Thread: "slicing up MagicTile puzzles without triangle vertex figures"

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Thanks,
a very beautiful demo !
Ed

----- Original Message -----=20
From: Brandon Enright=20
To: 4D_Cubing@yahoogroups.com=20
Cc: roice3@gmail.com ; bmenrigh@ucsd.edu=20
Sent: Friday, February 25, 2011 4:45 AM
Subject: Re: [MC4D] slicing up MagicTile puzzles without triangle vertex =
figures


=20=20=20=20
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On Tue, 22 Feb 2011 22:18:57 -0600
Roice Nelson wrote:

> Hi all,
>=20
> I made a toy to help study the problem of how to slice up (face
> turning) MagicTile puzzles that do not have triangle vertex figures,
> and wanted to share. Honestly, my initial impression is that I wish
> the slicing turned out to be more elegant in the general case.
> Instead, there seem to be a huge number of possible puzzles for
> tilings like the {3,7}, none of them which feel particularly natural
> to me. You can play with the study tool directly in a web browser if
> you have Silverlight installed (or are willing to install it). I
> seem to be overtaxing the Silverlight drawing a bit, and some of the
> spherical puzzles aren't perfect due to things projecting to
> infinity, but it serves the purpose I wanted pretty well.
>=20
> www.gravitation3d.com/magictile/slicing_study.html
>=20
> Here are a few thoughts I had, but I'd really love other opinions on
> what would be the best puzzles for the next iteration of MagicTile.
>=20
> - Starting with a small circle size and increasing, the transition
> between puzzle types happens at points where new intersections begin
> between sets of two or more slicing circles (it reminds me of Venn
> diagrams). All the possible ways in which this can happen are very
> complicated. As you increase the circle size, there can be *a lot*
> of puzzle "phase transitions".
[...]


Hey Roice,

I rarely have easy access to a Windows box so I don't have
Silver light (I'm looking forward to getting Moonlight+Mono working
though).

Based on what you are describing in words, I think you might be
interested in some similar work done by Carl Hoff.

Here is an animation of varying the cut depth for a face turning
dodecahedron:
http://twistypuzzles.com/forum/viewtopic.php?p=3D179381&#p179381

As you can see that puzzle stays pretty simple.

Carl also did a similar animation for edge-turning cuts:
http://twistypuzzles.com/forum/viewtopic.php?p=3D189458#p189458

As you can see, certain depths cause a huge number of tiny pieces to
spring in an out of existence ("phase transition").

As for solving experience, I prefer semi-deep cuts with no more than
two different types of tiny pieces. I think Schuma (Nan) likes crazy
challenges with tons of hard-to-isolate tiny pieces.

This weekend I'll find a Windows machine and play with your slicing
study.

Best,

Brandon

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>

Hi,

After playing it for a while, I found it a great tool to explore puzzles. T=
hank you for sharing it. It's similar in spirit to Jaap's applet:

http://www.jaapsch.net/puzzles/sphere.htm

But Roice's handles more general geometries other than only the sphere.=20

(1) In {4,4}, when the size of circles is right, and only two circles are a=
llowed to rotate, you get a real puzzle: Rashkey
http://www.jaapsch.net/puzzles/rashkey.htm
which is a neat and hard puzzle to solve.

(2) Roice said:
> - On the {3,6}, if you make the circles larger than the parent cell, you =
can
> slice into 3-per-side by making the slicing circle go 2/3rds the way acro=
ss
> some adjacent cell edges (which simultaneously puts it 1/3rd the way acro=
ss
> some incident cell edges). This feels like a nice puzzle to me, with a
> pretty star pattern in the middle of each cell. You can do the same thin=
g
> on the {3,5} icosahedron, but in that case, the cuts are not evenly space=
d
> along an edge and the star patten is not quite as regular.=20=20

This way of slicing {3,5} should be precisely Gelatinbrain's (2.1.4):
http://users.skynet.be/gelatinbrain/Applets/Magic%20Polyhedra/icosa_f1.gif

(3) In {5,5}, when I increase the size of circles close to the maximum valu=
e, suddenly all the cuts jump to the outside of the hyperbolic plane...... =
Is there a particular reason or just a bug?

(4) Roice said:
> - There is a very cool midpoint slicing of the {3,4}. The doubled-up
> slicing circles form a cuboctahedron, so this is a case where things do f=
it
> together quite nicely.

This puzzle is the Skewb Diamond (a shape mod of Skewb). It's interesting t=
hat it can be viewed as a cuboctahedron.=20

(5) A {6,3} puzzle with large circles is simulated by Gelatinbrain (7.1.1, =
7.1.2, 7.1.3, with different repeating patterns).

In general, I'd like to see some of the puzzles of this kind, especially in=
the hyperbolic plane. Since there is no macro function, I don't think I ca=
n solve puzzles with too many small pieces. Brandon, I'm not that crazy... =
=20

Nan

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Hi Brandon,

That's really cool, thanks for sharing. You're right, Carl's animations are
similar to this study, and his edge turning animation reminds me of some of
the puzzles I lamented being so complicated at times. His face turning
animation made me realize I should allow the circles to get bigger for the
spherical puzzles (so I've extended the range possible).

Of course, the twistypuzzles thread has further intriguing aspects to it
(impressive job on solving the multidodecahedron using the applets!).
Carl's scaling of the dodecahedron he used to move through the various
puzzles has a definite "4D feel" to it, as Nan hinted at in one post. The
central projection of a dodecahedron being moved through a 4th dimension
towards a camera would change size like that, and the resulting hypervolume
the dodecahedron moves through is a dodecahedral prism. So this
multidodecahedron discussion made me wonder if there are some unique
slicings possible for our prism puzzles which we haven't yet considered.
This Type II guy leads me to believe you can slice up the dodecahedral prism
so that the two dodecahedra at the two ends of the prism would get sliced
differently (say one looking like a Megaminx and the other like a Pyraminx
Crystal). The slicing planes would no longer be perpendicular to the cells,
and the big thing you'd give up is that after a twist, the shape of the
puzzle would change! It would no longer be a dodecahedral prism, since some
of the stickers on the two ends would be interchanged, and those swapped
stickers will have different shapes. Maybe it's not really possible to
scramble it much though. I'd need to think more about it (thinking of the
3D analogue of a pentagonal prism puzzle is helpful).

I suppose I should have considered shape-changing 4D puzzles before now, as
I'm sure they are inevitable in the evolution of things. (Maybe I haven't
because I've never really liked these puzzles very much myself.) Anyway,
cool links! And thanks too for the feedback of the characteristics of
puzzles you prefer.

Take Care,
Roice


On Thu, Feb 24, 2011 at 9:45 PM, Brandon Enright wrote:

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>
> On Tue, 22 Feb 2011 22:18:57 -0600
> Roice Nelson wrote:
>
> > Hi all,
> >
> > I made a toy to help study the problem of how to slice up (face
> > turning) MagicTile puzzles that do not have triangle vertex figures,
> > and wanted to share. Honestly, my initial impression is that I wish
> > the slicing turned out to be more elegant in the general case.
> > Instead, there seem to be a huge number of possible puzzles for
> > tilings like the {3,7}, none of them which feel particularly natural
> > to me. You can play with the study tool directly in a web browser if
> > you have Silverlight installed (or are willing to install it). I
> > seem to be overtaxing the Silverlight drawing a bit, and some of the
> > spherical puzzles aren't perfect due to things projecting to
> > infinity, but it serves the purpose I wanted pretty well.
> >
> > www.gravitation3d.com/magictile/slicing_study.html
> >
> > Here are a few thoughts I had, but I'd really love other opinions on
> > what would be the best puzzles for the next iteration of MagicTile.
> >
> > - Starting with a small circle size and increasing, the transition
> > between puzzle types happens at points where new intersections begin
> > between sets of two or more slicing circles (it reminds me of Venn
> > diagrams). All the possible ways in which this can happen are very
> > complicated. As you increase the circle size, there can be *a lot*
> > of puzzle "phase transitions".
> [...]
>
>
> Hey Roice,
>
> I rarely have easy access to a Windows box so I don't have
> Silver light (I'm looking forward to getting Moonlight+Mono working
> though).
>
> Based on what you are describing in words, I think you might be
> interested in some similar work done by Carl Hoff.
>
> Here is an animation of varying the cut depth for a face turning
> dodecahedron:
> http://twistypuzzles.com/forum/viewtopic.php?p=179381&#p179381
>
> As you can see that puzzle stays pretty simple.
>
> Carl also did a similar animation for edge-turning cuts:
> http://twistypuzzles.com/forum/viewtopic.php?p=189458#p189458
>
> As you can see, certain depths cause a huge number of tiny pieces to
> spring in an out of existence ("phase transition").
>
> As for solving experience, I prefer semi-deep cuts with no more than
> two different types of tiny pieces. I think Schuma (Nan) likes crazy
> challenges with tons of hard-to-isolate tiny pieces.
>
> This weekend I'll find a Windows machine and play with your slicing
> study.
>
> Best,
>
> Brandon
>
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Thank you Nan, for the feedback and positive comments.

On the {5,5}, yep that was a problem. For hyperbolic puzzles, I incorrectly
allowed the slicing circles to get scaled beyond the boundary of the
Poincare disk, which is not something that really makes a whole lot of
sense, since the circles then lived outside the bounds of the model of the
space. I fixed this when adjusting the scaling I mentioned in my email to
Brandon, and now the maximum radius for those circles will put it on the
disk boundary (meaning they would have infinite radius). These changes led
to the tool no longer defaulting to having the circles go through the
vertices, but I think the new behavior is better. (Btw, using Chrome, I had
to manually clear my browser cache to get the new version to load :S, though
in IE a refresh of the page was enough.)

Thanks for telling me about the Skewb Diamond. With the sizing newly
available on the spherical puzzles, I see now you can do something analogous
with the {5,3}. The midpoint slicing that doubles up slicing circles
results in a circle set that is an icosidodecahedron (looks like this is the
"Pentultimate"). The result on the dual {3,5} looks a little stranger,
having some hexagonal facets (It's pretty to have both these turned on at
the same time). I don't know what shape the latter is, but it's not in
wiki's list of Archimedean solids, so maybe those hexagons aren't regular or
are skew or something.

One last thought your email sparked. I like that gelatinbrain has labels
for all his puzzles that people can point to. I only wish they were somehow
more algorithmic and/or descriptive of the puzzle. This has me trying to
think of some natural naming scheme which could generally cover
possibilities. My initial thought (which I don't like, but may help someone
come up with a better idea) is the Schlafli in combination with a number for
the puzzle type, say the number you get by counting the sequence of
possibilities as slicing circle size increases. So the descriptive name
would be something like "p.q.n", and the Skewb Diamond in this scheme would
be 3.4.6 (I think). What I don't like about it is the last number, since it
is difficult to even count these, and though algorithmically producible,
isn't descriptive in a friendly way.

seeya,
Roice


On Sat, Feb 26, 2011 at 2:03 AM, schuma wrote:

> Hi,
>
> After playing it for a while, I found it a great tool to explore puzzles.
> Thank you for sharing it. It's similar in spirit to Jaap's applet:
>
> http://www.jaapsch.net/puzzles/sphere.htm
>
> But Roice's handles more general geometries other than only the sphere.
>
> (1) In {4,4}, when the size of circles is right, and only two circles are
> allowed to rotate, you get a real puzzle: Rashkey
> http://www.jaapsch.net/puzzles/rashkey.htm
> which is a neat and hard puzzle to solve.
>
> (2) Roice said:
> > - On the {3,6}, if you make the circles larger than the parent cell, you
> can
> > slice into 3-per-side by making the slicing circle go 2/3rds the way
> across
> > some adjacent cell edges (which simultaneously puts it 1/3rd the way
> across
> > some incident cell edges). This feels like a nice puzzle to me, with a
> > pretty star pattern in the middle of each cell. You can do the same
> thing
> > on the {3,5} icosahedron, but in that case, the cuts are not evenly
> spaced
> > along an edge and the star patten is not quite as regular.
>
> This way of slicing {3,5} should be precisely Gelatinbrain's (2.1.4):
> http://users.skynet.be/gelatinbrain/Applets/Magic%20Polyhedra/icosa_f1.gif
>
> (3) In {5,5}, when I increase the size of circles close to the maximum
> value, suddenly all the cuts jump to the outside of the hyperbolic
> plane...... Is there a particular reason or just a bug?
>
> (4) Roice said:
> > - There is a very cool midpoint slicing of the {3,4}. The doubled-up
> > slicing circles form a cuboctahedron, so this is a case where things do
> fit
> > together quite nicely.
>
> This puzzle is the Skewb Diamond (a shape mod of Skewb). It's interesting
> that it can be viewed as a cuboctahedron.
>
> (5) A {6,3} puzzle with large circles is simulated by Gelatinbrain (7.1.1,
> 7.1.2, 7.1.3, with different repeating patterns).
>
> In general, I'd like to see some of the puzzles of this kind, especially in
> the hyperbolic plane. Since there is no macro function, I don't think I can
> solve puzzles with too many small pieces. Brandon, I'm not that crazy...
>
> Nan
>
>
>

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Thank you Nan, for the feedback and positive comments.

I've had some fun with the slicing program, it makes some pretty shapes and=
patterns. I would love for arbitrary slices which can be created to be pl=
ayable as puzzles, though this would probably be a lot of work to implement=
. One thought I have is that Jaap's Sphere also allows slices parallel to =
the faces of a rhombic dodecahedron to be used, equivalent to the edges of =
a cube/octahedron (similarly for the triacontahedron), and maybe interestin=
g puzzles could be created by a similar means for hyperbolic tilings, by ro=
tating about edges. Another is to use the "fudging" concept (see http://ww=
w.shapeways.com/model/204721/futtminx.html?gid=3Dug13603 for one example, t=
hough the idea is a little different here) started by Oscar van Deventer to=
tweak the geometry and remove small pieces if so desired, if any ideas fro=
m this are implemented. No doubt people like Brandon and Nan will have no =
problems with these small pieces though.

--- In 4D_Cubing@yahoogroups.com, Roice Nelson wrote:

> Thanks for telling me about the Skewb Diamond. With the sizing newly
> available on the spherical puzzles, I see now you can do something analog=
ous
> with the {5,3}. The midpoint slicing that doubles up slicing circles
> results in a circle set that is an icosidodecahedron (looks like this is =
the
> "Pentultimate"). The result on the dual {3,5} looks a little stranger,
> having some hexagonal facets (It's pretty to have both these turned on at
> the same time). I don't know what shape the latter is, but it's not in
> wiki's list of Archimedean solids, so maybe those hexagons aren't regular=
or
> are skew or something.

2.1.5 in Gelatinbrain shows the result for the {3,5}, although 1.2.9 may be=
easier to follow as in the dual shape of the {5,3} it looks identical to a=
megaminx, the hexagons corresponding to the corners there, if that result =
helps any. The applets don't seem to be loading for me just now, so I apol=
ogise if I mention the wrong puzzles, but the result is the same.=20=20

Matt