again expanding the limits on twisty puzzles!
>>> Matt Galla
>>> PS How many moves counts as an official scramble so I can start
>>> submitting my solves? :)
>>> On Sat, Jan 30, 2010 at 9:04 PM, Roice Nelson wrote:
>>>
>>>>
>>>>
>>>> Thanks to everyone for the thoughtful feedback on my question this week.
>>>> I appreciate it, and it was good to get your perspectives.
>>>>
>>>> I think I'm ready enough to share a first pass of the new Rubik analogue
>>>> I started playing with before the MC4D 4.0 fun, which I mentioned the
>>>> possibility of heresome time ago. While you might observe it doesn't quite fall into the
>>>> category of hyperpuzzles, it does in at least once sense mentioned below :D
>>>> Here is the page with the download, pictures, and a video.
>>>> To describe the analogue idea, I'll just quote the beginning of the
>>>> explanation on that page:
>>>>
>>>>
>>>>> This program aims to support twisty puzzles based on regular polygonal
>>>>> tilings having Schlafli
>>>>> symbols of the form
>>>>> {p,3} for any p>=2. That is, all regular tilings of polygons with two or
>>>>> more sides, where three tiles (puzzle faces) meet at a vertex. The Rubik's
>>>>> cube is the special case where faces are squares (p=4). The other familiar
>>>>> special cases are the Megaminx (p=5) and the Pyraminx (p=3), although you'll
>>>>> discover the last takes a slightly different form under this abstraction
>>>>> (akin to Jing's Pyraminx ).
>>>>> All the other puzzles are new as far as I know, and some may be surprising,
>>>>> e.g. the puzzles based on digons
>>>>> (p=2).
>>>>
>>>>
>>>>> Each 2D tiling admits a particular constant curvature (homogenous)
>>>>> geometry. The geometry is Spherical for p=2 to p=5, Euclidean (flat) for
>>>>> p=6, and Hyperbolic for p>=7. Since you can't "isometrically embed"
>>>>> the entire hyperbolic plane in 3-space,
>>>>> I have a connection to hyperpuzzling even
>>>>> though I'm talking about 2D tilings!
>>>>
>>>> ...
>>>>
>>>>
>>>>
>>>>
>>>> I've actually only solved the 3x3x3 on it so far, and I wonder if it may
>>>> be more fun to watch than play! I've been calling it MagicTile, though
>>>> perhaps there could be something better? As with everything, it is a known
>>>> work in progress (the length of the task list has grown to scary
>>>> proportions). I have no plans for further development at the moment, though
>>>> I'll happily fix any glaring bugs.
>>>>
>>>> Enjoy!
>>>> Roice
>>>>
>>>> P.S. This is the only "twisty puzzle" group I'm active in, so if any of
>>>> you are also members of other groups and think they would be interested to
>>>> hear about these new puzzles, I'll appreciate the exposure :)
>>>>
>>>>
>>>
>>>
>>>
>>
>>
>>
>

--000325559052c5b0a9047e843458
Content-Type: text/html; charset=ISO-8859-1
Content-Transfer-Encoding: quoted-printable

This is embarrassing, but I was just playing with MagicTile and need to mak=
e another correction to something I claimed - I probably should let my emai=
ls sit a while before sending them! =A0My fingers are crossed that this is =
the last spam I feel compelled to make on this.



I was off track about vertex-centered twists on the trigonal=
puzzle being the right ones for a length-2 version. =A0It actually appears=
there is no good (non-trivial and non-overlapping) twist for a length-2 tr=
igonal puzzle. =A0The twists based on great circles that would slice faces =
in half would have order 1, meaning they would have to rotate 360 degrees b=
efore the puzzle could fit back together (sort of a trivial twist - there w=
ere some of these in MC4D too, which we ended up disallowing). =A0I think t=
he trivial order is related to the fact that the center of these twists don=
't correspond to a face center, edge, or vertex.


To at least try to make a useful observation, for puzzl=
es with Schlafli symbol { p, q }, face-centered twists will have order p an=
d vertex-centered twists will have order q. =A0Edge-centered twists will ha=
ve order 2, since two faces meet at each edge...


Have a nice week all,
Roice

>
On Sun, Jan 31, 2010 at 5:52 PM, Roice Nelson <=
span dir=3D"ltr"><roice3@gmail.coma>> wrote:

x #ccc solid;padding-left:1ex;">Greetings again,

I wante=
d to make a minor clarification. =A0I didn't describe my desired order-=
2 twist for a length-2 Megaminx well, and incorrectly wrote that this twist=
would swap opposite faces. =A0It would swap half the material for two pair=
s of opposite faces. =A0It would also completely swap a further two pairs o=
f faces (one pair is of adjacent faces, one is not, but neither are opposit=
e). =A0Grab a Megaminx and picture an entire half of the puzzle being rotat=
ed 180 degrees to see the twist. =A0Hopefully the goal was clear enough tho=
ugh... a length-2 Megaminx that wouldn't suffer from the weirdness of t=
he Impossiball. =A0It seems a worthy aesthetic goal for puzzles to have no =
overlapping material when twisting, but I'm always curious of other opi=
nions on things like that.



Cheers,
Roice
=3D"h5">


On Sun, Jan 31, 2010 at 3:2=
6 PM, Roice Nelson <
" target=3D"_blank">roice3@gmail.com> wrote:


x #ccc solid;padding-left:1ex">Thanks Matt!

I agree with=
your thoughts about inverting the outermost face, and have added it to &qu=
ot;the scary list". =A0The same issue comes up in the 4D puzzles (at l=
east in my implementation of Magic120Cell since I allow showing cells mirro=
red by the projection). =A0I had originally reversed the twisting of those =
faces, but ended up reverting that when I discovered some bug entropy relat=
ed to it. =A0Also, I worried it could lead to confusion, like "hey, th=
at's not counterclockwise!". =A0On the other hand, it makes the pr=
ojection effects explicit. =A0It might actually be nice to have an option f=
or this, which I think I'll do when I get to it.




The two layer puzzles are an interesting topic for sure=
. =A0I put them in the list even though they don't currently do anythin=
g (except in the Megaminx family), because they look pretty. =A0I was on th=
e fence about enabling the Megaminx behavior as is, but did so solely becau=
se of the Impossiball. =A0It is the only puzzle in the list right now which=
overlaps material when twisting. =A0I actually feel there is a better leng=
th-2 analogy for the Megaminx than the Impossiball, which is twisting a sli=
cing circle that is a "great circle" (cuts the unprojected puzzle=
sphere in half). =A0The order of this twist is 2 instead of 5 and swaps op=
posite faces. =A0It also twists without overlapping any material, which is =
why I prefer it. =A0The reason I didn't include this as the length-2 ve=
rsion is that this twist is edge-centered rather than face-centered, and I =
had made the (arbitrary) decision to restrict to the latter in the first ve=
rsion. =A0I have to keep myself sane somehow :)




The situation is similar in other puzzles. =A0Check out the =
length-2 digonal and trigonal puzzles and note that the nicest twists there=
(which don't overlap material) are vertex-centered. =A0To answer your =
question about supporting all even length puzzles vs. just length-2, absolu=
tely. =A0The guts are capable, I just didn't expose those as menu optio=
ns until the behavior is better worked out. =A0And I welcome further discus=
sion of the specification of these puzzles! =A0In particular, what would be=
a good way to specify edge/vertex twists?




When you get to the infinite tilings, thinking about ev=
en-layers becomes even stranger, and there is analogizing work to be done :=
) =A0In the hexagonal case, I think the right twist of the length-2 puzzle =
would be a translation of half of the puzzle! =A0(do you agree?) =A0Again, =
the reason I favor this is because no material overlaps. =A0The problem I r=
an into was how to specify such twists elegantly. =A0Try thinking about it =
and backing yourself into some corners :) =A0One thing I can say is that a =
{ clicked cell + direction } simply isn't enough information to specify=
it. =A0The same is true in the hyperbolic cases.




Related to the last paragraph, it is interesting that o=
uter twists for the infinite tilings don't make sense regardless of whe=
ther the puzzle is even or odd. =A0What would a slice-2 twist on a length-3=
hexagonal puzzle do? =A0The topology restricts the movement, which (I thin=
k) makes sense to me if I picture the hexagonal puzzle on a torus instead o=
f unrolled as in MagicTile. =A0Anyway, I'm perhaps getting a little off=
topic, but the reason to mention this is that we won't be able to spec=
ify reorientations for the infinite tilings as a twist "with all slice=
s down". =A0However, I'd love to see reorientations (both view rot=
ations and panning) done some other way in both the spherical/hyperbolic ca=
ses. =A0To see a really nice example of panning hyperbolic space, check out=
Don's =3D"_blank">hyperbolic tessellation applet! =A0The are some big challen=
ges to get this working in MagicTile, and performance is one of the largest=
, since so much needs to be drawn.




In regards to your question about the tiling patterns, =
there is a simple procedure I used to get the current list, and it involved=
specifying two numbers. =A0First was the number of reflections from a &quo=
t;fundamental" cell to an "orbit" cell (I actually called th=
em masters/slaves in the code). =A0Second was which polygon segment to do t=
his reflection across. =A0So for example, take a look at the 6-colored octa=
gonal puzzle. =A0This would be 2,4 (equivalently 4,2). =A0The white center =
cell is reflected twice across the 4th segment of each adjacent cell, and r=
ecursively thereafter. =A0I wish I understood this all better, as I actuall=
y just had the program run through a loop to see which of the configuration=
s "converged". =A0Some end up fitting together and some don't=
. =A0Math Magic! =A0Hope this helped clarify. =A0Tilings.org has some paper=
s with lists that would probably help expose the magic more. =A0Btw, the pa=
tterns that work in the hexagonal case end up producing puzzles where the n=
umber of colors are perfect squares, go figure! =A0And I wasn't able to=
find working patterns for polygons with 11 and 13 sides, though I wonder i=
f they exist and I wasn't recursing deeply enough.




So there is plenty more discussion that could happen, b=
ut I don't want to overload it right off the bat. =A0One last thing tho=
ugh related to your final comment. =A0I put a feature in the program just f=
or you Matt! =A0Under "Options -> Edit Settings...", play with=
the "Slicing Circles Expansion Factor". =A0This is analogous to =
"deepening the cuts" on the original puzzles, which I know you=
9;ve wanted in the 4D puzzles. =A0It is a fully experimental setting and I =
know cases where it doesn't work well, but often it does. =A0Try 1.4 on=
a Megaminx for a more difficult puzzle!




Also, if anyone feels some of this discussion shouldn&#=
39;t be on the hypercubing mailing list, let me know. =A0I'm a little w=
orried some might feel the hyperpuzzling connection a little too tenuous.div>



All the best,
Roice
>


On Sun, Jan 31, 2010 at=
3:37 AM, Matthew Galla <ity.edu" target=3D"_blank">mgalla@trinity.edu> wrote:



x #ccc solid;padding-left:1ex">






=20=20=20=20=20=20=20=20
















Very nice, Roice.

=A0

Although the options are clearly limited, this program is a work of ar=
t. The hyperbolic face patterns combined with the scrambled colors are abso=
lutely beautiful.

=A0

I played around with some puzzles I already understood, and it takes a=
while to get used to the little quirks in your program, but well worth it!=
I did notice that the outermost face for=A0the cube and megaminx series is=
controlled opposite to my intuition. If I am thinking in terms of macros a=
nd try to apply a macro I know works near the center of the puzzle on the o=
utermost face, I find that I must invert every move on the outermost face. =
A closer look reveals that this is because the outermost face is inverted. =
Now this is just an idea, but have you considered inverting the movement fo=
r just the outermost face? Although it my confuse some things visually, I t=
hink it may be an overall improvement solving-wise.





=A0

Also, most of your 2-layer puzzles are currently not working (which I&=
#39;m sure you already know). Are you looking into correcting this function=
of the program? If so, can we expect puzzles with an even number of layers=
>2? For puzzles with even layers (excluding cube) the visual pieces wil=
l have to pass under/over/through each other. This is an inevitable behavio=
r if you restrict the exterior shape of a puzzle (which your program does b=
ecause it forces it to be drawn on a hyperplane). However, as you demonstra=
ted with the two-layered megaminx (impossiball) this is clearly do-able.iv>




=A0

I am also looking forward to an updatewhere we can reorient some of th=
ese puzzles! This is allowable on the cubical and dodecahedral puzzles by h=
olding down every layer number, but I would love to watch some these hyperp=
lane tesselations shift!





=A0

My favorite thing about your program, however, is the identical puzzle=
s with different sticker patterns. I am very interested to know how you cam=
e up with the different patterns of colors on say, {6,3}, as well as the ot=
her puzzles with multiple color-pattern options.





=A0

All in all, an excellent program that opens up a world of puzzles I ha=
d never considered before! Although, I should say that none of the puzzles =
in your program are very hard ;)

=A0

Thank you for once again expanding the limits on twisty puzzles!

Matt Galla

PS How many moves counts as an official scramble so I can start submit=
ting my solves? :)

On Sat, Jan 30, 2010 at 9:04 PM, Roice Nelson pan dir=3D"ltr"><r=
oice3@gmail.com
> wrote:

dding-left:1ex" class=3D"gmail_quote">
=A0=20



Thanks to everyone for the thoughtful feedback on my question this week.=
=A0I appreciate it, and it was good to get your perspectives.=20


I think I'm ready enough to share a first pass of the new =
Rubik analogue I started playing with before the MC4D 4.0 fun, which I ref=3D"http://games.groups.yahoo.com/group/4D_Cubing/message/541" target=3D=
"_blank">mentioned the possibility of here
some time ago.=A0 While you =
might observe it doesn't quite fall into the category of hyperpuzzles, =
it does in at least once sense mentioned below=A0:D =A0Here is the =3D"http://www.gravitation3d.com/magictile" target=3D"_blank">page with the=
download, pictures, and a video
.=A0 To describe the analogue idea, I&#=
39;ll just quote the beginning of the explanation on that page:







_quote">

_quote">This program aims to support twisty puzzles based on=A0ttp://en.wikipedia.org/wiki/Regular_tessellation" target=3D"_blank">olor=3D"#000000">regular polygonal til=
ings
=A0having lafli_symbol" target=3D"_blank">-decoration:none">Schlafli symbols=A0of the form {p,3} fo=
r any p>=3D2. That is, all regular tilings of polygons with two or more =
sides, where three tiles (puzzle faces) meet at a vertex. The Rubik's c=
ube is the special case where faces are squares (p=3D4). The other familiar=
special cases are the Megaminx (p=3D5) and the Pyraminx (p=3D3), although =
you'll discover the last takes a slightly different form under this abs=
traction (akin to=A0" target=3D"_blank">none">Jing's Pyraminx). All the other puzzles are new=
as far as I know, and some may be surprising, e.g. the puzzles based on href=3D"http://en.wikipedia.org/wiki/Digon" target=3D"_blank">=3D"#000000">digons=
=A0(p=3D2).





_quote">

_quote">Each 2D tiling admits a particular constant curvature (homogenous) =
geometry. The geometry is Spherical for p=3D2 to p=3D5, Euclidean (flat) fo=
r p=3D6, and Hyperbolic for p>=3D7. Since=A0ornell.edu/~dwh/papers/crochet/crochet.html" target=3D"_blank">=3D"#000000">you can't "isome=
trically embed" the entire hyperbolic plane in 3-space
a>, I have a connection to=A0
/4D_Cubing/" target=3D"_blank">decoration:none">hyperpuzzling=A0even though I'm talk=
ing about 2D tilings!=A0





_quote">...

_quote">=A0


I've actually only solved the 3x3x3 on it so far, and I wonder=
if it may be more fun to watch than play! =A0I've been calling it Magi=
cTile, though perhaps there could be something better? =A0As with everythin=
g, it is a known work in progress (the length of the task list has grown to=
scary proportions). =A0I have no plans for further development at the mome=
nt, though I'll happily fix any glaring bugs.






Enjoy!
Roice

P.S. This is the only "twisty puzzle&q=
uot; group I'm active in, so if any of you are also members of other gr=
oups and think they would be interested to hear about these new puzzles,=A0=
I'll appreciate the exposure :)








te>











=20=20=20=20
=20=20=20=20

















--000325559052c5b0a9047e843458--




From: "spel_werdz_rite" <spel_werdz_rite@yahoo.com>
Date: Mon, 01 Feb 2010 10:12:00 -0000
Subject: Re: Introducing "MagicTile"



Roice... Great work! There really is nothing more than I can say. I'm very =
impressed with the mere idea of a heptagonal analog to the megaminx ever ex=
isting. That really threw me off.

Anyway, you have me email, but I'd like to just make public my opinion of l=
og files and that I've already solved the Klein's Quartic size 3 puzzle. To=
anyone who might have solved it too, my official finish time was January 1=
st, 2010 at 1:14AM (Pacific Time).

Good luck to all the solvers out there.




From: <alexander.sage@jacks.sdstate.edu>
Date: Mon, 1 Feb 2010 02:35:20 -0600
Subject: Re: Introducing "MagicTile"



=20=20



This is embarrassing, but I was just playing with MagicTile and need to mak=
e another correction to something I claimed - I probably should let my emai=
ls sit a while before sending them! My fingers are crossed that this is th=
e last spam I feel compelled to make on this.


I was off track about vertex-centered twists on the trigonal puzzle being t=
he right ones for a length-2 version. It actually appears there is no good=
(non-trivial and non-overlapping) twist for a length-2 trigonal puzzle. T=
he twists based on great circles that would slice faces in half would have =
order 1, meaning they would have to rotate 360 degrees before the puzzle co=
uld fit back together (sort of a trivial twist - there were some of these i=
n MC4D too, which we ended up disallowing). I think the trivial order is r=
elated to the fact that the center of these twists don't correspond to a fa=
ce center, edge, or vertex.


To at least try to make a useful observation, for puzzles with Schlafli sym=
bol { p, q }, face-centered twists will have order p and vertex-centered tw=
ists will have order q. Edge-centered twists will have order 2, since two =
faces meet at each edge...


Have a nice week all,
Roice



On Sun, Jan 31, 2010 at 5:52 PM, Roice Nelson wrote:

Greetings again,


I wanted to make a minor clarification. I didn't describe my desired order=
-2 twist for a length-2 Megaminx well, and incorrectly wrote that this twis=
t would swap opposite faces. It would swap half the material for two pairs=
of opposite faces. It would also completely swap a further two pairs of f=
aces (one pair is of adjacent faces, one is not, but neither are opposite).=
Grab a Megaminx and picture an entire half of the puzzle being rotated 18=
0 degrees to see the twist. Hopefully the goal was clear enough though... =
a length-2 Megaminx that wouldn't suffer from the weirdness of the Impossib=
all. It seems a worthy aesthetic goal for puzzles to have no overlapping m=
aterial when twisting, but I'm always curious of other opinions on things l=
ike that.


Cheers,
Roice






On Sun, Jan 31, 2010 at 3:26 PM, Roice Nelson wrote:

Thanks Matt!


I agree with your thoughts about inverting the outermost face, and have add=
ed it to "the scary list". The same issue comes up in the 4D puzzles (at l=
east in my implementation of Magic120Cell since I allow showing cells mirro=
red by the projection). I had originally reversed the twisting of those fa=
ces, but ended up reverting that when I discovered some bug entropy related=
to it. Also, I worried it could lead to confusion, like "hey, that's not =
counterclockwise!". On the other hand, it makes the projection effects exp=
licit. It might actually be nice to have an option for this, which I think=
I'll do when I get to it.


The two layer puzzles are an interesting topic for sure. I put them in the=
list even though they don't currently do anything (except in the Megaminx =
family), because they look pretty. I was on the fence about enabling the M=
egaminx behavior as is, but did so solely because of the Impossiball. It i=
s the only puzzle in the list right now which overlaps material when twisti=
ng. I actually feel there is a better length-2 analogy for the Megaminx th=
an the Impossiball, which is twisting a slicing circle that is a "great cir=
cle" (cuts the unprojected puzzle sphere in half). The order of this twist=
is 2 instead of 5 and swaps opposite faces. It also twists without overla=
pping any material, which is why I prefer it. The reason I didn't include =
this as the length-2 version is that this twist is edge-centered rather tha=
n face-centered, and I had made the (arbitrary) decision to restrict to the=
latter in the first version. I have to keep myself sane somehow :)


The situation is similar in other puzzles. Check out the length-2 digonal =
and trigonal puzzles and note that the nicest twists there (which don't ove=
rlap material) are vertex-centered. To answer your question about supporti=
ng all even length puzzles vs. just length-2, absolutely. The guts are cap=
able, I just didn't expose those as menu options until the behavior is bett=
er worked out. And I welcome further discussion of the specification of th=
ese puzzles! In particular, what would be a good way to specify edge/verte=
x twists?


When you get to the infinite tilings, thinking about even-layers becomes ev=
en stranger, and there is analogizing work to be done :) In the hexagonal =
case, I think the right twist of the length-2 puzzle would be a translation=
of half of the puzzle! (do you agree?) Again, the reason I favor this is=
because no material overlaps. The problem I ran into was how to specify s=
uch twists elegantly. Try thinking about it and backing yourself into some=
corners :) One thing I can say is that a { clicked cell + direction } sim=
ply isn't enough information to specify it. The same is true in the hyperb=
olic cases.


Related to the last paragraph, it is interesting that outer twists for the =
infinite tilings don't make sense regardless of whether the puzzle is even =
or odd. What would a slice-2 twist on a length-3 hexagonal puzzle do? The=
topology restricts the movement, which (I think) makes sense to me if I pi=
cture the hexagonal puzzle on a torus instead of unrolled as in MagicTile. =
Anyway, I'm perhaps getting a little off topic, but the reason to mention =
this is that we won't be able to specify reorientations for the infinite ti=
lings as a twist "with all slices down". However, I'd love to see reorient=
ations (both view rotations and panning) done some other way in both the sp=
herical/hyperbolic cases. To see a really nice example of panning hyperbol=
ic space, check out Don's hyperbolic tessellation applet! The are some big=
challenges to get this working in MagicTile, and performance is one of the=
largest, since so much needs to be drawn.


In regards to your question about the tiling patterns, there is a simple pr=
ocedure I used to get the current list, and it involved specifying two numb=
ers. First was the number of reflections from a "fundamental" cell to an "=
orbit" cell (I actually called them masters/slaves in the code). Second wa=
s which polygon segment to do this reflection across. So for example, take=
a look at the 6-colored octagonal puzzle. This would be 2,4 (equivalently=
4,2). The white center cell is reflected twice across the 4th segment of =
each adjacent cell, and recursively thereafter. I wish I understood this a=
ll better, as I actually just had the program run through a loop to see whi=
ch of the configurations "converged". Some end up fitting together and som=
e don't. Math Magic! Hope this helped clarify. Tilings.org has some pape=
rs with lists that would probably help expose the magic more. Btw, the pat=
terns that work in the hexagonal case end up producing puzzles where the nu=
mber of colors are perfect squares, go figure! And I wasn't able to find w=
orking patterns for polygons with 11 and 13 sides, though I wonder if they =
exist and I wasn't recursing deeply enough.


So there is plenty more discussion that could happen, but I don't want to o=
verload it right off the bat. One last thing though related to your final =
comment. I put a feature in the program just for you Matt! Under "Options=
-> Edit Settings...", play with the "Slicing Circles Expansion Factor". T=
his is analogous to "deepening the cuts" on the original puzzles, which I k=
now you've wanted in the 4D puzzles. It is a fully experimental setting an=
d I know cases where it doesn't work well, but often it does. Try 1.4 on a=
Megaminx for a more difficult puzzle!


Also, if anyone feels some of this discussion shouldn't be on the hypercubi=
ng mailing list, let me know. I'm a little worried some might feel the hyp=
erpuzzling connection a little too tenuous.


All the best,
Roice







On Sun, Jan 31, 2010 at 3:37 AM, Matthew Galla wrote:





Very nice, Roice.
=20
Although the options are clearly limited, this program is a work of art. Th=
e hyperbolic face patterns combined with the scrambled colors are absolutel=
y beautiful.
=20
I played around with some puzzles I already understood, and it takes a whil=
e to get used to the little quirks in your program, but well worth it! I di=
d notice that the outermost face for the cube and megaminx series is contro=
lled opposite to my intuition. If I am thinking in terms of macros and try =
to apply a macro I know works near the center of the puzzle on the outermos=
t face, I find that I must invert every move on the outermost face. A close=
r look reveals that this is because the outermost face is inverted. Now thi=
s is just an idea, but have you considered inverting the movement for just =
the outermost face? Although it my confuse some things visually, I think it=
may be an overall improvement solving-wise.
=20
Also, most of your 2-layer puzzles are currently not working (which I'm sur=
e you already know). Are you looking into correcting this function of the p=
rogram? If so, can we expect puzzles with an even number of layers >2? For =
puzzles with even layers (excluding cube) the visual pieces will have to pa=
ss under/over/through each other. This is an inevitable behavior if you res=
trict the exterior shape of a puzzle (which your program does because it fo=
rces it to be drawn on a hyperplane). However, as you demonstrated with the=
two-layered megaminx (impossiball) this is clearly do-able.
=20
I am also looking forward to an updatewhere we can reorient some of these p=
uzzles! This is allowable on the cubical and dodecahedral puzzles by holdin=
g down every layer number, but I would love to watch some these hyperplane =
tesselations shift!
=20
My favorite thing about your program, however, is the identical puzzles wit=
h different sticker patterns. I am very interested to know how you came up =
with the different patterns of colors on say, {6,3}, as well as the other p=
uzzles with multiple color-pattern options.
=20
All in all, an excellent program that opens up a world of puzzles I had nev=
er considered before! Although, I should say that none of the puzzles in yo=
ur program are very hard ;)
=20
Thank you for once again expanding the limits on twisty puzzles!
Matt Galla

PS How many moves counts as an official scramble so I can start submitting =
my solves? :)




On Sat, Jan 30, 2010 at 9:04 PM, Roice Nelson wrote:


=20=20



Thanks to everyone for the thoughtful feedback on my question this week. I=
appreciate it, and it was good to get your perspectives.=20


I think I'm ready enough to share a first pass of the new Rubik analogue I =
started playing with before the MC4D 4.0 fun, which I mentioned the possibi=
lity of here some time ago. While you might observe it doesn't quite fall =
into the category of hyperpuzzles, it does in at least once sense mentioned=
below :D Here is the page with the download, pictures, and a video. To d=
escribe the analogue idea, I'll just quote the beginning of the explanation=
on that page:





This program aims to support twisty puzzles based on regular polygonal tili=
ngs having Schlafli symbols of the form {p,3} for any p>=3D2. That is, all =
regular tilings of polygons with two or more sides, where three tiles (puzz=
le faces) meet at a vertex. The Rubik's cube is the special case where face=
s are squares (p=3D4). The other familiar special cases are the Megaminx (p=
=3D5) and the Pyraminx (p=3D3), although you'll discover the last takes a s=
lightly different form under this abstraction (akin to Jing's Pyraminx). Al=
l the other puzzles are new as far as I know, and some may be surprising, e=
.g. the puzzles based on digons (p=3D2).


Each 2D tiling admits a particular constant curvature (homogenous) geometry=
. The geometry is Spherical for p=3D2 to p=3D5, Euclidean (flat) for p=3D6,=
and Hyperbolic for p>=3D7. Since you can't "isometrically embed" the entir=
e hyperbolic plane in 3-space, I have a connection to hyperpuzzling even th=
ough I'm talking about 2D tilings!=20
...
=20

I've actually only solved the 3x3x3 on it so far, and I wonder if it may be=
more fun to watch than play! I've been calling it MagicTile, though perha=
ps there could be something better? As with everything, it is a known work=
in progress (the length of the task list has grown to scary proportions). =
I have no plans for further development at the moment, though I'll happily=
fix any glaring bugs.

Enjoy!
Roice

P.S. This is the only "twisty puzzle" group I'm active in, so if any of you=
are also members of other groups and think they would be interested to hea=
r about these new puzzles, I'll appreciate the exposure :)


















=20=09=09=20=09=20=20=20=09=09=20=20
--_f9511af0-b08c-4ffd-8eee-c0f5282eb838_
Content-Type: text/html; charset="Windows-1252"
Content-Transfer-Encoding: quoted-printable






I love the idea of this program.  My roommate's girlfriend gave m=
e a pretty funny laugh when I told her that I had finally found a non-=
euclidian rubik's puzzle applet.  I'd never thought of a hex-tiling tw=
isty puzzle before, and I like the idea a lot.  I had a lot of fun pla=
ying with it and the hyperbolic puzzles.  once again, nice idea.

 

The program is, of course, new.  It desperately needs a way to fix ori=
entations for all of the tilings, and that would be nice to have on th=
e spherical puzzles too.  I would also like to see more visibility opt=
ions than just "performance" and "originals".  It would be nice to hav=
e a slider that determines how far from the original side other sides could=
be seen, somewhat like the magic 120 cell puzzle.

 

I would like to see more kinds of puzzle, specifically the flat square and =
triangle tilings.   depth-2 puzzles would also be nice.

In the more long term, a symbol reader (like "invent your own!" in MC4D) to=
make any possible puzzle would be pretty awesome, but I know that probably=
won't happen very soon.

 

Good job in making this program, it's really well written so far, and I hav=
en't found any real bugs with it yet.

 

 


 



To: 4D_Cubing@yahoogroups.com
From: roice3@gmail.com
Date: Mon, 1 Feb=
2010 00:36:32 -0600
Subject: Re: [MC4D] Introducing "MagicTile"

=
 =20



This is embarrassing, but I was just playing with MagicTile and need to mak=
e another correction to something I claimed - I probably should let my emai=
ls sit a while before sending them!  My fingers are crossed that this =
is the last spam I feel compelled to make on this.


I was off track about vertex-centered twists on the trigonal puzzle be=
ing the right ones for a length-2 version.  It actually appears there =
is no good (non-trivial and non-overlapping) twist for a length-2 trigonal =
puzzle.  The twists based on great circles that would slice faces in h=
alf would have order 1, meaning they would have to rotate 360 degrees befor=
e the puzzle could fit back together (sort of a trivial twist - there were =
some of these in MC4D too, which we ended up disallowing).  I thi=
nk the trivial order is related to the fact that the center of these twists=
don't correspond to a face center, edge, or vertex.



To at least try to make a useful observation, for puzzles with Schlafl=
i symbol { p, q }, face-centered twists will have order p and vertex-center=
ed twists will have order q.  Edge-centered twists will have order 2, =
since two faces meet at each edge...



Have a nice week all,

Roice




On Sun, Jan 31, 2010 at 5:52 PM, Roice Nelson <=
SPAN dir=3Dltr><roice3@gmail.co=
m
> wrote:

Gr=
eetings again,


I wanted to make a minor clarification.  I didn't describe my des=
ired order-2 twist for a length-2 Megaminx well, and incorrectly wrote that=
this twist would swap opposite faces.  It would swap half the materia=
l for two pairs of opposite faces.  It would also completely swap a fu=
rther two pairs of faces (one pair is of adjacent faces, one is not, but ne=
ither are opposite).  Grab a Megaminx and picture an entire half of th=
e puzzle being rotated 180 degrees to see the twist.  Hopefully the go=
al was clear enough though... a length-2 Megaminx that wouldn't suffer from=
the weirdness of the Impossiball.  It seems a worthy aesthetic goal f=
or puzzles to have no overlapping material when twisting, but I'm always cu=
rious of other opinions on things like that.



Cheers,

Roice







On Sun, Jan 31, 2010 at 3:26 PM, Roice Nelson <=
SPAN dir=3Dltr><roice3@gmail.co=
m
> wrote:

Th=
anks Matt!


I agree with your thoughts about inverting the outermost face, and hav=
e added it to "the scary list".  The same issue comes up in the 4D puz=
zles (at least in my implementation of Magic120Cell since I allow showing c=
ells mirrored by the projection).  I had originally reversed the twist=
ing of those faces, but ended up reverting that when I discovered some bug =
entropy related to it.  Also, I worried it could lead to confusion, li=
ke "hey, that's not counterclockwise!".  On the other hand, it makes t=
he projection effects explicit.  It might actually be nice to have an =
option for this, which I think I'll do when I get to it.



The two layer puzzles are an interesting topic for sure.  I put t=
hem in the list even though they don't currently do anything (except in the=
Megaminx family), because they look pretty.  I was on the fence about=
enabling the Megaminx behavior as is, but did so solely because of the Imp=
ossiball.  It is the only puzzle in the list right now which overlaps =
material when twisting.  I actually feel there is a better length-2 an=
alogy for the Megaminx than the Impossiball, which is twisting a slicing ci=
rcle that is a "great circle" (cuts the unprojected puzzle sphere in half).=
 The order of this twist is 2 instead of 5 and swaps opposite faces. =
 It also twists without overlapping any material, which is why I prefe=
r it.  The reason I didn't include this as the length-2 version is tha=
t this twist is edge-centered rather than face-centered, and I had made the=
(arbitrary) decision to restrict to the latter in the first version.  =
;I have to keep myself sane somehow :)


The situation is similar in other puzzles.  Check out the length-=
2 digonal and trigonal puzzles and note that the nicest twists there (which=
don't overlap material) are vertex-centered.  To answer your question=
about supporting all even length puzzles vs. just length-2, absolutely. &n=
bsp;The guts are capable, I just didn't expose those as menu options until =
the behavior is better worked out.  And I welcome further discussion o=
f the specification of these puzzles!  In particular, what would be a =
good way to specify edge/vertex twists?



When you get to the infinite tilings, thinking about even-layers becom=
es even stranger, and there is analogizing work to be done :)  In the =
hexagonal case, I think the right twist of the length-2 puzzle would be a t=
ranslation of half of the puzzle!  (do you agree?)  Again, the re=
ason I favor this is because no material overlaps.  The problem I ran =
into was how to specify such twists elegantly.  Try thinking about it =
and backing yourself into some corners :)  One thing I can say is that=
a { clicked cell + direction } simply isn't enough information to specify =
it.  The same is true in the hyperbolic cases.



Related to the last paragraph, it is interesting that outer twists for=
the infinite tilings don't make sense regardless of whether the puzzle is =
even or odd.  What would a slice-2 twist on a length-3 hexagonal puzzl=
e do?  The topology restricts the movement, which (I think) makes sens=
e to me if I picture the hexagonal puzzle on a torus instead of unrolled as=
in MagicTile.  Anyway, I'm perhaps getting a little off topic, but th=
e reason to mention this is that we won't be able to specify reorientations=
for the infinite tilings as a twist "with all slices down".  However,=
I'd love to see reorientations (both view rotations and panning) done some=
other way in both the spherical/hyperbolic cases.  To see a real=
ly nice example of panning hyperbolic space, check out Don's p://www.plunk.org/~hatch/HyperbolicApplet/">hyperbolic tessellation applet<=
/A>!  The are some big challenges to get this working in MagicTile, an=
d performance is one of the largest, since so much needs to be drawn.



In regards to your question about the tiling patterns, there is a simp=
le procedure I used to get the current list, and it involved specifying two=
numbers.  First was the number of reflections from a "fundamental" ce=
ll to an "orbit" cell (I actually called them masters/slaves in the code). =
 Second was which polygon segment to do this reflection across.  =
So for example, take a look at the 6-colored octagonal puzzle.  This w=
ould be 2,4 (equivalently 4,2).  The white center cell is reflected tw=
ice across the 4th segment of each adjacent cell, and recursively thereafte=
r.  I wish I understood this all better, as I actually just had the pr=
ogram run through a loop to see which of the configurations "converged". &n=
bsp;Some end up fitting together and some don't.  Math Magic!  Ho=
pe this helped clarify.  Tilings.org has some papers with lists that w=
ould probably help expose the magic more.  Btw, the patterns that work=
in the hexagonal case end up producing puzzles where the number of colors =
are perfect squares, go figure!  And I wasn't able to find working pat=
terns for polygons with 11 and 13 sides, though I wonder if they exist and =
I wasn't recursing deeply enough.



So there is plenty more discussion that could happen, but I don't want=
to overload it right off the bat.  One last thing though related to y=
our final comment.  I put a feature in the program just for you Matt! =
 Under "Options -> Edit Settings...", play with the "Slicing Circle=
s Expansion Factor".  This is analogous to "deepening the cuts" on the=
original puzzles, which I know you've wanted in the 4D puzzles.  It i=
s a fully experimental setting and I know cases where it doesn't work well,=
but often it does.  Try 1.4 on a Megaminx for a more difficult puzzle=
!



Also, if anyone feels some of this discussion shouldn't be on the hype=
rcubing mailing list, let me know.  I'm a little worried some might fe=
el the hyperpuzzling connection a little too tenuous.



All the best,

Roice








On Sun, Jan 31, 2010 at 3:37 AM, Matthew Galla =
<
mgalla@trinity.BR>edu> wrote:





Very nice, Roice.

 

Although the options are clearly limited, this program is a work of ar=
t. The hyperbolic face patterns combined with the scrambled colors are abso=
lutely beautiful.

 

I played around with some puzzles I already understood, and it takes a=
while to get used to the little quirks in your program, but well worth it!=
I did notice that the outermost face for the cube and megaminx series=
is controlled opposite to my intuition. If I am thinking in terms of macro=
s and try to apply a macro I know works near the center of the puzzle on th=
e outermost face, I find that I must invert every move on the outermost fac=
e. A closer look reveals that this is because the outermost face is inverte=
d. Now this is just an idea, but have you considered inverting the movement=
for just the outermost face? Although it my confuse some things visually, =
I think it may be an overall improvement solving-wise.

 

Also, most of your 2-layer puzzles are currently not working (which I'=
m sure you already know). Are you looking into correcting this function of =
the program? If so, can we expect puzzles with an even number of layers >=
;2? For puzzles with even layers (excluding cube) the visual pieces will ha=
ve to pass under/over/through each other. This is an inevitable behavior if=
you restrict the exterior shape of a puzzle (which your program does becau=
se it forces it to be drawn on a hyperplane). However, as you demonstrated =
with the two-layered megaminx (impossiball) this is clearly do-able.

 

I am also looking forward to an updatewhere we can reorient some of th=
ese puzzles! This is allowable on the cubical and dodecahedral puzzles by h=
olding down every layer number, but I would love to watch some these hyperp=
lane tesselations shift!

 

My favorite thing about your program, however, is the identical puzzle=
s with different sticker patterns. I am very interested to know how you cam=
e up with the different patterns of colors on say, {6,3}, as well as the ot=
her puzzles with multiple color-pattern options.

 

All in all, an excellent program that opens up a world of puzzles I ha=
d never considered before! Although, I should say that none of the puzzles =
in your program are very hard ;)

 

Thank you for once again expanding the limits on twisty puzzles!

Matt Galla

PS How many moves counts as an official scramble so I can start submit=
ting my solves? :)




On Sat, Jan 30, 2010 at 9:04 PM, Roice Nelson <=
SPAN dir=3Dltr><roice3@gmail.co=
m
> wrote:


 =20



Thanks to everyone for the thoughtful feedback on my question this week.=
 I appreciate it, and it was good to get your perspectives.


I think I'm ready enough to share a first pass of the new Rubik an=
alogue I started playing with before the MC4D 4.0 fun, which I ttp://games.groups.yahoo.com/group/4D_Cubing/message/541">mentioned the pos=
sibility of here
some time ago.  While you might observe it doesn'=
t quite fall into the category of hyperpuzzles, it does in at least once se=
nse mentioned below :D  Here is the tion3d.com/magictile">page with the download, pictures, and a video.&nb=
sp; To describe the analogue idea, I'll just quote the beginning of the exp=
lanation on that page:



ail_quote>

ail_quote>This program aims to support twisty puzzles based on =3D"http://en.wikipedia.org/wiki/Regular_tessellation">0>regular polygonal tilingsONT> having l">Schlafli sym=
bols
 of the form {p,3} for any p>=3D2. That is, a=
ll regular tilings of polygons with two or more sides, where three tiles (p=
uzzle faces) meet at a vertex. The Rubik's cube is the special case where f=
aces are squares (p=3D4). The other familiar special cases are the Megaminx=
(p=3D5) and the Pyraminx (p=3D3), although you'll discover the last takes =
a slightly different form under this abstraction (akin to ttp://www.youtube.com/watch?v=3DFuD3YwQTW2c">yle=3D"TEXT-DECORATION: none">Jing's Pyraminx). All the o=
ther puzzles are new as far as I know, and some may be surprising, e.g. the=
puzzles based on r=3D#000000>digons&=
nbsp;(p=3D2).

ail_quote>

ail_quote>Each 2D tiling admits a particular constant curvature (homogenous=
) geometry. The geometry is Spherical for p=3D2 to p=3D5, Euclidean (flat) =
for p=3D6, and Hyperbolic for p>=3D7. Since ath.cornell.edu/~dwh/papers/crochet/crochet.html">AN style=3D"TEXT-DECORATION: none">you can't "isometrically embed" the enti=
re hyperbolic plane in 3-space
, I have a connection to&nb=
sp;=3D#000000>hyperpuzzling> even though I'm talking about 2D tilings! 

ail_quote>...

ail_quote> 


I've actually only solved the 3x3x3 on it so far, and I wonder if =
it may be more fun to watch than play!  I've been calling it MagicTile=
, though perhaps there could be something better?  As with everything,=
it is a known work in progress (the length of the task list has grown to s=
cary proportions).  I have no plans for further development at th=
e moment, though I'll happily fix any glaring bugs.


Enjoy!
Roice

P.S. This is the only "twisty puzzle" group=
I'm active in, so if any of you are also members of other groups and think=
they would be interested to hear about these new puzzles, I'll apprec=
iate the exposure :)








<=
BR>


=

IV>






--_f9511af0-b08c-4ffd-8eee-c0f5282eb838_--




From: "spel_werdz_rite" <spel_werdz_rite@yahoo.com>
Date: Mon, 01 Feb 2010 18:50:24 -0000
Subject: Re: Introducing "MagicTile"



In a follow up with Roice, I'd like to share some more interesting details =
with the Klein's Quartic puzzle.

The strategy to solving it was very much similar to how one would solve a M=
egaminx (my method at least). Edges, then sides, then edges, working all th=
e way down to the bottom of the puzzle. Doing this method lead to me a very=
interesting discovery that, surprisingly, not even Roice new about. It tur=
ns out that Klein's Quartic has two "bottoms." By which I mean if you follo=
w this method of inserting pieces downward until you reach the bottom of th=
e puzzle, you will end up at 2 different faces. At this location, solving b=
ecame a bit of a new task, but still not much of a challenge. The first ste=
p was making sure the remaining 2C and 3C pieces were on their correspondin=
g face and oriented correctly. After that, I borrowed many techniques I use=
d for the Megaminx. However, due to some obvious differences, the end took =
a lot of guesswork. In the end, the puzzle took about 2.5 hours (factoring =
in my "hey let's get distracted a lot" variables).

My final thoughts. Very fun. It was a true joy to play a technical 3D puzzl=
e that technically couldn't exist in the 3D world.




From: "matthewsheerin" <damienturtle@hotmail.co.uk>
Date: Mon, 01 Feb 2010 18:57:44 -0000
Subject: Re: [MC4D] Introducing "MagicTile"



Roice, that is utterly mind-bending! I just gave the 3x3x3 a go, and it wa=
s surprising how difficult to was to cope with the visualisation. So needl=
ess to say, I love it :), and it should be fun having a look at hyperbolic =
puzzles. The best thing about hanging out round here is playing with puzzl=
es which I never even considered the possible existence of.

You're a genius Roice, and as for my take on selling the software, a small =
donation will be on its was soon.

Matthew

--- In 4D_Cubing@yahoogroups.com, wrote:
>
>=20
> I love the idea of this program. My roommate's girlfriend gave me a pret=
ty funny laugh when I told her that I had finally found a non-euclidian rub=
ik's puzzle applet. I'd never thought of a hex-tiling twisty puzzle before=
, and I like the idea a lot. I had a lot of fun playing with it and the hy=
perbolic puzzles. once again, nice idea.
>=20
>=20=20
>=20
> The program is, of course, new. It desperately needs a way to fix orient=
ations for all of the tilings, and that would be nice to have on the spheri=
cal puzzles too. I would also like to see more visibility options than jus=
t "performance" and "originals". It would be nice to have a slider that de=
termines how far from the original side other sides could be seen, somewhat=
like the magic 120 cell puzzle.
>=20
>=20=20
>=20
> I would like to see more kinds of puzzle, specifically the flat square an=
d triangle tilings. depth-2 puzzles would also be nice.
>=20
> In the more long term, a symbol reader (like "invent your own!" in MC4D) =
to make any possible puzzle would be pretty awesome, but I know that probab=
ly won't happen very soon.
>=20
>=20=20
>=20
> Good job in making this program, it's really well written so far, and I h=
aven't found any real bugs with it yet.
>=20
>=20=20
>=20
>=20=20
>=20
>=20
>=20=20
>=20
>=20
> To: 4D_Cubing@yahoogroups.com
> From: roice3@...
> Date: Mon, 1 Feb 2010 00:36:32 -0600
> Subject: Re: [MC4D] Introducing "MagicTile"
>=20
>=20=20=20
>=20
>=20
>=20
> This is embarrassing, but I was just playing with MagicTile and need to m=
ake another correction to something I claimed - I probably should let my em=
ails sit a while before sending them! My fingers are crossed that this is =
the last spam I feel compelled to make on this.
>=20
>=20
> I was off track about vertex-centered twists on the trigonal puzzle being=
the right ones for a length-2 version. It actually appears there is no go=
od (non-trivial and non-overlapping) twist for a length-2 trigonal puzzle. =
The twists based on great circles that would slice faces in half would hav=
e order 1, meaning they would have to rotate 360 degrees before the puzzle =
could fit back together (sort of a trivial twist - there were some of these=
in MC4D too, which we ended up disallowing). I think the trivial order is=
related to the fact that the center of these twists don't correspond to a =
face center, edge, or vertex.
>=20
>=20
> To at least try to make a useful observation, for puzzles with Schlafli s=
ymbol { p, q }, face-centered twists will have order p and vertex-centered =
twists will have order q. Edge-centered twists will have order 2, since tw=
o faces meet at each edge...
>=20
>=20
> Have a nice week all,
> Roice
>=20
>=20
>=20
> On Sun, Jan 31, 2010 at 5:52 PM, Roice Nelson wrote:
>=20
> Greetings again,
>=20
>=20
> I wanted to make a minor clarification. I didn't describe my desired ord=
er-2 twist for a length-2 Megaminx well, and incorrectly wrote that this tw=
ist would swap opposite faces. It would swap half the material for two pai=
rs of opposite faces. It would also completely swap a further two pairs of=
faces (one pair is of adjacent faces, one is not, but neither are opposite=
). Grab a Megaminx and picture an entire half of the puzzle being rotated =
180 degrees to see the twist. Hopefully the goal was clear enough though..=
. a length-2 Megaminx that wouldn't suffer from the weirdness of the Imposs=
iball. It seems a worthy aesthetic goal for puzzles to have no overlapping=
material when twisting, but I'm always curious of other opinions on things=
like that.
>=20
>=20
> Cheers,
> Roice
>=20
>=20
>=20
>=20
>=20
>=20
> On Sun, Jan 31, 2010 at 3:26 PM, Roice Nelson wrote:
>=20
> Thanks Matt!
>=20
>=20
> I agree with your thoughts about inverting the outermost face, and have a=
dded it to "the scary list". The same issue comes up in the 4D puzzles (at=
least in my implementation of Magic120Cell since I allow showing cells mir=
rored by the projection). I had originally reversed the twisting of those =
faces, but ended up reverting that when I discovered some bug entropy relat=
ed to it. Also, I worried it could lead to confusion, like "hey, that's no=
t counterclockwise!". On the other hand, it makes the projection effects e=
xplicit. It might actually be nice to have an option for this, which I thi=
nk I'll do when I get to it.
>=20
>=20
> The two layer puzzles are an interesting topic for sure. I put them in t=
he list even though they don't currently do anything (except in the Megamin=
x family), because they look pretty. I was on the fence about enabling the=
Megaminx behavior as is, but did so solely because of the Impossiball. It=
is the only puzzle in the list right now which overlaps material when twis=
ting. I actually feel there is a better length-2 analogy for the Megaminx =
than the Impossiball, which is twisting a slicing circle that is a "great c=
ircle" (cuts the unprojected puzzle sphere in half). The order of this twi=
st is 2 instead of 5 and swaps opposite faces. It also twists without over=
lapping any material, which is why I prefer it. The reason I didn't includ=
e this as the length-2 version is that this twist is edge-centered rather t=
han face-centered, and I had made the (arbitrary) decision to restrict to t=
he latter in the first version. I have to keep myself sane somehow :)
>=20
>=20
> The situation is similar in other puzzles. Check out the length-2 digona=
l and trigonal puzzles and note that the nicest twists there (which don't o=
verlap material) are vertex-centered. To answer your question about suppor=
ting all even length puzzles vs. just length-2, absolutely. The guts are c=
apable, I just didn't expose those as menu options until the behavior is be=
tter worked out. And I welcome further discussion of the specification of =
these puzzles! In particular, what would be a good way to specify edge/ver=
tex twists?
>=20
>=20
> When you get to the infinite tilings, thinking about even-layers becomes =
even stranger, and there is analogizing work to be done :) In the hexagona=
l case, I think the right twist of the length-2 puzzle would be a translati=
on of half of the puzzle! (do you agree?) Again, the reason I favor this =
is because no material overlaps. The problem I ran into was how to specify=
such twists elegantly. Try thinking about it and backing yourself into so=
me corners :) One thing I can say is that a { clicked cell + direction } s=
imply isn't enough information to specify it. The same is true in the hype=
rbolic cases.
>=20
>=20
> Related to the last paragraph, it is interesting that outer twists for th=
e infinite tilings don't make sense regardless of whether the puzzle is eve=
n or odd. What would a slice-2 twist on a length-3 hexagonal puzzle do? T=
he topology restricts the movement, which (I think) makes sense to me if I =
picture the hexagonal puzzle on a torus instead of unrolled as in MagicTile=
. Anyway, I'm perhaps getting a little off topic, but the reason to mentio=
n this is that we won't be able to specify reorientations for the infinite =
tilings as a twist "with all slices down". However, I'd love to see reorie=
ntations (both view rotations and panning) done some other way in both the =
spherical/hyperbolic cases. To see a really nice example of panning hyperb=
olic space, check out Don's hyperbolic tessellation applet! The are some b=
ig challenges to get this working in MagicTile, and performance is one of t=
he largest, since so much needs to be drawn.
>=20
>=20
> In regards to your question about the tiling patterns, there is a simple =
procedure I used to get the current list, and it involved specifying two nu=
mbers. First was the number of reflections from a "fundamental" cell to an=
"orbit" cell (I actually called them masters/slaves in the code). Second =
was which polygon segment to do this reflection across. So for example, ta=
ke a look at the 6-colored octagonal puzzle. This would be 2,4 (equivalent=
ly 4,2). The white center cell is reflected twice across the 4th segment o=
f each adjacent cell, and recursively thereafter. I wish I understood this=
all better, as I actually just had the program run through a loop to see w=
hich of the configurations "converged". Some end up fitting together and s=
ome don't. Math Magic! Hope this helped clarify. Tilings.org has some pa=
pers with lists that would probably help expose the magic more. Btw, the p=
atterns that work in the hexagonal case end up producing puzzles where the =
number of colors are perfect squares, go figure! And I wasn't able to find=
working patterns for polygons with 11 and 13 sides, though I wonder if the=
y exist and I wasn't recursing deeply enough.
>=20
>=20
> So there is plenty more discussion that could happen, but I don't want to=
overload it right off the bat. One last thing though related to your fina=
l comment. I put a feature in the program just for you Matt! Under "Optio=
ns -> Edit Settings...", play with the "Slicing Circles Expansion Factor". =
This is analogous to "deepening the cuts" on the original puzzles, which I=
know you've wanted in the 4D puzzles. It is a fully experimental setting =
and I know cases where it doesn't work well, but often it does. Try 1.4 on=
a Megaminx for a more difficult puzzle!
>=20
>=20
> Also, if anyone feels some of this discussion shouldn't be on the hypercu=
bing mailing list, let me know. I'm a little worried some might feel the h=
yperpuzzling connection a little too tenuous.
>=20
>=20
> All the best,
> Roice
>=20
>=20
>=20
>=20
>=20
>=20
>=20
> On Sun, Jan 31, 2010 at 3:37 AM, Matthew Galla wrote:
>=20
>=20
>=20
>=20
>=20
> Very nice, Roice.
>=20=20
> Although the options are clearly limited, this program is a work of art. =
The hyperbolic face patterns combined with the scrambled colors are absolut=
ely beautiful.
>=20=20
> I played around with some puzzles I already understood, and it takes a wh=
ile to get used to the little quirks in your program, but well worth it! I =
did notice that the outermost face for the cube and megaminx series is cont=
rolled opposite to my intuition. If I am thinking in terms of macros and tr=
y to apply a macro I know works near the center of the puzzle on the outerm=
ost face, I find that I must invert every move on the outermost face. A clo=
ser look reveals that this is because the outermost face is inverted. Now t=
his is just an idea, but have you considered inverting the movement for jus=
t the outermost face? Although it my confuse some things visually, I think =
it may be an overall improvement solving-wise.
>=20=20
> Also, most of your 2-layer puzzles are currently not working (which I'm s=
ure you already know). Are you looking into correcting this function of the=
program? If so, can we expect puzzles with an even number of layers >2? Fo=
r puzzles with even layers (excluding cube) the visual pieces will have to =
pass under/over/through each other. This is an inevitable behavior if you r=
estrict the exterior shape of a puzzle (which your program does because it =
forces it to be drawn on a hyperplane). However, as you demonstrated with t=
he two-layered megaminx (impossiball) this is clearly do-able.
>=20=20
> I am also looking forward to an updatewhere we can reorient some of these=
puzzles! This is allowable on the cubical and dodecahedral puzzles by hold=
ing down every layer number, but I would love to watch some these hyperplan=
e tesselations shift!
>=20=20
> My favorite thing about your program, however, is the identical puzzles w=
ith different sticker patterns. I am very interested to know how you came u=
p with the different patterns of colors on say, {6,3}, as well as the other=
puzzles with multiple color-pattern options.
>=20=20
> All in all, an excellent program that opens up a world of puzzles I had n=
ever considered before! Although, I should say that none of the puzzles in =
your program are very hard ;)
>=20=20
> Thank you for once again expanding the limits on twisty puzzles!
> Matt Galla
>=20
> PS How many moves counts as an official scramble so I can start submittin=
g my solves? :)
>=20
>=20
>=20
>=20
> On Sat, Jan 30, 2010 at 9:04 PM, Roice Nelson wrote:
>=20
>=20
>=20=20=20
>=20
>=20
>=20
> Thanks to everyone for the thoughtful feedback on my question this week. =
I appreciate it, and it was good to get your perspectives.=20
>=20
>=20
> I think I'm ready enough to share a first pass of the new Rubik analogue =
I started playing with before the MC4D 4.0 fun, which I mentioned the possi=
bility of here some time ago. While you might observe it doesn't quite fal=
l into the category of hyperpuzzles, it does in at least once sense mention=
ed below :D Here is the page with the download, pictures, and a video. To=
describe the analogue idea, I'll just quote the beginning of the explanati=
on on that page:
>=20
>=20
>=20
>=20
>=20
> This program aims to support twisty puzzles based on regular polygonal ti=
lings having Schlafli symbols of the form {p,3} for any p>=3D2. That is, al=
l regular tilings of polygons with two or more sides, where three tiles (pu=
zzle faces) meet at a vertex. The Rubik's cube is the special case where fa=
ces are squares (p=3D4). The other familiar special cases are the Megaminx =
(p=3D5) and the Pyraminx (p=3D3), although you'll discover the last takes a=
slightly different form under this abstraction (akin to Jing's Pyraminx). =
All the other puzzles are new as far as I know, and some may be surprising,=
e.g. the puzzles based on digons (p=3D2).
>=20
>=20
> Each 2D tiling admits a particular constant curvature (homogenous) geomet=
ry. The geometry is Spherical for p=3D2 to p=3D5, Euclidean (flat) for p=3D=
6, and Hyperbolic for p>=3D7. Since you can't "isometrically embed" the ent=
ire hyperbolic plane in 3-space, I have a connection to hyperpuzzling even =
though I'm talking about 2D tilings!=20
> ...
>=20=20
>=20
> I've actually only solved the 3x3x3 on it so far, and I wonder if it may =
be more fun to watch than play! I've been calling it MagicTile, though per=
haps there could be something better? As with everything, it is a known wo=
rk in progress (the length of the task list has grown to scary proportions)=
. I have no plans for further development at the moment, though I'll happi=
ly fix any glaring bugs.
>=20
> Enjoy!
> Roice
>=20
> P.S. This is the only "twisty puzzle" group I'm active in, so if any of y=
ou are also members of other groups and think they would be interested to h=
ear about these new puzzles, I'll appreciate the exposure :)
>




From: Melinda Green <melinda@superliminal.com>
Date: Mon, 01 Feb 2010 19:39:21 -0800
Subject: Re: [MC4D] Introducing "MagicTile"



--------------070802050701060508070509
Content-Type: text/plain; charset=ISO-8859-1; format=flowed
Content-Transfer-Encoding: 7bit

Wow, Roice does it again!!!
What a gem this is. It's amazing how these things look infinite but
they're not.

* Hyperbolic panning is the most obvious missing control, but you noted
that in Help > Mouse Commands. It's not strictly needed since there are
no hidden sides, but it feels incomplete without it. Don't sweat it if
it's hard. I'm sure it's easy to get used to without it though in the
meantime I would suggest disabling panning.

* There seem to be some twists on the more complex puzzles that I can't
seem to get to. That could be because I'm not sure where to click. A
mode to highlight all of the pieces that will move if clicked would be
great. Just highlighting the circle border would be plenty helpful.

* I recommend supporting resizing via the mouse wheel, even if it's
available via other gestures.

* The alert sound on all solved states gets annoying. I suggest only
doing that for true solutions even if only partial scrambles, but not
when a pristine puzzle is given one twist and then immediately the
inverse twist. Alert on Help > About seems unneeded.

* I suggest defaulting the window to a square main panel. Doesn't seem
to be a reason for a landscape layout unless you add the preferences
panel there.

* Hotkeys for all of the Options menu items would be nice.

* Is there a reason that the Properties editor is modal? If it can be
made modeless that would make it easier to experiment with the settings.

* Line thickness of zero doesn't seem to work. BTW, this is the only
actual bug that I've seen so far which means that it's *really* solid.

* Please put the puzzle name & size in the title bar.

* Save & Open log files. Useful for difficult puzzles and needed if you
plan to support records for first & shortests.

* Number One on my wishlist? The {7,3} duel of the physically possible
{3,7} . this
{3,7} is by far my favorite infinite polyhedron.

Great work, Roice!
-Melinda

p.s. No problem discussing this on the 4D list. Even tangentially
related subjects are fine so really anything regarding twisty puzzles is
perfectly appropriate here.


Roice Nelson wrote:
>
>
> Thanks to everyone for the thoughtful feedback on my question this
> week. I appreciate it, and it was good to get your perspectives.
>
> I think I'm ready enough to share a first pass of the new Rubik
> analogue I started playing with before the MC4D 4.0 fun, which I
> mentioned the possibility of here
> some time
> ago. While you might observe it doesn't quite fall into the category
> of hyperpuzzles, it does in at least once sense mentioned below :D
> Here is the page with the download, pictures, and a video
> . To describe the analogue
> idea, I'll just quote the beginning of the explanation on that page:
>
>
> This program aims to support twisty puzzles based on regular
> polygonal tilings
> having
> Schlafli symbols of
> the form {p,3} for any p>=2. That is, all regular tilings of
> polygons with two or more sides, where three tiles (puzzle faces)
> meet at a vertex. The Rubik's cube is the special case where faces
> are squares (p=4). The other familiar special cases are the
> Megaminx (p=5) and the Pyraminx (p=3), although you'll discover
> the last takes a slightly different form under this abstraction
> (akin to Jing's Pyraminx
> ). All the other
> puzzles are new as far as I know, and some may be surprising, e.g.
> the puzzles based on digons
> (p=2).
>
>
> Each 2D tiling admits a particular constant curvature (homogenous)
> geometry. The geometry is Spherical for p=2 to p=5, Euclidean
> (flat) for p=6, and Hyperbolic for p>=7. Since you can't
> "isometrically embed" the entire hyperbolic plane in 3-space
> ,
> I have a connection to hyperpuzzling
> even though I'm
> talking about 2D tilings!
>
> ...
>
>
>
>
> I've actually only solved the 3x3x3 on it so far, and I wonder if it
> may be more fun to watch than play! I've been calling it MagicTile,
> though perhaps there could be something better? As with everything,
> it is a known work in progress (the length of the task list has grown
> to scary proportions). I have no plans for further development at the
> moment, though I'll happily fix any glaring bugs.
>
> Enjoy!
> Roice
>
> P.S. This is the only "twisty puzzle" group I'm active in, so if any
> of you are also members of other groups and think they would be
> interested to hear about these new puzzles, I'll appreciate the
> exposure :)
>
>
>
> __._,_.__

--------------070802050701060508070509
Content-Type: text/html; charset=ISO-8859-1
Content-Transfer-Encoding: 7bit







Wow, Roice does it again!!!

What a gem this is. It's amazing how these things look infinite but
they're not.



* Hyperbolic panning is the most obvious missing control, but you noted
that in Help > Mouse Commands. It's not strictly needed since there
are
no hidden sides, but it feels incomplete without it. Don't sweat it if
it's hard. I'm sure it's easy to get used to without it though in the
meantime I would suggest disabling panning.



* There seem to be some twists on the more complex puzzles that I can't
seem
to get to. That could be because I'm not sure where to click. A mode to
highlight all of the pieces that will move if clicked would be great.
Just highlighting the circle border would be plenty helpful.



* I recommend supporting resizing via the mouse wheel, even if it's
available via other gestures.



* The alert sound on all solved states gets annoying. I suggest only
doing that for true solutions even if only partial scrambles, but not
when a pristine puzzle is given one twist and then immediately the
inverse twist. Alert on Help > About seems unneeded.



* I suggest defaulting the window to a square main panel. Doesn't seem
to be a reason for a landscape layout unless you add the preferences
panel there.



* Hotkeys for all of the Options menu items would be nice.



* Is there a reason that the Properties editor is modal? If it can be
made modeless that would make it easier to experiment with the settings.



* Line thickness of zero doesn't seem to work. BTW, this is the only
actual bug that I've seen so far which means that it's *really* solid.



* Please put the puzzle name & size in the title bar.



* Save & Open log files. Useful for difficult puzzles and needed if
you plan to support records for
first & shortests.



* Number One on my wishlist? The {7,3} duel of the physically possible href="http://www.superliminal.com/geometry/infinite/3_7a.htm">{3,7}.
this {3,7} is by far my favorite infinite polyhedron.



Great work, Roice!

-Melinda



p.s. No problem discussing this on the 4D list. Even tangentially
related subjects are fine so really anything regarding twisty puzzles
is perfectly appropriate here.





Roice Nelson wrote:
cite="mid:b5979e761001301904w10ea3499o11bfe4ad2ef26661@mail.gmail.com"
type="cite">


Thanks to everyone for the thoughtful feedback on my question this
week.  I appreciate it, and it was good to get your perspectives.



I think I'm ready enough to share a first pass of the new Rubik
analogue I started playing with before the MC4D 4.0 fun, which I moz-do-not-send="true"
href="http://games.groups.yahoo.com/group/4D_Cubing/message/541"
target="_blank">mentioned the possibility of here some time ago. 
While you might observe it doesn't quite fall into the category of
hyperpuzzles, it does in at least once sense mentioned below :D  Here
is the href="http://www.gravitation3d.com/magictile" target="_blank">page
with the download, pictures, and a video
.  To describe the analogue
idea, I'll just quote the beginning of the explanation on that page:





style="border-left: 1px solid rgb(204, 204, 204); margin: 0px 0px 0px 0.8ex; padding-left: 1ex;">


style="border-left: 1px solid rgb(204, 204, 204); margin: 0px 0px 0px 0.8ex; padding-left: 1ex;">This
program aims to support twisty puzzles based on  moz-do-not-send="true"
href="http://en.wikipedia.org/wiki/Regular_tessellation"
target="_blank"> style="text-decoration: none;">regular polygonal tilings having
href="http://en.wikipedia.org/wiki/Schlafli_symbol" target="_blank"> color="#000000">Schlafli symbols
 of
the form {p,3} for any p>=2. That is, all regular tilings of
polygons with two or more sides, where three tiles (puzzle faces) meet
at a vertex. The Rubik's cube is the special case where faces are
squares (p=4). The other familiar special cases are the Megaminx (p=5)
and the Pyraminx (p=3), although you'll discover the last takes a
slightly different form under this abstraction (akin to  moz-do-not-send="true"
href="http://www.youtube.com/watch?v=FuD3YwQTW2c" target="_blank"> color="#000000">Jing's Pyraminx
).
All the other puzzles are new as far as I know, and some may be
surprising, e.g. the puzzles based on href="http://en.wikipedia.org/wiki/Digon" target="_blank"> color="#000000">digons (p=2).

style="border-left: 1px solid rgb(204, 204, 204); margin: 0px 0px 0px 0.8ex; padding-left: 1ex;">



style="border-left: 1px solid rgb(204, 204, 204); margin: 0px 0px 0px 0.8ex; padding-left: 1ex;">Each
2D tiling admits a particular constant curvature (homogenous) geometry.
The geometry is Spherical for p=2 to p=5, Euclidean (flat) for p=6, and
Hyperbolic for p>=7. Since  href="http://www.math.cornell.edu/%7Edwh/papers/crochet/crochet.html"
target="_blank"> style="text-decoration: none;">you can't "isometrically embed" the
entire hyperbolic plane in 3-space
, I have a
connection to  href="http://games.groups.yahoo.com/group/4D_Cubing/" target="_blank"> color="#000000">hyperpuzzling even
though I'm talking about 2D tilings! 

style="border-left: 1px solid rgb(204, 204, 204); margin: 0px 0px 0px 0.8ex; padding-left: 1ex;">...

style="border-left: 1px solid rgb(204, 204, 204); margin: 0px 0px 0px 0.8ex; padding-left: 1ex;"> 




I've actually only solved the 3x3x3 on it so far, and I wonder if it
may be more fun to watch than play!  I've been calling it MagicTile,
though perhaps there could be something better?  As with everything, it
is a known work in progress (the length of the task list has grown to
scary proportions).  I have no plans for further development at the
moment, though I'll happily fix any glaring bugs.



Enjoy!

Roice



P.S. This is the only "twisty puzzle" group I'm active in, so if any of
you are also members of other groups and think they would be interested
to hear about these new puzzles, I'll appreciate the exposure :)






__._,_.__





--------------070802050701060508070509--




From: Roice Nelson <roice3@gmail.com>
Date: Tue, 2 Feb 2010 19:26:28 -0600
Subject: Re: [MC4D] Introducing "MagicTile"



--0015173ff316888d15047ea81b07
Content-Type: text/plain; charset=ISO-8859-1

Hi Melinda,

Thanks for all the great feedback as usual! These suggestions will all
really help the polish, and most should be easy to knock out (though I was
thinking to take at least a couple weeks rest from coding first). It's
funny, the solve beep was definitely annoying me, yet I never considered to
do anything about it.

I've responded offline to some, but a public thanks to the others who have
been giving feedback and support as well! I always appreciate the positive
responses :)

I'm a bit bummed about how impossible the hyperbolic panning feels at this
point. I had significant drawing performance issues on the infinite
tilings, and the way I got around it was with OpenGL rendering caching
(display lists). When a twist happens, that caching is invalidated for only
the affected stickers. This is why performance on the puzzles with fewer
colors is poorer, since a twist invalidates more and requires more to be
drawn without the cache. Perhaps what could be done is to only show cell
outlines during a pan. Another thought I had was to try to use texture maps
(create needed textures of the current state on the fly to then use for the
panning), though that sounds quite involved. The hyperbolic panning and
drawing for the hyperbolic games at geometrygames.org are ultra smooth, and
I can tell they are using textures. Anyway, I'm worried I've gone too far
down the wrong path to handle this fundamental feature well, which is
disappointing. There's another hurdle as well I won't bother going into,
but maybe some good solutions will present themselves.

On the plus side, it seems the {3,7} and other puzzles without simplex
vertex figures likely won't require a great deal of work. After Alexander's
suggestion of the {4,4} and {3,6}, I made a one line change to quickly look
at the possibility of those, and while there were plenty of problems, I was
surprised how well it went without any real effort... only a partial
explosion :)

Also, I'm not sure, but I wonder if the "Klein's Quartic" puzzle is the dual
{7,3} you are looking for? As far as I know, the only way to fit together a
repeating set of heptagons is with 1 or 24 in the set, and Klein's Quartic
does have 56 vertices (corresponding to the number of triangles at the link
you sent). Or maybe you are looking for a Euclidean {7,3} "infinite regular
polyhedron", though I have no idea if such a thing is even possible. (?)

Lastly, I'm curious more about your twisting problems on the complex
puzzles. It is only the clicked face that matters (and faces are currently
demarked by yellow lines, though I'd like to make that configurable). The
particular stickers clicked don't matter, and slices are controlled with
number keys. Sorry I didn't follow that particular comment better.

Hope all is well!
Roice


On 2/1/10, Melinda Green wrote:
>
>
>
> Wow, Roice does it again!!!
> What a gem this is. It's amazing how these things look infinite but they're
> not.
>
> * Hyperbolic panning is the most obvious missing control, but you noted
> that in Help > Mouse Commands. It's not strictly needed since there are no
> hidden sides, but it feels incomplete without it. Don't sweat it if it's
> hard. I'm sure it's easy to get used to without it though in the meantime I
> would suggest disabling panning.
>
> * There seem to be some twists on the more complex puzzles that I can't
> seem to get to. That could be because I'm not sure where to click. A mode to
> highlight all of the pieces that will move if clicked would be great. Just
> highlighting the circle border would be plenty helpful.
>
> * I recommend supporting resizing via the mouse wheel, even if it's
> available via other gestures.
>
> * The alert sound on all solved states gets annoying. I suggest only doing
> that for true solutions even if only partial scrambles, but not when a
> pristine puzzle is given one twist and then immediately the inverse twist.
> Alert on Help > About seems unneeded.
>
> * I suggest defaulting the window to a square main panel. Doesn't seem to
> be a reason for a landscape layout unless you add the preferences panel
> there.
>
> * Hotkeys for all of the Options menu items would be nice.
>
> * Is there a reason that the Properties editor is modal? If it can be made
> modeless that would make it easier to experiment with the settings.
>
> * Line thickness of zero doesn't seem to work. BTW, this is the only actual
> bug that I've seen so far which means that it's *really* solid.
>
> * Please put the puzzle name & size in the title bar.
>
> * Save & Open log files. Useful for difficult puzzles and needed if you
> plan to support records for first & shortests.
>
> * Number One on my wishlist? The {7,3} duel of the physically possible
> {3,7} . this {3,7}
> is by far my favorite infinite polyhedron.
>
> Great work, Roice!
> -Melinda
>
> p.s. No problem discussing this on the 4D list. Even tangentially related
> subjects are fine so really anything regarding twisty puzzles is perfectly
> appropriate here.
>
>
> Roice Nelson wrote:
>
> Thanks to everyone for the thoughtful feedback on my question this week. I
> appreciate it, and it was good to get your perspectives.
>
> I think I'm ready enough to share a first pass of the new Rubik analogue I
> started playing with before the MC4D 4.0 fun, which I mentioned the
> possibility of heresome time ago. While you might observe it doesn't quite fall into the
> category of hyperpuzzles, it does in at least once sense mentioned below :D
> Here is the page with the download, pictures, and a video.
> To describe the analogue idea, I'll just quote the beginning of the
> explanation on that page:
>
>
>
>>
>> This program aims to support twisty puzzles based on regular polygonal
>> tilings having Schlafli
>> symbols of the form {p,3}
>> for any p>=2. That is, all regular tilings of polygons with two or more
>> sides, where three tiles (puzzle faces) meet at a vertex. The Rubik's cube
>> is the special case where faces are squares (p=4). The other familiar
>> special cases are the Megaminx (p=5) and the Pyraminx (p=3), although you'll
>> discover the last takes a slightly different form under this abstraction
>> (akin to Jing's Pyraminx ).
>> All the other puzzles are new as far as I know, and some may be surprising,
>> e.g. the puzzles based on digons
>> (p=2).
>
>
>> Each 2D tiling admits a particular constant curvature (homogenous)
>> geometry. The geometry is Spherical for p=2 to p=5, Euclidean (flat) for
>> p=6, and Hyperbolic for p>=7. Since you can't "isometrically embed" the
>> entire hyperbolic plane in 3-space,
>> I have a connection to hyperpuzzling even
>> though I'm talking about 2D tilings!
>
> ...
>
>
>
>
> I've actually only solved the 3x3x3 on it so far, and I wonder if it may be
> more fun to watch than play! I've been calling it MagicTile, though perhaps
> there could be something better? As with everything, it is a known work in
> progress (the length of the task list has grown to scary proportions). I
> have no plans for further development at the moment, though I'll happily fix
> any glaring bugs.
>
> Enjoy!
> Roice
>
> P.S. This is the only "twisty puzzle" group I'm active in, so if any of you
> are also members of other groups and think they would be interested to hear
> about these new puzzles, I'll appreciate the exposure :)
>
>
>
> __._,_.__
>
>
>
>

--0015173ff316888d15047ea81b07
Content-Type: text/html; charset=ISO-8859-1
Content-Transfer-Encoding: quoted-printable

Hi Melinda,

=A0

Thanks for all the great feedback as usual!=A0 These suggestions will =
all really help the polish, and most should be easy to knock out (though I =
was thinking to take at least a=A0couple weeks rest from coding first). =A0=
It's funny, the solve beep was definitely annoying me, yet I never cons=
idered to do anything about it. =A0


I've responded offline to some, but a public thanks=
to the others who have been giving feedback and support as well! =A0I alwa=
ys appreciate the positive responses :)


=A0

I'm a bit bummed about how impossible the hyperbolic panning feels=
at this point.=A0 I had significant drawing performance issues on the infi=
nite tilings, and the way I got around it was with OpenGL rendering caching=
(display lists).=A0 When a twist happens, that caching is invalidated for =
only the affected stickers.=A0 This is why performance on the puzzles with =
fewer colors is poorer, since a twist invalidates more and requires more to=
be drawn without the cache.=A0 Perhaps what could be done is to only show =
cell outlines during a pan.=A0 Another thought I had was to try to use text=
ure maps (create=A0needed textures of the current state on the fly to then =
use for the panning), though that sounds quite involved.=A0 The hyperbolic =
panning and drawing for the hyperbolic games at ames.org/" target=3D"_blank">geometrygames.org are ultra smooth, and I =
can tell they are using textures. =A0Anyway, I'm worried I've gone =
too far down the wrong path to handle this fundamental feature well, which =
is disappointing. =A0There's another hurdle as well I won't bother =
going into, but=A0maybe some good solutions will present themselves.


=A0

On the plus side,=A0it seems=A0the {3,7} and other puzzles without sim=
plex vertex figures likely won't require a great deal of work.=A0 After=
Alexander's suggestion of the {4,4} and {3,6}, I made a one line chang=
e to quickly look at the possibility of those, and while there were plenty =
of problems, I was surprised how well it went without any real effort... on=
ly a partial explosion :)



=A0

Also, I'm not sure, but I wonder if the "Klein's Quartic&=
quot;=A0puzzle is the dual {7,3} you are looking for?=A0 As far as I know, =
the only way to fit together a repeating set of heptagons is with 1 or 24 i=
n the set, and Klein's Quartic does have 56 vertices (corresponding to =
the number of triangles at the link you sent).=A0 Or maybe you are looking =
for a Euclidean {7,3}=A0"infinite regular polyhedron", though I h=
ave no idea if such a thing is even possible. (?)




Lastly, I'm curious more about your twisting problems on the c=
omplex puzzles.=A0 It is only the clicked face that matters (and faces are =
currently demarked by yellow lines, though I'd like to make that config=
urable).=A0 The particular stickers clicked don't matter, and slices ar=
e controlled with number keys.=A0 Sorry I didn't follow that particular=
comment better.



=A0

Hope all is well!

Roice

=A0

=A0

On 2/1/10, M=
elinda Green
=A0wrote:
=20
0px 0.8ex;border-left:#ccc 1px solid">


Wow, Roice does it again!=
!!
What a gem this is. It's amazing how these things look infinite b=
ut they're not.

* Hyperbolic panning is the most obvious missing=
control, but you noted that in Help > Mouse Commands. It's not stri=
ctly needed since there are no hidden sides, but it feels incomplete withou=
t it. Don't sweat it if it's hard. I'm sure it's easy to ge=
t used to without it though in the meantime I would suggest disabling panni=
ng.



* There seem to be some twists on the more complex puzzles that I can&#=
39;t seem to get to. That could be because I'm not sure where to click.=
A mode to highlight all of the pieces that will move if clicked would be g=
reat. Just highlighting the circle border would be plenty helpful.



* I recommend supporting resizing via the mouse wheel, even if it's=
available via other gestures.

* The alert sound on all solved state=
s gets annoying. I suggest only doing that for true solutions even if only =
partial scrambles, but not when a pristine puzzle is given one twist and th=
en immediately the inverse twist. Alert on Help > About seems unneeded.<=
br>


* I suggest defaulting the window to a square main panel. Doesn't s=
eem to be a reason for a landscape layout unless you add the preferences pa=
nel there.

* Hotkeys for all of the Options menu items would be nice=
.



* Is there a reason that the Properties editor is modal? If it can be m=
ade modeless that would make it easier to experiment with the settings.
=

* Line thickness of zero doesn't seem to work. BTW, this is the onl=
y actual bug that I've seen so far which means that it's *really* s=
olid.



* Please put the puzzle name & size in the title bar.

* Save=
& Open log files. Useful for difficult puzzles and needed if you plan =
to support records for first & shortests.

* Number One on my wis=
hlist? The {7,3} duel of the physically possible rliminal.com/geometry/infinite/3_7a.htm" target=3D"_blank">{3,7}. this =
{3,7} is by far my favorite infinite polyhedron.



Great work, Roice!
-Melinda

p.s. No problem discussing this o=
n the 4D list. Even tangentially related subjects are fine so really anythi=
ng regarding twisty puzzles is perfectly appropriate here.



Roice Nelson wrote:=20

Thanks to everyone for the thoughtful feedback on my question th=
is week. =A0I appreciate it, and it was good to get your perspectives.=20

I think I'm ready enough to share a first pass of the new Rubi=
k analogue I started playing with before the MC4D 4.0 fun, which I =3D"http://games.groups.yahoo.com/group/4D_Cubing/message/541" target=3D"_b=
lank">mentioned the possibility of here
some time ago.=A0 While you mig=
ht observe it doesn't quite fall into the category of hyperpuzzles, it =
does in at least once sense mentioned below=A0:D =A0Here is the http://www.gravitation3d.com/magictile" target=3D"_blank">page with the dow=
nload, pictures, and a video
.=A0 To describe the analogue idea, I'l=
l just quote the beginning of the explanation on that page:



=A0


0px 0.8ex;border-left:rgb(204,204,204) 1px solid">

0px 0.8ex;border-left:rgb(204,204,204) 1px solid">This program aims to supp=
ort twisty puzzles based on=A0ar_tessellation" target=3D"_blank">ext-decoration:none">regular polygonal tilings=A0having <=
a href=3D"http://en.wikipedia.org/wiki/Schlafli_symbol" target=3D"_blank"><=
font color=3D"#000000">Schlafli symbol=
s
=A0of the form {p,3} for any p>=3D2. That is, all reg=
ular tilings of polygons with two or more sides, where three tiles (puzzle =
faces) meet at a vertex. The Rubik's cube is the special case where fac=
es are squares (p=3D4). The other familiar special cases are the Megaminx (=
p=3D5) and the Pyraminx (p=3D3), although you'll discover the last take=
s a slightly different form under this abstraction (akin to=A0tp://www.youtube.com/watch?v=3DFuD3YwQTW2c" target=3D"_blank">=3D"#000000">Jing's Pyraminx>). All the other puzzles are new as far as I know, and some may=
be surprising, e.g. the puzzles based on g/wiki/Digon" target=3D"_blank">-decoration:none">digons=A0(p=3D2).



0px 0.8ex;border-left:rgb(204,204,204) 1px solid">

0px 0.8ex;border-left:rgb(204,204,204) 1px solid">Each 2D tiling admits a p=
articular constant curvature (homogenous) geometry. The geometry is Spheric=
al for p=3D2 to p=3D5, Euclidean (flat) for p=3D6, and Hyperbolic for p>=
=3D7. Since=A0crochet.html" target=3D"_blank">-decoration:none">you can't "isometrically embed" the entire =
hyperbolic plane in 3-space
, I have a connection to=A0href=3D"http://games.groups.yahoo.com/group/4D_Cubing/" target=3D"_blank"><=
font color=3D"#000000">hyperpuzzlingspan>
=A0even though I'm talking about 2D tilings!=A0uote>


0px 0.8ex;border-left:rgb(204,204,204) 1px solid">...

0px 0.8ex;border-left:rgb(204,204,204) 1px solid">=A0


I've actually only solved the 3x3x3 on it so far, and I wonder=
if it may be more fun to watch than play! =A0I've been calling it Magi=
cTile, though perhaps there could be something better? =A0As with everythin=
g, it is a known work in progress (the length of the task list has grown to=
scary proportions). =A0I have no plans for further development at the mome=
nt, though I'll happily fix any glaring bugs.




Enjoy!
Roice

P.S. This is the only "twisty puzzle&q=
uot; group I'm active in, so if any of you are also members of other gr=
oups and think they would be interested to hear about these new puzzles,=A0=
I'll appreciate the exposure :)


=A0


=A0

__._,_.__
te>


>


--0015173ff316888d15047ea81b07--




From: Melinda Green <melinda@superliminal.com>
Date: Tue, 02 Feb 2010 18:11:11 -0800
Subject: Re: [MC4D] Introducing "MagicTile"



Roice Nelson wrote:
>
>
> Hi Melinda,
>
> Thanks for all the great feedback as usual! These suggestions will
> all really help the polish, and most should be easy to knock out
> (though I was thinking to take at least a couple weeks rest from
> coding first). It's funny, the solve beep was definitely annoying me,
> yet I never considered to do anything about it.

Yeah, well we had lots of the same issue in MC4D and I'm probably going
to start sounding like a broken record when I start suggesting that
similar actions in this puzzle follow the MC4D patterns... In this case
MC4D gives a beep when solving a partial scramble, and a fanfare on a
full solve, but no action for otherwise creating a pristine state.

>
> I've responded offline to some, but a public thanks to the others who
> have been giving feedback and support as well! I always appreciate
> the positive responses :)
>
> I'm a bit bummed about how impossible the hyperbolic panning feels at
> this point. I had significant drawing performance issues on the
> infinite tilings, and the way I got around it was with OpenGL
> rendering caching (display lists). When a twist happens, that caching
> is invalidated for only the affected stickers. This is why
> performance on the puzzles with fewer colors is poorer, since a twist
> invalidates more and requires more to be drawn without the cache.
> Perhaps what could be done is to only show cell outlines during a
> pan. Another thought I had was to try to use texture maps
> (create needed textures of the current state on the fly to then use
> for the panning), though that sounds quite involved. The hyperbolic
> panning and drawing for the hyperbolic games at geometrygames.org
> are ultra smooth, and I can tell they are
> using textures. Anyway, I'm worried I've gone too far down the wrong
> path to handle this fundamental feature well, which is disappointing.
> There's another hurdle as well I won't bother going into, but maybe
> some good solutions will present themselves.

Seriously, I think that you'll be fine if you just disable all panning.
Doing it "the right way" seems way out of scope for its importance. That
said, I very much like your idea of only drawing outlines during
panning, so if that's relatively easy, then I endorse that product
and/or service. One more possible approach: The MC4D way! :-)
Ctrl-click a face to snap it to the center. Maybe confusing without the
animation, so I'm liking the outline idea more and more.

>
> On the plus side, it seems the {3,7} and other puzzles without simplex
> vertex figures likely won't require a great deal of work. After
> Alexander's suggestion of the {4,4} and {3,6}, I made a one line
> change to quickly look at the possibility of those, and while there
> were plenty of problems, I was surprised how well it went without any
> real effort... only a partial explosion :)
>
> Also, I'm not sure, but I wonder if the "Klein's Quartic" puzzle is
> the dual {7,3} you are looking for? As far as I know, the only way to
> fit together a repeating set of heptagons is with 1 or 24 in the set,
> and Klein's Quartic does have 56 vertices (corresponding to the number
> of triangles at the link you sent). Or maybe you are looking for a
> Euclidean {7,3} "infinite regular polyhedron", though I have no idea
> if such a thing is even possible. (?)

The Quartic thing should have triggered that memory. Indeed I'm sure
that this is indeed the duel I was looking for. If you can support the
{3,7} version, that's great but not important.

> Lastly, I'm curious more about your twisting problems on the complex
> puzzles. It is only the clicked face that matters (and faces are
> currently demarked by yellow lines, though I'd like to make that
> configurable). The particular stickers clicked don't matter, and
> slices are controlled with number keys. Sorry I didn't follow that
> particular comment better.

I just found myself very confused sometimes with highly scrambled
puzzles, often getting twists that I didn't expect. Just like how
highlighting in MC4D is helpful feedback for what will come, I would
love to see the "active" face highlighted. Simply thickening its edge,
or drawing that edge in a highlight color, or brightening all tiles in
that face (the MC4D way) all work for me!

Fun stuff!!
-Melinda

>
> On 2/1/10, *Melinda Green* wrote:
>
>
>
> Wow, Roice does it again!!!
> What a gem this is. It's amazing how these things look infinite
> but they're not.
>
> * Hyperbolic panning is the most obvious missing control, but you
> noted that in Help > Mouse Commands. It's not strictly needed
> since there are no hidden sides, but it feels incomplete without
> it. Don't sweat it if it's hard. I'm sure it's easy to get used to
> without it though in the meantime I would suggest disabling panning.
>
> * There seem to be some twists on the more complex puzzles that I
> can't seem to get to. That could be because I'm not sure where to
> click. A mode to highlight all of the pieces that will move if
> clicked would be great. Just highlighting the circle border would
> be plenty helpful.
>
> * I recommend supporting resizing via the mouse wheel, even if
> it's available via other gestures.
>
> * The alert sound on all solved states gets annoying. I suggest
> only doing that for true solutions even if only partial scrambles,
> but not when a pristine puzzle is given one twist and then
> immediately the inverse twist. Alert on Help > About seems unneeded.
>
> * I suggest defaulting the window to a square main panel. Doesn't
> seem to be a reason for a landscape layout unless you add the
> preferences panel there.
>
> * Hotkeys for all of the Options menu items would be nice.
>
> * Is there a reason that the Properties editor is modal? If it can
> be made modeless that would make it easier to experiment with the
> settings.
>
> * Line thickness of zero doesn't seem to work. BTW, this is the
> only actual bug that I've seen so far which means that it's
> *really* solid.
>
> * Please put the puzzle name & size in the title bar.
>
> * Save & Open log files. Useful for difficult puzzles and needed
> if you plan to support records for first & shortests.
>
> * Number One on my wishlist? The {7,3} duel of the physically
> possible {3,7}
> . this
> {3,7} is by far my favorite infinite polyhedron.
>
> Great work, Roice!
> -Melinda
>
> p.s. No problem discussing this on the 4D list. Even tangentially
> related subjects are fine so really anything regarding twisty
> puzzles is perfectly appropriate here.
>
>
> Roice Nelson wrote:
>> Thanks to everyone for the thoughtful feedback on my question
>> this week. I appreciate it, and it was good to get your
>> perspectives.
>>
>> I think I'm ready enough to share a first pass of the new Rubik
>> analogue I started playing with before the MC4D 4.0 fun, which I
>> mentioned the possibility of here
>> some
>> time ago. While you might observe it doesn't quite fall into the
>> category of hyperpuzzles, it does in at least once sense
>> mentioned below :D Here is the page with the download, pictures,
>> and a video . To
>> describe the analogue idea, I'll just quote the beginning of the
>> explanation on that page:
>>
>>
>>
>>
>> This program aims to support twisty puzzles based on regular
>> polygonal tilings
>> having
>> Schlafli symbols
>> of the form
>> {p,3} for any p>=2. That is, all regular tilings of polygons
>> with two or more sides, where three tiles (puzzle faces) meet
>> at a vertex. The Rubik's cube is the special case where faces
>> are squares (p=4). The other familiar special cases are the
>> Megaminx (p=5) and the Pyraminx (p=3), although you'll
>> discover the last takes a slightly different form under this
>> abstraction (akin to Jing's Pyraminx
>> ). All the other
>> puzzles are new as far as I know, and some may be surprising,
>> e.g. the puzzles based on digons
>> (p=2).
>>
>>
>> Each 2D tiling admits a particular constant curvature
>> (homogenous) geometry. The geometry is Spherical for p=2 to
>> p=5, Euclidean (flat) for p=6, and Hyperbolic for p>=7.
>> Since you can't "isometrically embed" the entire hyperbolic
>> plane in 3-space
>> ,
>> I have a connection to hyperpuzzling
>> even though
>> I'm talking about 2D tilings!
>>
>> ...
>>
>>
>>
>>
>> I've actually only solved the 3x3x3 on it so far, and I wonder if
>> it may be more fun to watch than play! I've been calling it
>> MagicTile, though perhaps there could be something better? As
>> with everything, it is a known work in progress (the length of
>> the task list has grown to scary proportions). I have no plans
>> for further development at the moment, though I'll happily fix
>> any glaring bugs.
>>
>> Enjoy!
>> Roice
>>
>> P.S. This is the only "twisty puzzle" group I'm active in, so if
>> any of you are also members of other groups and think they would
>> be interested to hear about these new puzzles, I'll appreciate
>> the exposure :)
>>
>>
>>
>> __._,_.__
>
>
>
>
>
>




From: Roice Nelson <roice3@gmail.com>
Date: Wed, 3 Feb 2010 00:03:53 -0600
Subject: Re: [MC4D] Re: Introducing "MagicTile"



--0015175884fea9af83047eabfbb1
Content-Type: text/plain; charset=ISO-8859-1

I found the "two bottoms" observation extremely interesting :D I tried to
figure out the why of this last night by rereading John Baez's
articleon Klein's Quartic,
but didn't have much luck finding the insight I was
looking for (though the two cells in the last layer is mentioned there, and
the article is full of other neat information). It felt like the behavior
should be related to the topology and the fact that a 3-holed-torus (genus
3, which is the topology of the puzzle) is not simply
connected.
But the 12-colored octagonal puzzle is also genus 3, and it doesn't behave
the same.

I found some more info this evening, and it turns out there is an entire
book on Klein's Quartic! Amazon has
it,
but you can download a free version
online(however, the
pictures seem to be missing). In the first section by
Thurston, he notes "The infinite hyperbolic honeycomb is divided into 3
kinds of groups of 8 cells each, where each group is composed of a heptagon
together with its 7 neighbors.". Together, these groups account for the 24
cells, and after labeling the 3 groups red/green/white, he writes:


> It is interesting to watch what happens when you rotate the pattern by a
> 1/7 revolution about the central tile: red groups go to red groups, green
> groups go to green groups and white groups go to white groups. The person in
> the center of a green group rotates by 2/7 revolution, and the person in the
> center of a red group rotates by 4/7 revolution. The interpretation on the
> surface is that the 24 cells are grouped into 8 affinity groups of 3 each.
> The symmetries of the surface always take affinity groups to affinity
> groups. This is analogous to the dodecahedron, whose twelve pentagonal faces
> are divided into 6 affinity groups of 2 each, consisting of pairs of
> opposite faces.


So I think it has more to do with the symmetries of the object than the
topology (though perhaps there is some interrelation). I think what Nelson
found was one of these 8 affinity groups. Btw, by editing colors, you
should be able to use the program to more easily see the reg/green/white
groups described above - I'll have to try this.

Also, I did note to myself last night that it is possible to solve the {7,3}
cells in an order such that you'd be left with one cell at the end instead
of two, but you wouldn't be working "layer-by-layer" in that case.

Very cool discovery of this unusual behavior Nelson!

Roice


On 2/1/10, spel_werdz_rite wrote:
>
> In a follow up with Roice, I'd like to share some more interesting details
> with the Klein's Quartic puzzle.
>
> The strategy to solving it was very much similar to how one would solve a
> Megaminx (my method at least). Edges, then sides, then edges, working all
> the way down to the bottom of the puzzle. Doing this method lead to me a
> very interesting discovery that, surprisingly, not even Roice new about. It
> turns out that Klein's Quartic has two "bottoms." By which I mean if you
> follow this method of inserting pieces downward until you reach the bottom
> of the puzzle, you will end up at 2 different faces. At this location,
> solving became a bit of a new task, but still not much of a challenge. The
> first step was making sure the remaining 2C and 3C pieces were on their
> corresponding face and oriented correctly. After that, I borrowed many
> techniques I used for the Megaminx. However, due to some obvious
> differences, the end took a lot of guesswork. In the end, the puzzle took
> about 2.5 hours (factoring in my "hey let's get distracted a lot"
> variables).
>
> My final thoughts. Very fun. It was a true joy to play a technical 3D
> puzzle that technically couldn't exist in the 3D world.
>
>

--0015175884fea9af83047eabfbb1
Content-Type: text/html; charset=ISO-8859-1
Content-Transfer-Encoding: quoted-printable




=A0

I found some more info this evening, and it turns out there is an enti=
re book on Klein's Quartic!=A0 uct/0521004195?ie=3DUTF8&tag=3Dgravit-20&linkCode=3Das2&camp=3D=
1789&creative=3D390957&creativeASIN=3D0521004195">Amazon has it
=
, but you can download a Book35/">free version online (however, the pictures seem to be missing)=
.=A0 In the first section by Thurston, he notes "The infinite hyperbol=
ic honeycomb is divided into 3 kinds of groups of 8 cells each, where each =
group is composed of a heptagon together with its 7 neighbors.".=A0 To=
gether, these groups=A0account for the 24 cells, and=A0after labeling the 3=
groups red/green/white, he writes:


=A0

0px 0.8ex;border-left:#ccc 1px solid">It is interesting to watch what happe=
ns when you rotate the pattern by a 1/7 revolution about the central tile: =
red groups go to red groups, green groups go to green groups and white grou=
ps go to white groups. The person in the center of a green group rotates by=
2/7 revolution, and the person in the center of a red group rotates by 4/7=
revolution. The interpretation on the surface is that the 24 cells are gro=
uped into 8 affinity groups of 3 each. The symmetries of the surface always=
take affinity groups to affinity groups. This is analogous to the dodecahe=
dron, whose twelve pentagonal faces are divided into 6 affinity groups of 2=
each, consisting of pairs of opposite faces.



So I think it has more to do with the symmetries of the object than the=
topology (though perhaps there is some interrelation). =A0I think what Nel=
son found was one of these 8 affinity groups. =A0Btw, by editing colors, yo=
u should be able to use the program to more easily see the reg/green/white =
groups described above - I'll have to try this.

=A0
Also, I did note to myself last night that it is possible to solve t=
he {7,3} cells in an order such that you'd be left with one cell at the=
end instead of two, but you wouldn't be working "layer-by-layer&q=
uot; in that case.


=A0

Very cool discovery of this unusual behavior Nelson!

=A0

Roice

=A0

On 2/1/10, s=
pel_werdz_rite
wrote:
=20
0px 0.8ex;border-left:#ccc 1px solid">In a follow up with Roice, I'd li=
ke to share some more interesting details with the Klein's Quartic puzz=
le.



The strategy to solving it was very much similar to how one would solve=
a Megaminx (my method at least). Edges, then sides, then edges, working al=
l the way down to the bottom of the puzzle. Doing this method lead to me a =
very interesting discovery that, surprisingly, not even Roice new about. It=
turns out that Klein's Quartic has two "bottoms." By which I=
mean if you follow this method of inserting pieces downward until you reac=
h the bottom of the puzzle, you will end up at 2 different faces. At this l=
ocation, solving became a bit of a new task, but still not much of a challe=
nge. The first step was making sure the remaining 2C and 3C pieces were on =
their corresponding face and oriented correctly. After that, I borrowed man=
y techniques I used for the Megaminx. However, due to some obvious differen=
ces, the end took a lot of guesswork. In the end, the puzzle took about 2.5=
hours (factoring in my "hey let's get distracted a lot" vari=
ables).



My final thoughts. Very fun. It was a true joy to play a technical 3D p=
uzzle that technically couldn't exist in the 3D world.

ote>



--0015175884fea9af83047eabfbb1--




From: Melinda Green <melinda@superliminal.com>
Date: Wed, 03 Feb 2010 23:09:27 -0800
Subject: Re: [MC4D] Re: Introducing "MagicTile"



--------------050701080006010605010805
Content-Type: text/plain; charset=ISO-8859-1; format=flowed
Content-Transfer-Encoding: 7bit

I think that the reason you end up with two bottoms is due to the
surface topology and not the polygonal symmetries. My page on infinite
regular polyhedra
describes
the topologies of these sorts of animals. Scroll down there and look at
the second black & white diagram showing a genus 3 IRP. In an infinitely
tiled construction you simply connect lots of those shapes in a cubic
lattice, but the more interesting case is when you have just one copy
which connects back onto itself as indicated by the arrows. You can do
that within a repeating *finite* space with the topology of a 3-torus
which is the 3D equivalent of the finite repeating space you may
remember from the old "Asteroids" video game. The modern game "Portal"
uses 3D spaces in this way.

To visualize the reason for the two bottoms of the Klein's Quartic
puzzle is even simpler: Just connect up the open mouths of a single copy
of the figure with simple tubes that follow the arrows. The {7,3} or its
{3,7} duel can be drawn on that surface and the twisty puzzle version
can be solved there. You can smoothly bend the shape around all you like
without changing the topology. You can even have the handles pass
through each other without interacting. Just imagine unfolding it until
you have the equivalent of a hollow ball with three tubular "handles".
Now imagine holding it by one handle and letting the rest dangle
downward. If you start your solution at the top of that top handle and
solve by crawling your way ever outward from there, you can see how your
solution can spread around the body of the ball and then down the other
two handles until finishing at the bottom of each of them. You
essentially end at two different local minima.

Here is a cross-eyed stereo photo of part of the infinite {3,7}
. You can easily imagine
how each of the triangles can be carved up into the twisty puzzle
version. I think that I'll rebuild that puzzle sometime such that it is
colored to show the heptagonal patches of Klein's Quartic. I probably
won't get to that right away but I'll post photos when I've done that.
Unfortunately I don't have 24 differently colored sets of triangles, but
hopefully I will find a nicely symmetric 4-coloring. If so I expect that
it will be rather handsome.

This stuff was a huge interest of mine for a long time and it's fun to
think about it again. I never would have dreamed that Rubik's cubes
would connect back to this stuff in any way, but there it is. Thanks for
plugging these two interests of mine together Roice. It gives me a
really nice feeling of completeness.

-Melinda

Roice Nelson wrote:
>
>
> I found the "two bottoms" observation extremely interesting :D I
> tried to figure out the why of this last night by rereading John
> Baez's article on Klein's
> Quartic, but didn't have much luck finding the insight I was looking
> for (though the two cells in the last layer is mentioned there, and
> the article is full of other neat information). It felt like the
> behavior should be related to the topology and the fact that a
> 3-holed-torus (genus 3, which is the topology of the puzzle) is not
> simply connected . But
> the 12-colored octagonal puzzle is also genus 3, and it doesn't behave
> the same.
>
> I found some more info this evening, and it turns out there is an
> entire book on Klein's Quartic! Amazon has it
> ,
> but you can download a free version online
> (however, the
> pictures seem to be missing). In the first section by Thurston, he
> notes "The infinite hyperbolic honeycomb is divided into 3 kinds of
> groups of 8 cells each, where each group is composed of a heptagon
> together with its 7 neighbors.". Together, these groups account for
> the 24 cells, and after labeling the 3 groups red/green/white, he writes:
>
>
> It is interesting to watch what happens when you rotate the
> pattern by a 1/7 revolution about the central tile: red groups go
> to red groups, green groups go to green groups and white groups go
> to white groups. The person in the center of a green group rotates
> by 2/7 revolution, and the person in the center of a red group
> rotates by 4/7 revolution. The interpretation on the surface is
> that the 24 cells are grouped into 8 affinity groups of 3 each.
> The symmetries of the surface always take affinity groups to
> affinity groups. This is analogous to the dodecahedron, whose
> twelve pentagonal faces are divided into 6 affinity groups of 2
> each, consisting of pairs of opposite faces.
>
>
> So I think it has more to do with the symmetries of the object than
> the topology (though perhaps there is some interrelation). I think
> what Nelson found was one of these 8 affinity groups. Btw, by editing
> colors, you should be able to use the program to more easily see the
> reg/green/white groups described above - I'll have to try this.
>
> Also, I did note to myself last night that it is possible to solve the
> {7,3} cells in an order such that you'd be left with one cell at the
> end instead of two, but you wouldn't be working "layer-by-layer" in
> that case.
>
> Very cool discovery of this unusual behavior Nelson!
>
> Roice
>
>
> On 2/1/10, *spel_werdz_rite* wrote:
>
> In a follow up with Roice, I'd like to share some more interesting
> details with the Klein's Quartic puzzle.
>
> The strategy to solving it was very much similar to how one would
> solve a Megaminx (my method at least). Edges, then sides, then
> edges, working all the way down to the bottom of the puzzle. Doing
> this method lead to me a very interesting discovery that,
> surprisingly, not even Roice new about. It turns out that Klein's
> Quartic has two "bottoms." By which I mean if you follow this
> method of inserting pieces downward until you reach the bottom of
> the puzzle, you will end up at 2 different faces. At this
> location, solving became a bit of a new task, but still not much
> of a challenge. The first step was making sure the remaining 2C
> and 3C pieces were on their corresponding face and oriented
> correctly. After that, I borrowed many techniques I used for the
> Megaminx. However, due to some obvious differences, the end took a
> lot of guesswork. In the end, the puzzle took about 2.5 hours
> (factoring in my "hey let's get distracted a lot" variables).
>
> My final thoughts. Very fun. It was a true joy to play a technical
> 3D puzzle that technically couldn't exist in the 3D world.
>
>
>
>
>

--------------050701080006010605010805
Content-Type: text/html; charset=ISO-8859-1
Content-Transfer-Encoding: 7bit







I think that the reason you end up with two bottoms is due to the
surface topology and not the polygonal symmetries. My page on href="http://www.superliminal.com/geometry/infinite/infinite.htm">infinite
regular polyhedra describes the topologies of these sorts of
animals. Scroll down there and look at the second black & white
diagram showing a genus 3 IRP. In an infinitely tiled construction you
simply connect lots of those shapes in a cubic lattice, but the more
interesting case is when you have just one copy which connects back
onto itself as indicated by the arrows. You can do that within a
repeating *finite* space with the topology of a 3-torus which is the 3D
equivalent of the finite repeating space you may remember from the old
"Asteroids" video game. The modern game "Portal" uses 3D spaces in this
way.



To visualize the reason for the two bottoms of the Klein's Quartic
puzzle is even simpler: Just connect up the open mouths of a single
copy of the figure with simple tubes that follow the arrows. The {7,3}
or its {3,7} duel can be drawn on that surface and the twisty puzzle
version can be solved there. You can smoothly bend the shape around all
you like without changing the topology. You can even have the handles
pass through each other without interacting. Just imagine unfolding it
until you have the equivalent of a hollow ball with three tubular
"handles". Now imagine holding it by one handle and letting the rest
dangle downward. If you start your solution at the top of that top
handle and solve by crawling your way ever outward from there, you can
see how your solution can spread around the body of the ball and then
down the other two handles until finishing at the bottom of each of
them. You essentially end at two different local minima.



Here is a cross-eyed stereo photo of href="http://www.superliminal.com/geometry/3_7st.jpg">part of the
infinite {3,7}. You can easily imagine how each of the triangles
can be carved up into the twisty puzzle version. I think that I'll
rebuild that puzzle sometime such that it is colored to show the
heptagonal patches of Klein's Quartic. I probably won't get to that
right away but I'll post photos when I've done that. Unfortunately I
don't have 24 differently colored sets of triangles, but hopefully I
will find a nicely symmetric 4-coloring. If so I expect that it will be
rather handsome.



This stuff was a huge interest of mine for a long time and it's fun to
think about it again. I never would have dreamed that Rubik's cubes
would connect back to this stuff in any way, but there it is. Thanks
for plugging these two interests of mine together Roice. It gives me a
really nice feeling of completeness.



-Melinda



Roice Nelson wrote:
cite="mid:b5979e761002022203p56d70b96g5947af860aabb507@mail.gmail.com"
type="cite">


I found the "two bottoms" observation extremely interesting :D 
I tried to figure out the why of this last night by rereading moz-do-not-send="true" href="http://math.ucr.edu/home/baez/klein.html">John
Baez's article on Klein's Quartic, but didn't have much luck
finding the insight I was looking for (though the two cells in the last
layer is mentioned there, and the article is full of other neat
information).  It felt like the behavior should be related to the
topology and the fact that a 3-holed-torus (genus 3, which is the
topology of the puzzle) is not href="http://en.wikipedia.org/wiki/Simply_connected">simply connected
But the 12-colored octagonal puzzle is also genus 3, and it doesn't
behave the same.

 

I found some more info this evening, and it turns out there is
an entire book on Klein's Quartic!  href="http://www.amazon.com/gp/product/0521004195?ie=UTF8&tag=gravit-20&linkCode=as2&camp=1789&creative=390957&creativeASIN=0521004195">Amazon
has it
, but you can download a href="http://www.msri.org/publications/books/Book35/">free version
online
(however, the pictures seem to be missing).  In the first
section by Thurston, he notes "The infinite hyperbolic honeycomb is
divided into 3 kinds of groups of 8 cells each, where each group is
composed of a heptagon together with its 7 neighbors.".  Together,
these groups account for the 24 cells, and after labeling the 3 groups
red/green/white, he writes:


 

style="border-left: 1px solid rgb(204, 204, 204); margin: 0px 0px 0px 0.8ex; padding-left: 1ex;">It
is interesting to watch what happens when you rotate the pattern by a
1/7 revolution about the central tile: red groups go to red groups,
green groups go to green groups and white groups go to white groups.
The person in the center of a green group rotates by 2/7 revolution,
and the person in the center of a red group rotates by 4/7 revolution.
The interpretation on the surface is that the 24 cells are grouped into
8 affinity groups of 3 each. The symmetries of the surface always take
affinity groups to affinity groups. This is analogous to the
dodecahedron, whose twelve pentagonal faces are divided into 6 affinity
groups of 2 each, consisting of pairs of opposite faces.



So I think it has more to do with the symmetries of the object than the
topology (though perhaps there is some interrelation).  I think what
Nelson found was one of these 8 affinity groups.  Btw, by editing
colors, you should be able to use the program to more easily see the
reg/green/white groups described above - I'll have to try this.

 

Also, I did note to myself last night that it is possible to solve the
{7,3} cells in an order such that you'd be left with one cell at the
end instead of two, but you wouldn't be working "layer-by-layer" in
that case.

 

Very cool discovery of this unusual behavior Nelson!

 

Roice



 

On 2/1/10, spel_werdz_rite
wrote:

style="border-left: 1px solid rgb(204, 204, 204); margin: 0px 0px 0px 0.8ex; padding-left: 1ex;">In
a follow up with Roice, I'd like to share some more interesting details
with the Klein's Quartic puzzle.



The strategy to solving it was very much similar to how one would solve
a Megaminx (my method at least). Edges, then sides, then edges, working
all the way down to the bottom of the puzzle. Doing this method lead to
me a very interesting discovery that, surprisingly, not even Roice new
about. It turns out that Klein's Quartic has two "bottoms." By which I
mean if you follow this method of inserting pieces downward until you
reach the bottom of the puzzle, you will end up at 2 different faces.
At this location, solving became a bit of a new task, but still not
much of a challenge. The first step was making sure the remaining 2C
and 3C pieces were on their corresponding face and oriented correctly.
After that, I borrowed many techniques I used for the Megaminx.
However, due to some obvious differences, the end took a lot of
guesswork. In the end, the puzzle took about 2.5 hours (factoring in my
"hey let's get distracted a lot" variables).



My final thoughts. Very fun. It was a true joy to play a technical 3D
puzzle that technically couldn't exist in the 3D world.













--------------050701080006010605010805--




From: <alexander.sage@jacks.sdstate.edu>
Date: Thu, 4 Feb 2010 01:22:56 -0600
Subject: RE: [MC4D] Re: Introducing "MagicTile"



=20=20




I found the "two bottoms" observation extremely interesting :D I tried to =
figure out the why of this last night by rereading John Baez's article on K=
lein's Quartic, but didn't have much luck finding the insight I was looking=
for (though the two cells in the last layer is mentioned there, and the ar=
ticle is full of other neat information). It felt like the behavior should=
be related to the topology and the fact that a 3-holed-torus (genus 3, whi=
ch is the topology of the puzzle) is not simply connected. But the 12-colo=
red octagonal puzzle is also genus 3, and it doesn't behave the same.
=20
I found some more info this evening, and it turns out there is an entire bo=
ok on Klein's Quartic! Amazon has it, but you can download a free version =
online (however, the pictures seem to be missing). In the first section by=
Thurston, he notes "The infinite hyperbolic honeycomb is divided into 3 ki=
nds of groups of 8 cells each, where each group is composed of a heptagon t=
ogether with its 7 neighbors.". Together, these groups account for the 24 =
cells, and after labeling the 3 groups red/green/white, he writes:
=20

It is interesting to watch what happens when you rotate the pattern by a 1/=
7 revolution about the central tile: red groups go to red groups, green gro=
ups go to green groups and white groups go to white groups. The person in t=
he center of a green group rotates by 2/7 revolution, and the person in the=
center of a red group rotates by 4/7 revolution. The interpretation on the=
surface is that the 24 cells are grouped into 8 affinity groups of 3 each.=
The symmetries of the surface always take affinity groups to affinity grou=
ps. This is analogous to the dodecahedron, whose twelve pentagonal faces ar=
e divided into 6 affinity groups of 2 each, consisting of pairs of opposite=
faces.
So I think it has more to do with the symmetries of the object than the top=
ology (though perhaps there is some interrelation). I think what Nelson fo=
und was one of these 8 affinity groups. Btw, by editing colors, you should=
be able to use the program to more easily see the reg/green/white groups d=
escribed above - I'll have to try this.
=20
Also, I did note to myself last night that it is possible to solve the {7,3=
} cells in an order such that you'd be left with one cell at the end instea=
d of two, but you wouldn't be working "layer-by-layer" in that case.
=20
Very cool discovery of this unusual behavior Nelson!
=20
Roice

=20
On 2/1/10, spel_werdz_rite wrote:=20
In a follow up with Roice, I'd like to share some more interesting details =
with the Klein's Quartic puzzle.

The strategy to solving it was very much similar to how one would solve a M=
egaminx (my method at least). Edges, then sides, then edges, working all th=
e way down to the bottom of the puzzle. Doing this method lead to me a very=
interesting discovery that, surprisingly, not even Roice new about. It tur=
ns out that Klein's Quartic has two "bottoms." By which I mean if you follo=
w this method of inserting pieces downward until you reach the bottom of th=
e puzzle, you will end up at 2 different faces. At this location, solving b=
ecame a bit of a new task, but still not much of a challenge. The first ste=
p was making sure the remaining 2C and 3C pieces were on their correspondin=
g face and oriented correctly. After that, I borrowed many techniques I use=
d for the Megaminx. However, due to some obvious differences, the end took =
a lot of guesswork. In the end, the puzzle took about 2.5 hours (factoring =
in my "hey let's get distracted a lot" variables).

My final thoughts. Very fun. It was a true joy to play a technical 3D puzzl=
e that technically couldn't exist in the 3D world.






=20=09=09=20=09=20=20=20=09=09=20=20
--_5f7217d1-7eb3-44c4-bfa2-347489e7eed9_
Content-Type: text/html; charset="Windows-1252"
Content-Transfer-Encoding: quoted-printable






I played with the hyperbolic puzzles more, and realized that this is c=
ertainly a well written program.  I love this idea of playing wit=
h twisty puzzles that lack the spatial ability to exist, including&nbs=
p;both these hyperbolic polyhedra, and the polychora.  This is a =
really ingenious idea, one that I would probably never thought of in my lif=
e, if not for you.

 

I'm glad to hear that {4,4} is coming along nicely.

 

I first noticed the double bottoms on the hex tiling, when I was using=
9 colors.  needless to say, I was rather suprised.  I think=
that the only regular euclidian polyhedron that has anythin=
g like double-bottoms is the tetrahedron, but that is obviously an&nbs=
p;exception. 

 

Having used the program more, I have a few more words of advice.&=
nbsp; A save function would be very nice.  Right now, even the profess=
or's cube is really hard to do in one sitting.  You said tha=
t it would be difficult to change the hyperbolic viewpoint, but we rea=
lly don't need to see animation for it.  It might be hard, but ev=
en if the change was instant, that feature would be very helpful on some of=
the bigger puzzles.  One of the other features I would really like to=
see is a simple rotation, x-y plane.  Sometimes it's nice not to=
have to turn your head sideways to try to figure out if an algoritm i=
s going to have your desired effect.  (turning one's head is faster th=
an figuring it out spatially.)

 

Two more suggestions.  (If you haven't already,) Put a paypal box=
up on your website.  Your programs are very deserving of so=
me compensation.  Second, get more people interested in this.&nbs=
p; I know that it's hard, having tried myself (with no good results), but i=
f you're smart enough to envision these programs, you probably can think&nb=
sp;of something.  ;)
 



To: 4D_Cubing@yahoogroups.com
From: roice3@gmail.com
Date: Wed, 3 Feb=
2010 00:03:53 -0600
Subject: Re: [MC4D] Re: Introducing "MagicTile"
=

 =20




I found the "two bottoms" observation extremely interesting :>D  I tried to figure out the why of this last night by rereading ref=3D"http://math.ucr.edu/home/baez/klein.html">John Baez's article on=
Klein's Quartic, but didn't have much luck finding the insight I was =
looking for (though the two cells in the last layer is mentioned there=
, and the article is full of other neat information).  It fe=
lt like the behavior should be related to the topology and the fact that a =
3-holed-torus (genus 3, which is the topology of the puzzle) is not =3D"http://en.wikipedia.org/wiki/Simply_connected">simply connected.&nb=
sp; But the 12-colored octagonal puzzle is also genus 3, and it doesn't beh=
ave the same.

 

I found some more info this evening, and it turns out there is an enti=
re book on Klein's Quartic!  ct/0521004195?ie=3DUTF8&tag=3Dgravit-20&linkCode=3Das2&camp=3D1=
789&creative=3D390957&creativeASIN=3D0521004195">Amazon has it
,=
but you can download a ook35/">free version online (however, the pictures seem to be missing).=
  In the first section by Thurston, he notes "The infinite hyperbolic =
honeycomb is divided into 3 kinds of groups of 8 cells each, where each gro=
up is composed of a heptagon together with its 7 neighbors.".  Togethe=
r, these groups account for the 24 cells, and after labeling the =
3 groups red/green/white, he writes:

 

It=
is interesting to watch what happens when you rotate the pattern by a 1/7 =
revolution about the central tile: red groups go to red groups, green group=
s go to green groups and white groups go to white groups. The person in the=
center of a green group rotates by 2/7 revolution, and the person in the c=
enter of a red group rotates by 4/7 revolution. The interpretation on the s=
urface is that the 24 cells are grouped into 8 affinity groups of 3 each. T=
he symmetries of the surface always take affinity groups to affinity groups=
. This is analogous to the dodecahedron, whose twelve pentagonal faces are =
divided into 6 affinity groups of 2 each, consisting of pairs of opposite f=
aces.

So I think it has more to do with the symmetries of t=
he object than the topology (though perhaps there is some interrelation)R>.  I think what Nelson found was one of these 8 affinity groups. &nb=
sp;Btw, by editing colors, you should be able to use the program to more ea=
sily see the reg/green/white groups described above - I'll have to try this=
.
 
Also, I did note to myself last night that it is possible to=
solve the {7,3} cells in an order such that you'd be left with one cell at=
the end instead of two, but you wouldn't be working "layer-by-layer" in th=
at case.

 

Very cool discovery of this unusual behavior Nelson!

 

Roice

 

On 2/1/10, >spel_werdz_rite wrote:=20
In=
a follow up with Roice, I'd like to share some more interesting details wi=
th the Klein's Quartic puzzle.

The strategy to solving it was very m=
uch similar to how one would solve a Megaminx (my method at least). Edges, =
then sides, then edges, working all the way down to the bottom of the puzzl=
e. Doing this method lead to me a very interesting discovery that, surprisi=
ngly, not even Roice new about. It turns out that Klein's Quartic has two "=
bottoms." By which I mean if you follow this method of inserting pieces dow=
nward until you reach the bottom of the puzzle, you will end up at 2 differ=
ent faces. At this location, solving became a bit of a new task, but still =
not much of a challenge. The first step was making sure the remaining 2C an=
d 3C pieces were on their corresponding face and oriented correctly. After =
that, I borrowed many techniques I used for the Megaminx. However, due to s=
ome obvious differences, the end took a lot of guesswork. In the end, the p=
uzzle took about 2.5 hours (factoring in my "hey let's get distracted a lot=
" variables).

My final thoughts. Very fun. It was a true joy to play=
a technical 3D puzzle that technically couldn't exist in the 3D world.
=








--_5f7217d1-7eb3-44c4-bfa2-347489e7eed9_--




From: Chris Locke <project.eutopia@gmail.com>
Date: Thu, 4 Feb 2010 16:51:04 +0900
Subject: Re: [MC4D] Re: Introducing "MagicTile"



--0016e6471a62d04645047ec198a7
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: quoted-printable

Congratulations Roice on another jaw-dropping program! The community of
beyond physically realizable Rubik cubers I'm sure is both shocked and
pleased by the recent flurry of new puzzles. First we get tons new cool 4D
shapes to play with, and now hyperbolic puzzles?? What's next, 4D
equivalents of these hyperbolic puzzles? :D

First thing I would note is that you should move the "donate" button to a
more visible location. As you can see by Alexander's post it's not easy to
find. I didn't even know you had it on the page until recently when I
actually scrolled all the way to the bottom!

As for the puzzles themselves. I find that from my limited playing around
with the hexagonal and heptagonal tilings, that low color counts completely
baffle me, but as the number of colors increases, the amount of coupling
between the twists decreases, which makes it much easier to work with. The
3 and 4 color hexagonal tilings are just weird to try and solve :P. But in
the cases with many colors, it's feels like it's just a matter of patient t=
o
reduce it down to a last layer. From there, Megamix LL
algorithms/commutators can be generalized to finish them up.

As for the double bottoms thing, Melinda gave a pretty good explanation as
to how it's a result of the topology. In the case of the 3 holed torus
(Klein's quartic) it's pretty easy to see how spreading out from one
position can leave with with a case of two disconnected unsolved faces.
Furthermore, I don't think it's possible to end up with 3 disconnected face=
s
in this case, and also if you understand the topology from the tiling, it
should also always be possible to solve it such that you end up with just
one unsolved face. But that in itself is a challenge too :D.

While we're on the topic of suggestions: can you make it so that when you
change the performance setting, the puzzle doesn't reset? I was playing
with the Klein's quartic and halfway through my solve I tried that setting
and it resetted it :(. So I decided to solve the 9 color hexagonal tiling
instead as I didn't want to work on the same puzzle twice ^^. Less colors
is too hard, and more colors takes too long.

Thanks again for this sweet program!

Chris

2010/2/4

>
>
> I played with the hyperbolic puzzles more, and realized that this
> is certainly a well written program. I love this idea of playing with
> twisty puzzles that lack the spatial ability to exist, including both the=
se
> hyperbolic polyhedra, and the polychora. This is a really ingenious idea=
,
> one that I would probably never thought of in my life, if not for you.
>
> I'm glad to hear that {4,4} is coming along nicely.
>
> I first noticed the double bottoms on the hex tiling, when I was using 9
> colors. needless to say, I was rather suprised. I think that the only
> regular euclidian polyhedron that has anything like double-bottoms is the
> tetrahedron, but that is obviously an exception.
>
> Having used the program more, I have a few more words of advice. A save
> function would be very nice. Right now, even the professor's cube is rea=
lly
> hard to do in one sitting. You said that it would be difficult to change
> the hyperbolic viewpoint, but we really don't need to see animation for i=
t.
> It might be hard, but even if the change was instant, that feature would =
be
> very helpful on some of the bigger puzzles. One of the other features I
> would really like to see is a simple rotation, x-y plane. Sometimes it's
> nice not to have to turn your head sideways to try to figure out if an
> algoritm is going to have your desired effect. (turning one's head is
> faster than figuring it out spatially.)
>
> Two more suggestions. (If you haven't already,) Put a paypal box up on
> your website. Your programs are very deserving of some compensation.
> Second, get more people interested in this. I know that it's hard, havin=
g
> tried myself (with no good results), but if you're smart enough to envisi=
on
> these programs, you probably can think of something. ;)
>
> ------------------------------
> To: 4D_Cubing@yahoogroups.com
> From: roice3@gmail.com
> Date: Wed, 3 Feb 2010 00:03:53 -0600
> Subject: Re: [MC4D] Re: Introducing "MagicTile"
>
>
>
> I found the "two bottoms" observation extremely interesting :D I tried
> to figure out the why of this last night by rereading John Baez's article=
on Klein's Quartic, but didn't ha=
ve much luck finding the insight I was
> looking for (though the two cells in the last layer is mentioned there, a=
nd
> the article is full of other neat information). It felt like the behavio=
r
> should be related to the topology and the fact that a 3-holed-torus (genu=
s
> 3, which is the topology of the puzzle) is not simply connected.wikipedia.org/wiki/Simply_connected>.
> But the 12-colored octagonal puzzle is also genus 3, and it doesn't behav=
e
> the same.
>
> I found some more info this evening, and it turns out there is an entire
> book on Klein's Quartic! Amazon has it0521004195?ie=3DUTF8&tag=3Dgravit-20&linkCode=3Das2&camp=3D1789&creative=3D=
390957&creativeASIN=3D0521004195>,
> but you can download a free version onlinens/books/Book35/>(however, the pictures seem to be missing). In the first =
section by
> Thurston, he notes "The infinite hyperbolic honeycomb is divided into 3
> kinds of groups of 8 cells each, where each group is composed of a heptag=
on
> together with its 7 neighbors.". Together, these groups account for the =
24
> cells, and after labeling the 3 groups red/green/white, he writes:
>
>
> It is interesting to watch what happens when you rotate the pattern by a
> 1/7 revolution about the central tile: red groups go to red groups, green
> groups go to green groups and white groups go to white groups. The person=
in
> the center of a green group rotates by 2/7 revolution, and the person in =
the
> center of a red group rotates by 4/7 revolution. The interpretation on th=
e
> surface is that the 24 cells are grouped into 8 affinity groups of 3 each=
.
> The symmetries of the surface always take affinity groups to affinity
> groups. This is analogous to the dodecahedron, whose twelve pentagonal fa=
ces
> are divided into 6 affinity groups of 2 each, consisting of pairs of
> opposite faces.
>
>
> So I think it has more to do with the symmetries of the object than the
> topology (though perhaps there is some interrelation). I think what Nels=
on
> found was one of these 8 affinity groups. Btw, by editing colors, you
> should be able to use the program to more easily see the reg/green/white
> groups described above - I'll have to try this.
>
> Also, I did note to myself last night that it is possible to solve the
> {7,3} cells in an order such that you'd be left with one cell at the end
> instead of two, but you wouldn't be working "layer-by-layer" in that case=
.
>
> Very cool discovery of this unusual behavior Nelson!
>
> Roice
>
>
> On 2/1/10, *spel_werdz_rite* wrote:
>
> In a follow up with Roice, I'd like to share some more interesting detail=
s
> with the Klein's Quartic puzzle.
>
> The strategy to solving it was very much similar to how one would solve a
> Megaminx (my method at least). Edges, then sides, then edges, working all
> the way down to the bottom of the puzzle. Doing this method lead to me a
> very interesting discovery that, surprisingly, not even Roice new about. =
It
> turns out that Klein's Quartic has two "bottoms." By which I mean if you
> follow this method of inserting pieces downward until you reach the botto=
m
> of the puzzle, you will end up at 2 different faces. At this location,
> solving became a bit of a new task, but still not much of a challenge. Th=
e
> first step was making sure the remaining 2C and 3C pieces were on their
> corresponding face and oriented correctly. After that, I borrowed many
> techniques I used for the Megaminx. However, due to some obvious
> differences, the end took a lot of guesswork. In the end, the puzzle took
> about 2.5 hours (factoring in my "hey let's get distracted a lot"
> variables).
>
> My final thoughts. Very fun. It was a true joy to play a technical 3D
> puzzle that technically couldn't exist in the 3D world.
>
>
>
>=20=20=20
>

--0016e6471a62d04645047ec198a7
Content-Type: text/html; charset=UTF-8
Content-Transfer-Encoding: quoted-printable

Congratulations Roice on another jaw-dropping program!=C2=A0 The community =
of beyond physically realizable Rubik cubers I'm sure is both shocked a=
nd pleased by the recent flurry of new puzzles.=C2=A0 First we get tons new=
cool 4D shapes to play with, and now hyperbolic puzzles??=C2=A0 What's=
next, 4D equivalents of these hyperbolic puzzles? :D


First thing I would note is that you should move the "donate"=
button to a more visible location.=C2=A0 As you can see by Alexander's=
post it's not easy to find.=C2=A0 I didn't even know you had it on=
the page until recently when I actually scrolled all the way to the bottom=
!


As for the puzzles themselves.=C2=A0 I find that from my limited playin=
g around with the hexagonal and heptagonal tilings, that low color counts c=
ompletely baffle me, but as the number of colors increases, the amount of c=
oupling between the twists decreases, which makes it much easier to work wi=
th.=C2=A0 The 3 and 4 color hexagonal tilings are just weird to try and sol=
ve :P.=C2=A0 But in the cases with many colors, it's feels like it'=
s just a matter of patient to reduce it down to a last layer.=C2=A0 From th=
ere, Megamix LL algorithms/commutators can be generalized to finish them up=
.


As for the double bottoms thing, Melinda gave a pretty good explanation=
as to how it's a result of the topology.=C2=A0 In the case of the 3 ho=
led torus (Klein's quartic) it's pretty easy to see how spreading o=
ut from one position can leave with with a case of two disconnected unsolve=
d faces.=C2=A0 Furthermore, I don't think it's possible to end up w=
ith 3 disconnected faces in this case, and also if you understand the topol=
ogy from the tiling, it should also always be possible to solve it such tha=
t you end up with just one unsolved face.=C2=A0 But that in itself is a cha=
llenge too :D.


While we're on the topic of suggestions: can you make it so that wh=
en you change the performance setting, the puzzle doesn't reset?=C2=A0 =
I was playing with the Klein's quartic and halfway through my solve I t=
ried that setting and it resetted it :(.=C2=A0 So I decided to solve the 9 =
color hexagonal tiling instead as I didn't want to work on the same puz=
zle twice ^^.=C2=A0 Less colors is too hard, and more colors takes too long=
.


Thanks again for this sweet program!

Chris

gmail_quote">2010/2/4 <ge@jacks.sdstate.edu">alexander.sage@jacks.sdstate.edu>
lockquote class=3D"gmail_quote" style=3D"border-left: 1px solid rgb(204, 20=
4, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">















=C2=A0







=20=20=20=20=20=20
=20=20=20=20=20=20




I played with the hyperbolic puzzles more, and realized that this is=C2=A0c=
ertainly a well=C2=A0written program.=C2=A0 I love this idea of playing wit=
h twisty puzzles that lack the spatial ability to exist,=C2=A0including=C2=
=A0both these hyperbolic polyhedra, and=C2=A0the polychora.=C2=A0 This is a=
really ingenious idea, one that I would probably never thought of in my li=
fe, if not for you.


=C2=A0

I'm glad to hear that {4,4} is coming along nicely.

=C2=A0

I first=C2=A0noticed the double bottoms on the hex tiling, when I was using=
9 colors.=C2=A0 needless to say, I was=C2=A0rather suprised.=C2=A0 I think=
that=C2=A0the only regular=C2=A0euclidian polyhedron that=C2=A0has anythin=
g like=C2=A0double-bottoms is the tetrahedron, but that is obviously an=C2=
=A0exception.=C2=A0


=C2=A0

Having used the program more,=C2=A0I have a few more=C2=A0words of advice.=
=C2=A0 A save function would be very nice.=C2=A0 Right now, even the profes=
sor's cube is really hard to=C2=A0do in one=C2=A0sitting.=C2=A0 You sai=
d that it would be difficult to change the=C2=A0hyperbolic viewpoint, but w=
e really don't need to see animation for it.=C2=A0 It might be=C2=A0har=
d, but even if the change was instant, that feature would be very helpful o=
n some of the bigger puzzles.=C2=A0 One of the other features I would reall=
y like to see is a simple rotation, x-y plane.=C2=A0 Sometimes it's nic=
e=C2=A0not to have to turn your head sideways=C2=A0to try to figure out if =
an algoritm is going to have your desired effect.=C2=A0 (turning one's =
head is faster than figuring it out spatially.)


=C2=A0

Two more suggestions.=C2=A0=C2=A0(If you haven't already,) Put a paypal=
box up on your website.=C2=A0=C2=A0Your programs are=C2=A0very deserving o=
f some compensation.=C2=A0 Second,=C2=A0get more people interested in this.=
=C2=A0 I know that it's hard, having tried myself (with no good results=
), but if you're smart enough to envision these programs, you probably =
can think=C2=A0of something.=C2=A0 ;)

=C2=A0


From: rget=3D"_blank">roice3@gmail.com
Date: Wed, 3 Feb 2010 00:03:53 -060=
0

Subject: Re: [MC4D] Re: Introducing "MagicTile"
iv class=3D"h5">

=C2=A0=20




I found the "two bottoms" observation extremely interesting=
=C2=A0:D=C2=A0 I tried to figure out the why of this last night by rereadin=
g Jo=
hn Baez's article
on Klein's Quartic, but didn't have much =
luck finding the=C2=A0insight I was looking for=C2=A0(though the two cells =
in the last layer is mentioned there, and the article is full of other neat=
information).=C2=A0 It=C2=A0felt like the behavior should be related to th=
e topology and the fact that a 3-holed-torus (genus 3, which is the topolog=
y of the puzzle) is not ected" target=3D"_blank">simply connected.=C2=A0 But the 12-colored oct=
agonal puzzle is also genus 3, and it doesn't behave the same.


=C2=A0

I found some more info this evening, and it turns out there is an enti=
re book on Klein's Quartic!=C2=A0 roduct/0521004195?ie=3DUTF8&tag=3Dgravit-20&linkCode=3Das2&camp=
=3D1789&creative=3D390957&creativeASIN=3D0521004195" target=3D"_bla=
nk">Amazon has it
, but you can download a g/publications/books/Book35/" target=3D"_blank">free version online (ho=
wever, the pictures seem to be missing).=C2=A0 In the first section by Thur=
ston, he notes "The infinite hyperbolic honeycomb is divided into 3 ki=
nds of groups of 8 cells each, where each group is composed of a heptagon t=
ogether with its 7 neighbors.".=C2=A0 Together, these groups=C2=A0acco=
unt for the 24 cells, and=C2=A0after labeling the 3 groups red/green/white,=
he writes:


=C2=A0

It is inte=
resting to watch what happens when you rotate the pattern by a 1/7 revoluti=
on about the central tile: red groups go to red groups, green groups go to =
green groups and white groups go to white groups. The person in the center =
of a green group rotates by 2/7 revolution, and the person in the center of=
a red group rotates by 4/7 revolution. The interpretation on the surface i=
s that the 24 cells are grouped into 8 affinity groups of 3 each. The symme=
tries of the surface always take affinity groups to affinity groups. This i=
s analogous to the dodecahedron, whose twelve pentagonal faces are divided =
into 6 affinity groups of 2 each, consisting of pairs of opposite faces.lockquote>

So I think it has more to do with the symmetries of the object than the=
topology (though perhaps there is some interrelation). =C2=A0I think what =
Nelson found was one of these 8 affinity groups. =C2=A0Btw, by editing colo=
rs, you should be able to use the program to more easily see the reg/green/=
white groups described above - I'll have to try this.

=C2=A0
Also, I did note to myself last night that it is possible to solv=
e the {7,3} cells in an order such that you'd be left with one cell at =
the end instead of two, but you wouldn't be working "layer-by-laye=
r" in that case.


=C2=A0

Very cool discovery of this unusual behavior Nelson!

=C2=A0

Roice

=C2=A0

On 2/1/10, spel_werdz_rite wrote:=20
In a follo=
w up with Roice, I'd like to share some more interesting details with t=
he Klein's Quartic puzzle.

The strategy to solving it was very m=
uch similar to how one would solve a Megaminx (my method at least). Edges, =
then sides, then edges, working all the way down to the bottom of the puzzl=
e. Doing this method lead to me a very interesting discovery that, surprisi=
ngly, not even Roice new about. It turns out that Klein's Quartic has t=
wo "bottoms." By which I mean if you follow this method of insert=
ing pieces downward until you reach the bottom of the puzzle, you will end =
up at 2 different faces. At this location, solving became a bit of a new ta=
sk, but still not much of a challenge. The first step was making sure the r=
emaining 2C and 3C pieces were on their corresponding face and oriented cor=
rectly. After that, I borrowed many techniques I used for the Megaminx. How=
ever, due to some obvious differences, the end took a lot of guesswork. In =
the end, the puzzle took about 2.5 hours (factoring in my "hey let'=
;s get distracted a lot" variables).


My final thoughts. Very fun. It was a true joy to play a technical 3D p=
uzzle that technically couldn't exist in the 3D world.

ote>





=20=09=09=20=09=20=20=20=09=09=20=20



=20=20=20=20=20

=20=20=20=20







=20=20









--0016e6471a62d04645047ec198a7--




From: Melinda Green <melinda@superliminal.com>
Date: Thu, 04 Feb 2010 00:33:52 -0800
Subject: Re: [MC4D] Re: Introducing "MagicTile"




Chris Locke wrote:
>
>
> Congratulations Roice on another jaw-dropping program! The community
> of beyond physically realizable Rubik cubers I'm sure is both shocked
> and pleased by the recent flurry of new puzzles. First we get tons
> new cool 4D shapes to play with, and now hyperbolic puzzles?? What's
> next, 4D equivalents of these hyperbolic puzzles? :D
>
> First thing I would note is that you should move the "donate" button
> to a more visible location. As you can see by Alexander's post it's
> not easy to find. I didn't even know you had it on the page until
> recently when I actually scrolled all the way to the bottom!
>
> As for the puzzles themselves. I find that from my limited playing
> around with the hexagonal and heptagonal tilings, that low color
> counts completely baffle me, but as the number of colors increases,
> the amount of coupling between the twists decreases, which makes it
> much easier to work with. The 3 and 4 color hexagonal tilings are
> just weird to try and solve :P. But in the cases with many colors,
> it's feels like it's just a matter of patient to reduce it down to a
> last layer. From there, Megamix LL algorithms/commutators can be
> generalized to finish them up.

I think that this phenomenon is the same one that makes the megaminx
easier to solve than the cube because there's more "room" on the surface
of the puzzles with more faces in which to squirrel away some parts
while working on others. The smaller, puzzles are "tighter" and cause
every action to affect just about everything else. I have a funny
feeling that there is some sort of natural difficulty metric in which
the original 3^3 Rubik's cube will turn out to be the hardest of all the
similar puzzles in all dimensions, but I can't quite see how to define
that metric. I only feel it.

> As for the double bottoms thing, Melinda gave a pretty good
> explanation as to how it's a result of the topology. In the case of
> the 3 holed torus (Klein's quartic) it's pretty easy to see how
> spreading out from one position can leave with with a case of two
> disconnected unsolved faces. Furthermore, I don't think it's possible
> to end up with 3 disconnected faces in this case, and also if you
> understand the topology from the tiling, it should also always be
> possible to solve it such that you end up with just one unsolved
> face. But that in itself is a challenge too :D.

You'd need a genus 4 or higher surface to end up with 3 or more
disconnected unsolved faces.

I'm not sure if you are right that one should be able to follow a
modified layer-by-layer method on any of these puzzles and always end up
with a single unsolved face. The key would be to keep track of the
"outer edge" of your solved patch as you grow it. Never let one part of
that edge connect with any other part. In other words, you make sure to
keep that edge a simple closed curve. Any time a part of it wants to
connect with another part, just leave that area and work on some other
part of the border that has room to spread. The question is whether the
initially growing edge will eventually shrink back down to a single face.

At first I thought yes, but as I started to write the above, I'm now not
so sure. This is reminding me a lot of the short Wikipedia article that
Roice cited on simply connected
spaces. Imagine trying
the above solving strategy with a {4,4} puzzle defined on the surface of
a torus. I think you'll end up with a seam that you can't get around. I
just can't quite see it in my head without some good pictures.

> [...] Less colors is too hard, and more colors takes too long.

Each color is a single face. Each time that you see a red face, it's
always the same face just seen from a different perspective. Kind of
like how gravitational lensing can let you see multiple images of the
same distant galaxy in a single photograph. I agree that tighter puzzles
take more brain work and less tedium but when they become extremely
tight, they seem to get easier again in a way as you can begin to get
your head around the whole thing. In addition to wanting to know which
puzzle is hardest for it's size, I also want to know is which puzzle out
of all the twisty puzzles different people think has the challenge that
is "just right" for them.

-Melinda




From: Roice Nelson <roice3@gmail.com>
Date: Thu, 4 Feb 2010 21:19:13 -0600
Subject: Re: [MC4D] Re: Introducing "MagicTile"



--00032555a3167108c2047ed1eaa8
Content-Type: text/plain; charset=ISO-8859-1

What an great flow of ideas. I love it!

About solving towards a single cell, I think the program can help answer
this by using the setting to only show the fundamental set of tiles. I'm
not perfectly confident, but my intuition says it is possible to avoid
multiple disjoint unsolved cells at the end. My reasoning is that even if
the topology of the object as a whole (fundamental and orbit regions glued
up) is not simply connected, the fundamental region alone is simply
connected (as long as you never leave the boundary of it, so that it behaves
topologically as a disk). Looking at only the fundamental set of Klein's
Quartic, it is easy to pick a solve order leaving only one cell at the end,
by using sort of a "solve wave" to push unsolved cells from one end to the
other. This would be my strategy to "keep the edge a simple closed curve",
as Melinda nicely described things.

If I require that I solve cells in a connected fashion, where each cell
solved must be adjacent to the prior one, it appears I lose the ability to
end on a single cell. As an aside, I wonder if it is possible to have
fundamental sets of tiles which are not simply connected, which could
invalidate my logic above.

Anyway, you've all convince me that it is the topology that is important
factor in all this (though it still seems the symmetries and topology are
intertwined). After taking another look at the 12-colored octagonal, also
genus 3, I see it actually does behave the same.

Regarding which puzzle(s) may be "just right", I've only done a few so far,
but last night I did go through a solve of the 9-colored torus puzzle and
can say I found it really enjoyable - perfect for a solve in one sitting for
sure. And the repeating units caught me a couple times, which was fun. It
may be inevitable that the 3x3x3 will always be the most perfect puzzle in
my mind though :)

One small comment related to difficulty... The state-space of the
3-coloreds is small enough that they will often solve themselves if I just
randomly click.

Yesterday, I learned about some additional coloring possibilities on the
hexagonal puzzles which I think will be good, balanced puzzles when it comes
to the tradeoffs being discussed, e.g. a 7-colored (This was pointed out to
me by someone from the twistypuzzles forums). Applying the tiling algorithm
which produces those to the hyperbolic puzzles will probably allow some
further balanced variants there as well. The 6-12 color range feels nice to
me, both in terms of coupling and tedium.

Btw, a while back I realized repeating units can be applied to spherical
puzzles too, which will end up throwing even a few more into the mix. A
Megaminx with a 6-colored fundamental region and opposite sides being orbits
would topologically act as the
non-orientable real
projective plane (I
verified by playing with color settings). Due to the non-orientability of
that surface, orbits would rotate in an opposite sense to the fundamental
cells! One could do a non-orientable hexagonal version as well,
topologically a Klein bottle . I
definitely want to support these variants at some point, which will only
involve minor code changes to deal with reversing the twisting.

Re: Alexander... I bet doing a ctrl+click view rotation without animation
wouldn't be too bad, and something like that could likely be the first pass.
Also, I've been doing the same head twisting and so am sharing the desire
for rotation control of the projected puzzles too, which will be trivial to
add. And load/save is at the top of the list. I hope to address the
majority of the requests I've received so far, though I expect it will be at
least a month before I'll be able to provide any updates.
Regarding publicizing, I have been sending out an email or two each evening
to try to get the word out. Mathpuzzle gave
MagicTile a really nice shout out this morning. I want to do a blog post
too (both about MC4D and this), though that will probably just get the word
out to my mom and siblings :)

Re: Chris... so sorry about the loss of work :( I will definitely switch
this resetting behavior. It caught me the other night too, at which point I
noted the poor behavior in my list. I was lucky enough to not be half way
into a large puzzle. Also, on the 4D front, I do think it'd be cool to have
stereographically projected versions of the puzzles in MC4D. I even opened an
issue for ita
while back :D

Have a good night all,
Roice

P.S. I did move the donate button up (thanks for the suggestion on that
too, Chris and Alexander)


On 2/4/10, Melinda Green wrote:
>
>
> Chris Locke wrote:
> >
> >
> > Congratulations Roice on another jaw-dropping program! The community
> > of beyond physically realizable Rubik cubers I'm sure is both shocked
> > and pleased by the recent flurry of new puzzles. First we get tons
> > new cool 4D shapes to play with, and now hyperbolic puzzles?? What's
> > next, 4D equivalents of these hyperbolic puzzles? :D
> >
> > First thing I would note is that you should move the "donate" button
> > to a more visible location. As you can see by Alexander's post it's
> > not easy to find. I didn't even know you had it on the page until
> > recently when I actually scrolled all the way to the bottom!
> >
> > As for the puzzles themselves. I find that from my limited playing
> > around with the hexagonal and heptagonal tilings, that low color
> > counts completely baffle me, but as the number of colors increases,
> > the amount of coupling between the twists decreases, which makes it
> > much easier to work with. The 3 and 4 color hexagonal tilings are
> > just weird to try and solve :P. But in the cases with many colors,
> > it's feels like it's just a matter of patient to reduce it down to a
> > last layer. From there, Megamix LL algorithms/commutators can be
> > generalized to finish them up.
>
> I think that this phenomenon is the same one that makes the megaminx
> easier to solve than the cube because there's more "room" on the surface
> of the puzzles with more faces in which to squirrel away some parts
> while working on others. The smaller, puzzles are "tighter" and cause
> every action to affect just about everything else. I have a funny
> feeling that there is some sort of natural difficulty metric in which
> the original 3^3 Rubik's cube will turn out to be the hardest of all the
> similar puzzles in all dimensions, but I can't quite see how to define
> that metric. I only feel it.
>
> > As for the double bottoms thing, Melinda gave a pretty good
> > explanation as to how it's a result of the topology. In the case of
> > the 3 holed torus (Klein's quartic) it's pretty easy to see how
> > spreading out from one position can leave with with a case of two
> > disconnected unsolved faces. Furthermore, I don't think it's possible
> > to end up with 3 disconnected faces in this case, and also if you
> > understand the topology from the tiling, it should also always be
> > possible to solve it such that you end up with just one unsolved
> > face. But that in itself is a challenge too :D.
>
> You'd need a genus 4 or higher surface to end up with 3 or more
> disconnected unsolved faces.
>
> I'm not sure if you are right that one should be able to follow a
> modified layer-by-layer method on any of these puzzles and always end up
> with a single unsolved face. The key would be to keep track of the
> "outer edge" of your solved patch as you grow it. Never let one part of
> that edge connect with any other part. In other words, you make sure to
> keep that edge a simple closed curve. Any time a part of it wants to
> connect with another part, just leave that area and work on some other
> part of the border that has room to spread. The question is whether the
> initially growing edge will eventually shrink back down to a single face.
>
> At first I thought yes, but as I started to write the above, I'm now not
> so sure. This is reminding me a lot of the short Wikipedia article that
> Roice cited on simply connected
> spaces. Imagine trying
> the above solving strategy with a {4,4} puzzle defined on the surface of
> a torus. I think you'll end up with a seam that you can't get around. I
> just can't quite see it in my head without some good pictures.
>
> > [...] Less colors is too hard, and more colors takes too long.
>
> Each color is a single face. Each time that you see a red face, it's
> always the same face just seen from a different perspective. Kind of
> like how gravitational lensing can let you see multiple images of the
> same distant galaxy in a single photograph. I agree that tighter puzzles
> take more brain work and less tedium but when they become extremely
> tight, they seem to get easier again in a way as you can begin to get
> your head around the whole thing. In addition to wanting to know which
> puzzle is hardest for it's size, I also want to know is which puzzle out
> of all the twisty puzzles different people think has the challenge that
> is "just right" for them.
>
> -Melinda
>
>
> ------------------------------------
>
> Yahoo! Groups Links
>
>
>
>

--00032555a3167108c2047ed1eaa8
Content-Type: text/html; charset=ISO-8859-1
Content-Transfer-Encoding: quoted-printable

What an great flow of ideas.=A0 I love it!

=A0

About solving towards a single cell, I think the program can help answ=
er this by using the setting to only show the fundamental set of tiles.=A0 =
I'm not perfectly confident, but my intuition says it is possible to av=
oid multiple disjoint unsolved cells at the end.=A0 My reasoning is that ev=
en if the topology of the object as a whole (fundamental and orbit regions =
glued up) is not simply connected, the fundamental region alone is simply c=
onnected (as long as you never leave the boundary of it, so that it behaves=
topologically as a disk). =A0Looking at only the fundamental set of Klein&=
#39;s Quartic, it is easy to pick a solve order leaving only one cell at th=
e end, by using sort of a "solve wave" to push unsolved cells fro=
m one end to the other. =A0This would be my strategy to "keep the edge=
a simple closed curve", as=A0Melinda nicely described things. =A0v>

If=A0I require that I solve cells in a connected fashio=
n, where each cell solved must be adjacent to the prior one, it appears I l=
ose the ability to end on a single cell. =A0As an aside, I wonder if it is =
possible to have fundamental sets of tiles which are not simply connected, =
which could invalidate my logic above.



Anyway, you've all convince me that it is the topology that is imp=
ortant factor in all this (though it still seems the symmetries and topolog=
y are intertwined). =A0After taking another look at the 12-colored octagona=
l, also genus 3, I see it actually does behave the same.



=A0

Regarding which puzzle(s) may be "just right", I've only=
done a few so far, but last night I did go through a solve of the 9-colore=
d torus puzzle and can say I found it really enjoyable -=A0perfect for a=A0=
solve in one sitting for sure.=A0 And=A0the repeating units caught me a cou=
ple times, which was fun. =A0It may be inevitable that the 3x3x3 will alway=
s be the most perfect puzzle in my mind though :)


One small comment related to difficulty... =A0The state=
-space of the 3-coloreds is small enough that they will often solve themsel=
ves if I just randomly click.

Yesterday, I learned=
about some additional coloring possibilities on the hexagonal puzzles whic=
h I think will be good, balanced puzzles when it comes to the=A0tradeoffs=
=A0being discussed, e.g. a 7-colored=A0(This was pointed out to me by someo=
ne from the twistypuzzles forums). =A0Applying the tiling=A0algorithm which=
produces those=A0to the hyperbolic puzzles will probably allow some furthe=
r balanced variants there as well. =A0The 6-12 color range feels nice to me=
, both in terms of coupling and tedium.



Btw, a while back I realized repeating units can be applied to spheric=
al puzzles too, which will end up throwing even a few more into the mix.=A0=
A Megaminx with a 6-colored fundamental region and opposite sides being or=
bits would topologically act as the /Orientable">non-orientable l_projective_plane" target=3D"_blank">real projective plane=A0(I verifi=
ed by playing with color settings).=A0 Due to the non-orientability of that=
surface, orbits would rotate in an opposite sense to the fundamental cells=
!=A0 One could do a non-orientable hexagonal version as well, topologically=
a Klein bottle.=
=A0 I definitely want to support these variants at some point, which will o=
nly involve minor code changes to deal with reversing the twisting.



Re: Alexander... I bet doing a ctrl+click view rotation without animat=
ion wouldn't be too bad, and something like that could likely be the fi=
rst pass. =A0Also, I've been doing the same head twisting and so am sha=
ring the desire for rotation control of the projected puzzles too, which wi=
ll be trivial to add.=A0 And load/save is at the top of the list.=A0 I hope=
to address the majority of the requests I've received so far, though I=
=A0expect it will be at least a=A0month before I'll be able to provide =
any updates. =A0Regarding=A0publicizing, I=A0have=A0been sending out an ema=
il or two each evening to try to get the word out.=A0 .mathpuzzle.com/" target=3D"_blank">Mathpuzzle gave MagicTile a really =
nice shout out this morning. =A0I want to do a blog post too (both about MC=
4D and this), though that will probably just get the word out to my mom and=
siblings :)



=A0

Re: Chris... so sorry about the loss of work=A0:( =A0I will definitely=
switch this resetting behavior.=A0 It caught me the other night too, at wh=
ich point I noted the poor behavior in my list. =A0I was lucky enough to no=
t be half way into a large puzzle.=A0 Also, on the 4D front, I do think it&=
#39;d be cool to have stereographically projected versions of the puzzles i=
n MC4D.=A0 I even opened sues/detail?id=3D14&q=3Dstereographic">an issue for it a while back=
:D



=A0

Have a good night all,

Roice

=A0

P.S.=A0 I did move the donate button up (thanks for the suggestion on =
that too, Chris and Alexander)

=A0

On 2/4/10, M=
elinda Green
<_blank">melinda@superliminal.com> wrote:
=20
0px 0.8ex;border-left:#ccc 1px solid">
Chris Locke wrote:
>
>=
;
> Congratulations Roice on another jaw-dropping program!=A0=A0The c=
ommunity


> of beyond physically realizable Rubik cubers I'm sure is both shoc=
ked
> and pleased by the recent flurry of new puzzles.=A0=A0First we =
get tons
> new cool 4D shapes to play with, and now hyperbolic puzzle=
s??=A0=A0What's


> next, 4D equivalents of these hyperbolic puzzles? :D
>
> F=
irst thing I would note is that you should move the "donate" butt=
on
> to a more visible location.=A0=A0As you can see by Alexander'=
;s post it's


> not easy to find.=A0=A0I didn't even know you had it on the page u=
ntil
> recently when I actually scrolled all the way to the bottom!r>>
> As for the puzzles themselves.=A0=A0I find that from my limi=
ted playing


> around with the hexagonal and heptagonal tilings, that low color
&g=
t; counts completely baffle me, but as the number of colors increases,
&=
gt; the amount of coupling between the twists decreases, which makes it


> much easier to work with.=A0=A0The 3 and 4 color hexagonal tilings are=

> just weird to try and solve :P.=A0=A0But in the cases with many co=
lors,
> it's feels like it's just a matter of patient to redu=
ce it down to a


> last layer.=A0=A0From there, Megamix LL algorithms/commutators can be<=
br>> generalized to finish them up.

I think that this phenomenon =
is the same one that makes the megaminx
easier to solve than the cube be=
cause there's more "room" on the surface


of the puzzles with more faces in which to squirrel away some parts
whil=
e working on others. The smaller, puzzles are "tighter" and cause=

every action to affect just about everything else. I have a funny


feeling that there is some sort of natural difficulty metric in which
th=
e original 3^3 Rubik's cube will turn out to be the hardest of all the<=
br>similar puzzles in all dimensions, but I can't quite see how to defi=
ne


that metric. I only feel it.

> As for the double bottoms thing, M=
elinda gave a pretty good
> explanation as to how it's a result o=
f the topology.=A0=A0In the case of
> the 3 holed torus (Klein's =
quartic) it's pretty easy to see how


> spreading out from one position can leave with with a case of two
&=
gt; disconnected unsolved faces.=A0=A0Furthermore, I don't think it'=
;s possible
> to end up with 3 disconnected faces in this case, and a=
lso if you


> understand the topology from the tiling, it should also always be
&=
gt; possible to solve it such that you end up with just one unsolved
>=
; face.=A0=A0But that in itself is a challenge too :D.

You'd nee=
d a genus 4 or higher surface to end up with 3 or more


disconnected unsolved faces.

I'm not sure if you are right that =
one should be able to follow a
modified layer-by-layer method on any of =
these puzzles and always end up
with a single unsolved face. The key wou=
ld be to keep track of the


"outer edge" of your solved patch as you grow it. Never let one p=
art of
that edge connect with any other part. In other words, you make s=
ure to
keep that edge a simple closed curve. Any time a part of it wants=
to


connect with another part, just leave that area and work on some other
p=
art of the border that has room to spread. The question is whether the
i=
nitially growing edge will eventually shrink back down to a single face.>


At first I thought yes, but as I started to write the above, I'm no=
w not
so sure. This is reminding me a lot of the short Wikipedia article=
that
Roice cited on simply connected
<dia.org/wiki/Simply_connected" target=3D"_blank">http://en.wikipedia.org/wi=
ki/Simply_connected
> spaces. Imagine trying


the above solving strategy with a {4,4} puzzle defined on the surface of>a torus. I think you'll end up with a seam that you can't get arou=
nd. I
just can't quite see it in my head without some good pictures.=




> [...] Less colors is too hard, and more colors takes too long.
=

Each color is a single face. Each time that you see a red face, it'=
s
always the same face just seen from a different perspective. Kind ofr>

like how gravitational lensing can let you see multiple images of the
sa=
me distant galaxy in a single photograph. I agree that tighter puzzles
t=
ake more brain work and less tedium but when they become extremely

tight, they seem to get easier again in a way as you can begin to get

your head around the whole thing. In addition to wanting to know which
p=
uzzle is hardest for it's size, I also want to know is which puzzle out=

of all the twisty puzzles different people think has the challenge that=



is "just right" for them.

-Melinda


------------=
------------------------

Yahoo! Groups Links

<*> To vis=
it your group on the web, go to:
=A0=A0 om/group/4D_Cubing/" target=3D"_blank">http://groups.yahoo.com/group/4D_Cub=
ing/




<*> Your email settings:
=A0=A0 Individual Email | Traditional=


<*> To change settings online go to:
=A0=A0 p://groups.yahoo.com/group/4D_Cubing/join" target=3D"_blank">http://groups.=
yahoo.com/group/4D_Cubing/join



=A0=A0 (Yahoo! ID required)

<*> To change settings via email:<=
br>=A0=A0 ank">4D_Cubing-digest@yahoogroups.com

=A0=A0 _blank">4D_Cubing-fullfeatured@yahoogroups.com

<*> To unsu=
bscribe from this group, send an email to:

=A0=A0 blank">4D_Cubing-unsubscribe@yahoogroups.com

<*> Your use =
of Yahoo! Groups is subject to:

=A0=A0 http=
://docs.yahoo.com/info/terms/





--00032555a3167108c2047ed1eaa8--




From: Melinda Green <melinda@superliminal.com>
Date: Thu, 04 Feb 2010 22:29:45 -0800
Subject: Re: [MC4D] Re: Introducing "MagicTile"



--------------050204020708080202040602
Content-Type: text/plain; charset=ISO-8859-1; format=flowed
Content-Transfer-Encoding: 7bit

Roice Nelson wrote:
>
>
> What an great flow of ideas. I love it!
>
> About solving towards a single cell, I think the program can help
> answer this by using the setting to only show the fundamental set of
> tiles. I'm not perfectly confident, but my intuition says it is
> possible to avoid multiple disjoint unsolved cells at the end. My
> reasoning is that even if the topology of the object as a whole
> (fundamental and orbit regions glued up) is not simply connected, the
> fundamental region alone is simply connected (as long as you never
> leave the boundary of it, so that it behaves topologically as a disk).
> Looking at only the fundamental set of Klein's Quartic, it is easy to
> pick a solve order leaving only one cell at the end, by using sort of
> a "solve wave" to push unsolved cells from one end to the other. This
> would be my strategy to "keep the edge a simple closed curve",
> as Melinda nicely described things.

Can you solve a puzzle on a torus this way? It may be different than on
surfaces of higher higher genus but it's a good first test of the idea.

>
> If I require that I solve cells in a connected fashion, where each
> cell solved must be adjacent to the prior one, it appears I lose the
> ability to end on a single cell. As an aside, I wonder if it is
> possible to have fundamental sets of tiles which are not simply
> connected, which could invalidate my logic above.
>
> Anyway, you've all convince me that it is the topology that is
> important factor in all this (though it still seems the symmetries and
> topology are intertwined). After taking another look at the
> 12-colored octagonal, also genus 3, I see it actually does behave the
> same.
>
> Regarding which puzzle(s) may be "just right", I've only done a few so
> far, but last night I did go through a solve of the 9-colored torus
> puzzle and can say I found it really enjoyable - perfect for a solve
> in one sitting for sure. And the repeating units caught me a couple
> times, which was fun. It may be inevitable that the 3x3x3 will always
> be the most perfect puzzle in my mind though :)

For me the 3^3 seems the hardest relative to it's size but something
about the megaminx really appeals to me more these days. Maybe a little
too far in the tedious direction but I find the geometry very elegant.

> One small comment related to difficulty... The state-space of the
> 3-coloreds is small enough that they will often solve themselves if I
> just randomly click.
>
> Yesterday, I learned about some additional coloring possibilities on
> the hexagonal puzzles which I think will be good, balanced puzzles
> when it comes to the tradeoffs being discussed, e.g. a 7-colored (This
> was pointed out to me by someone from the twistypuzzles forums).
> Applying the tiling algorithm which produces those to the hyperbolic
> puzzles will probably allow some further balanced variants there as
> well. The 6-12 color range feels nice to me, both in terms of
> coupling and tedium.
>
> Btw, a while back I realized repeating units can be applied to
> spherical puzzles too, which will end up throwing even a few more into
> the mix. A Megaminx with a 6-colored fundamental region and opposite
> sides being orbits would topologically act as the non-orientable
> real projective plane
> (I verified by
> playing with color settings). Due to the non-orientability of that
> surface, orbits would rotate in an opposite sense to the fundamental
> cells! One could do a non-orientable hexagonal version as well,
> topologically a Klein bottle
> . I definitely want to
> support these variants at some point, which will only involve minor
> code changes to deal with reversing the twisting.

Absolutely!
Non-orientable puzzles will work really well here. Their
self-intersections in physical models really detract from their beauty,
but in your presentation they'll look extra cool as they twist. I'd love
to see Klein bottle puzzles and the even crazier Boy's surface
. BTW, if you do make a
megaminx with opposite faces identified, then it will have genus 6 which
should give solvers all kinds of conniptions. I don't think that it
needs to be non-orientable, but either way it will be interesting to see
a high-genus puzzle. You should also be able to make twisty puzzles from
the duels of all the IRPs that I have listed in the first column of the
table on my IRP page
. A
particularly interesting one there is a genus 5 {3,8}
made out of
snub cubes. It was one of the hardest models to work out the 3D vertex
positions for.

Remember, all of the suggestions that we're making are because we're
excited, and not because anything is broken or missing. I hope that you
don't feel pressure to make additions just for us. Please take a
well-deserved rest and then come back and add puzzles and features when
you just can't wait to see the results.

-Melinda

--------------050204020708080202040602
Content-Type: text/html; charset=ISO-8859-1
Content-Transfer-Encoding: 7bit







Roice Nelson wrote:
cite="mid:b5979e761002041919k6faff61dp37b5f6c1b570d532@mail.gmail.com"
type="cite">


What an great flow of ideas.  I love it!

 

About solving towards a single cell, I think the program can
help answer this by using the setting to only show the fundamental set
of tiles.  I'm not perfectly confident, but my intuition says it is
possible to avoid multiple disjoint unsolved cells at the end.  My
reasoning is that even if the topology of the object as a whole
(fundamental and orbit regions glued up) is not simply connected, the
fundamental region alone is simply connected (as long as you never
leave the boundary of it, so that it behaves topologically as a disk).
 Looking at only the fundamental set of Klein's Quartic, it is easy to
pick a solve order leaving only one cell at the end, by using sort of a
"solve wave" to push unsolved cells from one end to the other.  This
would be my strategy to "keep the edge a simple closed curve",
as Melinda nicely described things. 





Can you solve a puzzle on a torus this way? It may be different than on
surfaces of higher higher genus but it's a good first test of the idea.



cite="mid:b5979e761002041919k6faff61dp37b5f6c1b570d532@mail.gmail.com"
type="cite">



If I require that I solve cells in a connected fashion, where
each cell solved must be adjacent to the prior one, it appears I lose
the ability to end on a single cell.  As an aside, I wonder if it is
possible to have fundamental sets of tiles which are not simply
connected, which could invalidate my logic above.




Anyway, you've all convince me that it is the topology that is
important factor in all this (though it still seems the symmetries and
topology are intertwined).  After taking another look at the 12-colored
octagonal, also genus 3, I see it actually does behave the same.

 

Regarding which puzzle(s) may be "just right", I've only done a
few so far, but last night I did go through a solve of the 9-colored
torus puzzle and can say I found it really enjoyable - perfect for
a solve in one sitting for sure.  And the repeating units caught me a
couple times, which was fun.  It may be inevitable that the 3x3x3 will
always be the most perfect puzzle in my mind though :)




For me the 3^3 seems the hardest relative to it's size but something
about the megaminx really appeals to me more these days. Maybe a little
too far in the tedious direction but I find the geometry very elegant.



cite="mid:b5979e761002041919k6faff61dp37b5f6c1b570d532@mail.gmail.com"
type="cite">
One small comment related to difficulty...  The state-space of
the 3-coloreds is small enough that they will often solve themselves if
I just randomly click.




Yesterday, I learned about some additional coloring
possibilities on the hexagonal puzzles which I think will be good,
balanced puzzles when it comes to the tradeoffs being discussed, e.g. a
7-colored (This was pointed out to me by someone from the twistypuzzles
forums).  Applying the tiling algorithm which produces those to the
hyperbolic puzzles will probably allow some further balanced variants
there as well.  The 6-12 color range feels nice to me, both in terms of
coupling and tedium.




Btw, a while back I realized repeating units can be applied to
spherical puzzles too, which will end up throwing even a few more into
the mix.  A Megaminx with a 6-colored fundamental region and opposite
sides being orbits would topologically act as the moz-do-not-send="true" href="http://en.wikipedia.org/wiki/Orientable">non-orientable
href="http://en.wikipedia.org/wiki/Real_projective_plane"
target="_blank">real projective plane
 (I verified by playing with
color settings).  Due to the non-orientability of that surface, orbits
would rotate in an opposite sense to the fundamental cells!  One could
do a non-orientable hexagonal version as well, topologically a moz-do-not-send="true" href="http://en.wikipedia.org/wiki/Klein_bottle">Klein
bottle.  I definitely want to support these variants at some point,
which will only involve minor code changes to deal with reversing the
twisting.




Absolutely!

Non-orientable puzzles will work really well here. Their
self-intersections in physical models really detract from their beauty,
but in your presentation they'll look extra cool as they twist. I'd
love to see Klein bottle puzzles and the even crazier href="http://en.wikipedia.org/wiki/Boy%27s_surface">Boy's surface.
BTW, if you do make a megaminx with opposite faces identified, then it
will have genus 6 which should give solvers all kinds of conniptions. I
don't think that it needs to be non-orientable, but either way it will
be interesting to see a high-genus puzzle. You should also be able to
make twisty puzzles from the duels of all the IRPs that I have listed
in the first column of the table on my href="http://www.superliminal.com/geometry/infinite/infinite.htm">IRP
page. A particularly interesting one there is a href="http://www.superliminal.com/geometry/infinite/3_8b.htm">genus 5
{3,8} made out of snub cubes. It was one of the hardest models to
work out the 3D vertex positions for.



Remember, all of the suggestions that we're making are because we're
excited, and not because anything is broken or missing. I hope that you
don't feel pressure to make additions just for us. Please take a
well-deserved rest and then come back and add puzzles and features when
you just can't wait to see the results.



-Melinda




--------------050204020708080202040602--




From: Roice Nelson <roice3@gmail.com>
Date: Sun, 7 Feb 2010 22:01:18 -0600
Subject: Re: [MC4D] Re: Introducing "MagicTile"



--002215401c127f216a047f0eda92
Content-Type: text/plain; charset=ISO-8859-1

On Fri, Feb 5, 2010 at 12:29 AM, Melinda Green wrote:

>
> Roice Nelson wrote:
>
> About solving towards a single cell, I think the program can help answer
> this by using the setting to only show the fundamental set of tiles. I'm
> not perfectly confident, but my intuition says it is possible to avoid
> multiple disjoint unsolved cells at the end. My reasoning is that even if
> the topology of the object as a whole (fundamental and orbit regions glued
> up) is not simply connected, the fundamental region alone is simply
> connected (as long as you never leave the boundary of it, so that it behaves
> topologically as a disk). Looking at only the fundamental set of Klein's
> Quartic, it is easy to pick a solve order leaving only one cell at the end,
> by using sort of a "solve wave" to push unsolved cells from one end to the
> other. This would be my strategy to "keep the edge a simple closed curve",
> as Melinda nicely described things.
>
>
> Can you solve a puzzle on a torus this way? It may be different than on
> surfaces of higher higher genus but it's a good first test of the idea.
>
>
>
I went through a solve of the 12-color octagonal puzzle this weekend (fun!),
and have to say I've been thoroughly confusing myself thinking about the
group's discussion on finishing with a single unsolved cell :) This makes
me want to not make any claims about the torus puzzles!

My thinking above was flawed due to the fact that pieces (2C edges and 3C
corners) are connected across the border of the fundamental region. I was
picturing solving the stickers on cells without regard to these basic
connections.

This leaves me wondering what other people have been claiming though. From
a stickers-only perspective, it will always be impossible to have a single
unsolved cell at the end, due to the simple fact that if there is a sticker
unsolved on a cell, it had to come from another cell! So is it fair to say
that what we are talking about is whether a single unsolved "layer" is
possible, and how many of those there can be? (Sorry, this distinction may
be obvious to everyone.)

In that case, there is a much simpler demonstration that you can always end
up with one unsolved layer at the end - just do a single twist of a single
cell! This sounds a little trivial to say, but maybe that really does
answer the question. Admittedly, this "final single layer" can be very
weird on some puzzles, e.g. on the 4-colored torus, since a single twist
changes stickers on all 4 cells.

Continuing on the assumption that we're talking about layers, I think it is
easy to show that more than 3 disjoint unsolved layers can be had on Klein's
Quartic. Quickly experimenting, I seem to be able to make up to 5 disjoint
twists.

I hope I'm not being excessive in describing silly tautology-like things.
Am I off track?

And here is a related question for you all. Can you checkerboard Klein's
Quartic? If so, does it have to be a 3-cycle checkerboard, due to there
being "two bottoms" to every cell that are in-a-sense its opposites? As I
started investigating just now, I was able to easily 2-cycle checkerboard
both the 6-colored and 12-colored octagonal puzzles...

seeya,
Roice

--002215401c127f216a047f0eda92
Content-Type: text/html; charset=ISO-8859-1
Content-Transfer-Encoding: quoted-printable


On Fri, Feb 5, 2010 at 12:29 AM, Melinda Gre=
en <melind=
a@superliminal.com
>
wrote:
e" style=3D"margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;"=
>







=20=20=20=20=20=20=20=20



=20=20








Roice Nelson wrote:

=20=20
=20=20
=20=20
About solving towards a single cell, I think the program can
help answer this by using the setting to only show the fundamental set
of tiles.=A0 I'm not perfectly confident, but my intuition says it is
possible to avoid multiple disjoint unsolved cells at the end.=A0 My
reasoning is that even if the topology of the object as a whole
(fundamental and orbit regions glued up) is not simply connected, the
fundamental region alone is simply connected (as long as you never
leave the boundary of it, so that it behaves topologically as a disk).
=A0Looking at only the fundamental set of Klein's Quartic, it is easy t=
o
pick a solve order leaving only one cell at the end, by using sort of a
"solve wave" to push unsolved cells from one end to the other. =
=A0This
would be my strategy to "keep the edge a simple closed curve",
as=A0Melinda nicely described things.=A0





Can you solve a puzzle on a torus this way? It may be different than on
surfaces of higher higher genus but it's a good first test of the idea.=




I went through a solve of =
the 12-color octagonal puzzle this weekend (fun!), and have to say I've=
been=A0thoroughly=A0confusing myself thinking about the group's discus=
sion on finishing with a single unsolved cell :) =A0This makes me want to n=
ot make any claims about the torus puzzles!


My thinking above was flawed due to the fact that piece=
s (2C edges and 3C corners) are connected across the border of the fundamen=
tal region. =A0I was picturing solving the stickers on cells without regard=
to these basic connections.


This leaves me wondering what other people have been cl=
aiming though. =A0From a stickers-only perspective, it will always be impos=
sible to have a single unsolved cell at the end, due to the simple fact tha=
t if there is a sticker unsolved on a cell, it had to come from another cel=
l! =A0So is it fair to say that what we are talking about is whether a sing=
le unsolved "layer" is possible, and how many of those there can =
be? =A0(Sorry, this distinction may be obvious to everyone.)


In that case, there is a much simpler demonstration tha=
t you can always end up with one unsolved layer at the end - just do a sing=
le twist of a single cell! =A0This sounds a little trivial to say, but mayb=
e that really does answer the question. =A0Admittedly, this "final sin=
gle layer" can be very weird on some puzzles, e.g. on the 4-colored to=
rus, since a single twist changes stickers on all 4 cells.


Continuing on the assumption that we're talking abo=
ut layers, I think it is easy to show that more than 3 disjoint unsolved la=
yers can be had on Klein's Quartic. =A0Quickly experimenting, I seem to=
be able to make up to 5 disjoint twists.


I hope I'm not being excessive in describing silly=
=A0tautology-like things. =A0Am I off track?

And h=
ere is a related question for you all. =A0Can you checkerboard Klein's =
Quartic? =A0If so, does it have to be a 3-cycle checkerboard, due to there =
being "two bottoms" to every cell=A0that are in-a-sense its oppos=
ites? =A0As I started investigating just now, I was able to easily 2-cycle =
checkerboard both the 6-colored and 12-colored octagonal puzzles...


seeya,
Roice


--002215401c127f216a047f0eda92--




From: Melinda Green <melinda@superliminal.com>
Date: Sun, 07 Feb 2010 22:49:59 -0800
Subject: Re: [MC4D] Re: Introducing "MagicTile"



Roice Nelson wrote:
>
> On Fri, Feb 5, 2010 at 12:29 AM, Melinda Green
> > wrote:
>
>
> Roice Nelson wrote:
>> About solving towards a single cell, I think the program can help
>> answer this by using the setting to only show the fundamental set
>> of tiles. I'm not perfectly confident, but my intuition says it
>> is possible to avoid multiple disjoint unsolved cells at the
>> end. My reasoning is that even if the topology of the object as
>> a whole (fundamental and orbit regions glued up) is not simply
>> connected, the fundamental region alone is simply connected (as
>> long as you never leave the boundary of it, so that it behaves
>> topologically as a disk). Looking at only the fundamental set of
>> Klein's Quartic, it is easy to pick a solve order leaving only
>> one cell at the end, by using sort of a "solve wave" to push
>> unsolved cells from one end to the other. This would be my
>> strategy to "keep the edge a simple closed curve", as Melinda
>> nicely described things.
>
> Can you solve a puzzle on a torus this way? It may be different
> than on surfaces of higher higher genus but it's a good first test
> of the idea.
>
>
>
> I went through a solve of the 12-color octagonal puzzle this weekend
> (fun!), and have to say I've been thoroughly confusing myself thinking
> about the group's discussion on finishing with a single unsolved cell
> :) This makes me want to not make any claims about the torus puzzles!
>
> My thinking above was flawed due to the fact that pieces (2C edges and
> 3C corners) are connected across the border of the fundamental region.
> I was picturing solving the stickers on cells without regard to these
> basic connections.
>
> This leaves me wondering what other people have been claiming though.
> From a stickers-only perspective, it will always be impossible to
> have a single unsolved cell at the end, due to the simple fact that if
> there is a sticker unsolved on a cell, it had to come from another
> cell! So is it fair to say that what we are talking about is whether
> a single unsolved "layer" is possible, and how many of those there can
> be? (Sorry, this distinction may be obvious to everyone.)
>
> In that case, there is a much simpler demonstration that you can
> always end up with one unsolved layer at the end - just do a single
> twist of a single cell! This sounds a little trivial to say, but
> maybe that really does answer the question. Admittedly, this "final
> single layer" can be very weird on some puzzles, e.g. on the 4-colored
> torus, since a single twist changes stickers on all 4 cells.
>
> Continuing on the assumption that we're talking about layers, I think
> it is easy to show that more than 3 disjoint unsolved layers can be
> had on Klein's Quartic. Quickly experimenting, I seem to be able to
> make up to 5 disjoint twists.
>
> I hope I'm not being excessive in describing silly tautology-like
> things. Am I off track?

I think that you are right that we should use the term "layer" instead
of cell in this discussion, but I suspect that you may be off track when
it comes to the question at hand. Or rather the question that's in my
mind, in case we're not in sync. In my mind, the question is whether
it's possible to solve KQ or other puzzles of genus > 0 by using a
simple layer-by-layer method. The key word here is "simple" as in
"simple closed curve" of the expanding edge. Ignoring that constraint,
I'm sure that you can end up with all sorts of non-simple curves
including ones that are disjoint. The question instead is whether you
necessarily have to go through such phases in a layer-by-layer solution.

>
> And here is a related question for you all. Can you checkerboard
> Klein's Quartic? If so, does it have to be a 3-cycle checkerboard,
> due to there being "two bottoms" to every cell that are in-a-sense its
> opposites? As I started investigating just now, I was able to easily
> 2-cycle checkerboard both the 6-colored and 12-colored octagonal
> puzzles...

I haven't the slightest idea about the checkerboard possibilities but
would love to see some. :-)
-Melinda




From: <alexander.sage@jacks.sdstate.edu>
Date: Mon, 8 Feb 2010 11:10:09 -0600
Subject: RE: [MC4D] Re: Introducing "MagicTile"



=20=20



Roice Nelson wrote:
>
> On Fri, Feb 5, 2010 at 12:29 AM, Melinda Green=20
> > wrote:
>
>
> Roice Nelson wrote:
>> About solving towards a single cell, I think the program can help
>> answer this by using the setting to only show the fundamental set
>> of tiles. I'm not perfectly confident, but my intuition says it
>> is possible to avoid multiple disjoint unsolved cells at the
>> end. My reasoning is that even if the topology of the object as
>> a whole (fundamental and orbit regions glued up) is not simply
>> connected, the fundamental region alone is simply connected (as
>> long as you never leave the boundary of it, so that it behaves
>> topologically as a disk). Looking at only the fundamental set of
>> Klein's Quartic, it is easy to pick a solve order leaving only
>> one cell at the end, by using sort of a "solve wave" to push
>> unsolved cells from one end to the other. This would be my
>> strategy to "keep the edge a simple closed curve", as Melinda
>> nicely described things.=20
>
> Can you solve a puzzle on a torus this way? It may be different
> than on surfaces of higher higher genus but it's a good first test
> of the idea.
>
>
>
> I went through a solve of the 12-color octagonal puzzle this weekend=20
> (fun!), and have to say I've been thoroughly confusing myself thinking=20
> about the group's discussion on finishing with a single unsolved cell=20
> :) This makes me want to not make any claims about the torus puzzles!
>
> My thinking above was flawed due to the fact that pieces (2C edges and=20
> 3C corners) are connected across the border of the fundamental region.=20
> I was picturing solving the stickers on cells without regard to these=20
> basic connections.
>
> This leaves me wondering what other people have been claiming though.=20
> From a stickers-only perspective, it will always be impossible to=20
> have a single unsolved cell at the end, due to the simple fact that if=20
> there is a sticker unsolved on a cell, it had to come from another=20
> cell! So is it fair to say that what we are talking about is whether=20
> a single unsolved "layer" is possible, and how many of those there can=20
> be? (Sorry, this distinction may be obvious to everyone.)
>
> In that case, there is a much simpler demonstration that you can=20
> always end up with one unsolved layer at the end - just do a single=20
> twist of a single cell! This sounds a little trivial to say, but=20
> maybe that really does answer the question. Admittedly, this "final=20
> single layer" can be very weird on some puzzles, e.g. on the 4-colored=20
> torus, since a single twist changes stickers on all 4 cells.
>
> Continuing on the assumption that we're talking about layers, I think=20
> it is easy to show that more than 3 disjoint unsolved layers can be=20
> had on Klein's Quartic. Quickly experimenting, I seem to be able to=20
> make up to 5 disjoint twists.
>
> I hope I'm not being excessive in describing silly tautology-like=20
> things. Am I off track?

I think that you are right that we should use the term "layer" instead=20
of cell in this discussion, but I suspect that you may be off track when=20
it comes to the question at hand. Or rather the question that's in my=20
mind, in case we're not in sync. In my mind, the question is whether=20
it's possible to solve KQ or other puzzles of genus > 0 by using a=20
simple layer-by-layer method. The key word here is "simple" as in=20
"simple closed curve" of the expanding edge. Ignoring that constraint,=20
I'm sure that you can end up with all sorts of non-simple curves=20
including ones that are disjoint. The question instead is whether you=20
necessarily have to go through such phases in a layer-by-layer solution.

>
> And here is a related question for you all. Can you checkerboard=20
> Klein's Quartic? If so, does it have to be a 3-cycle checkerboard,=20
> due to there being "two bottoms" to every cell that are in-a-sense its=20
> opposites? As I started investigating just now, I was able to easily=20
> 2-cycle checkerboard both the 6-colored and 12-colored octagonal=20
> puzzles...

I haven't the slightest idea about the checkerboard possibilities but=20
would love to see some. :-)
-Melinda



=20=09=09=20=09=20=20=20=09=09=20=20
--_0adc4861-ed36-42b8-a524-bbac49b87831_
Content-Type: text/html; charset="Windows-1252"
Content-Transfer-Encoding: quoted-printable






 As far as klein's quartic, what is happening is that if you solv=
e a single face, then solve all layers bordering that face, and c=
ontinue, then on a megaminx (or any other platonic solid besides =
a pyramid) you will eventually get to a single layer left to solv=
e.  if you try to start from one face on the {7,3}, and solv=
e every face bordering it, then every face bordering what you have sol=
ved, you will end up with multiple unsolved faces.  essentially, there=
seems to be more than one face that is the maximum distance from the =
first face.

 

I noticed something unfortunate.  tthe random seed looks like it reset=
s, so I always seem to get the same puzzle every time I open the progr=
am, unless I continue to shuffle it by using a couple extra five twist=
s.  Could you make a random seed that saves itself?
 



To: 4D_Cubing@yahoogroups.com
From: melinda@superliminal.com
Date: Su=
n, 7 Feb 2010 22:49:59 -0800
Subject: Re: [MC4D] Re: Introducing "MagicT=
ile"

 =20



Roice Nelson wrote:
>
> On Fri, Feb 5, 2010 at 12:29 AM, Melind=
a Green
> <melinda@su=
perliminal.com
<mailto:om">melinda@superliminal.com>> wrote:
>
>
>=
; Roice Nelson wrote:
>> About solving towards a single cell, I th=
ink the program can help
>> answer this by using the setting to on=
ly show the fundamental set
>> of tiles. I'm not perfectly confide=
nt, but my intuition says it
>> is possible to avoid multiple disj=
oint unsolved cells at the
>> end. My reasoning is that even if th=
e topology of the object as
>> a whole (fundamental and orbit regi=
ons glued up) is not simply
>> connected, the fundamental region a=
lone is simply connected (as
>> long as you never leave the bounda=
ry of it, so that it behaves
>> topologically as a disk). Looking =
at only the fundamental set of
>> Klein's Quartic, it is easy to p=
ick a solve order leaving only
>> one cell at the end, by using so=
rt of a "solve wave" to push
>> unsolved cells from one end to the=
other. This would be my
>> strategy to "keep the edge a simple cl=
osed curve", as Melinda
>> nicely described things.
>
&g=
t; Can you solve a puzzle on a torus this way? It may be different
> =
than on surfaces of higher higher genus but it's a good first test
> =
of the idea.
>
>
>
> I went through a solve of the =
12-color octagonal puzzle this weekend
> (fun!), and have to say I'v=
e been thoroughly confusing myself thinking
> about the group's disc=
ussion on finishing with a single unsolved cell
> :) This makes me w=
ant to not make any claims about the torus puzzles!
>
> My thin=
king above was flawed due to the fact that pieces (2C edges and
> 3C=
corners) are connected across the border of the fundamental region.
&g=
t; I was picturing solving the stickers on cells without regard to these R>> basic connections.
>
> This leaves me wondering what oth=
er people have been claiming though.
> From a stickers-only perspect=
ive, it will always be impossible to
> have a single unsolved cell a=
t the end, due to the simple fact that if
> there is a sticker unsol=
ved on a cell, it had to come from another
> cell! So is it fair to =
say that what we are talking about is whether
> a single unsolved "l=
ayer" is possible, and how many of those there can
> be? (Sorry, thi=
s distinction may be obvious to everyone.)
>
> In that case, th=
ere is a much simpler demonstration that you can
> always end up wit=
h one unsolved layer at the end - just do a single
> twist of a sing=
le cell! This sounds a little trivial to say, but
> maybe that reall=
y does answer the question. Admittedly, this "final
> single layer" =
can be very weird on some puzzles, e.g. on the 4-colored
> torus, si=
nce a single twist changes stickers on all 4 cells.
>
> Continu=
ing on the assumption that we're talking about layers, I think
> it =
is easy to show that more than 3 disjoint unsolved layers can be
> h=
ad on Klein's Quartic. Quickly experimenting, I seem to be able to
>=
make up to 5 disjoint twists.
>
> I hope I'm not being excessi=
ve in describing silly tautology-like
> things. Am I off track?
<=
BR>I think that you are right that we should use the term "layer" instead <=
BR>of cell in this discussion, but I suspect that you may be off track when=

it comes to the question at hand. Or rather the question that's in my =

mind, in case we're not in sync. In my mind, the question is whether R>it's possible to solve KQ or other puzzles of genus > 0 by using a >simple layer-by-layer method. The key word here is "simple" as in
"sim=
ple closed curve" of the expanding edge. Ignoring that constraint,
I'm =
sure that you can end up with all sorts of non-simple curves
including =
ones that are disjoint. The question instead is whether you
necessarily=
have to go through such phases in a layer-by-layer solution.

>R>> And here is a related question for you all. Can you checkerboard >> Klein's Quartic? If so, does it have to be a 3-cycle checkerboard, R>> due to there being "two bottoms" to every cell that are in-a-sense i=
ts
> opposites? As I started investigating just now, I was able to e=
asily
> 2-cycle checkerboard both the 6-colored and 12-colored octag=
onal
> puzzles...

I haven't the slightest idea about the chec=
kerboard possibilities but
would love to see some. :-)
-Melinda
<=
BR>




--_0adc4861-ed36-42b8-a524-bbac49b87831_--