Thread: "Edge turning {3,7} IRP solved"

Nan found some double colors early on in the defaults, which=A0were co=
rrected.=A0 Looks like this latest one must be color 7, "Cyan" an=
d color 27, "Aqua".=A0 Different names, but visually they look id=
entical to me.=A0 I'll try to pick a better default for color 27, and p=
lease let me know if these weren't the ones that caused the confusion.<=
/div>



--e89a8f22bd89474b3804bfc5ae8b--

---

From: Melinda Green <melinda@superliminal.com>
Date: Fri, 11 May 2012 12:47:33 -0700
Subject: Re: [MC4D] Re: Edge turning {3,7} IRP solved

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Yes, that was it. I also adjusted another pair that was distinct but
very close. Maybe orange and 255,128,0?

You may like to look at the generateVisuallyDistinctColors method I
wrote for MC4D which I'm rather proud of. It finds N colors as different
from each other as possible in the YUV color space and with the
restriction that within each color, at least one component of its RGB
equivalent is greater than a given minimum and one that is less than a
given maximum. One of the best things about using a dynamically
generated list is that those cases that need fewer colors (the common
cases) will have more freedom to choose more contrasting colors than
those which require more.

-Melinda

On 5/11/2012 10:01 AM, Roice Nelson wrote:
>
>
> Nan found some double colors early on in the defaults, which were
> corrected. Looks like this latest one must be color 7, "Cyan" and
> color 27, "Aqua". Different names, but visually they look identical
> to me. I'll try to pick a better default for color 27, and please let
> me know if these weren't the ones that caused the confusion.
> Thanks!
> Roice
>
> On Fri, May 11, 2012 at 2:16 AM, Melinda Green
> > wrote:
>
>
>
> Oh, so that's what you meant! Roice had reminded me about your
> "aha" but we thought you might have meant something else. So it's
> just a bug in the default colors? It sure drove me batty for a
> while but I'm glad that I thought to check the colors.
>
> Your puzzle was the edge turning {4,4|3}, one of the "Harlequin"
> puzzles which are really the direct cousin of this one. By the
> way, toggle the "show as skew" setting for this one. It's really
> quite amazing.
>
> -Melinda
>
>
> On 5/10/2012 11:47 PM, Eduard Baumann wrote:
>> The doubled cyan color! This was my "AHA" I mentioned enigmaticly
>> in the comment of a smaller case.
>
>
>
>
>
>

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Yes, that was it. I also adjusted another pair that was distinct but
very close. Maybe orange and 255,128,0?



You may like to look at the generateVisuallyDistinctColors method I
wrote for MC4D which I'm rather proud of. It finds N colors as
different from each other as possible in the YUV color space and
with the restriction that within each color, at least one component
of its RGB equivalent is greater than a given minimum and one that
is less than a given maximum. One of the best things about using a
dynamically generated list is that those cases that need fewer
colors (the common cases) will have more freedom to choose more
contrasting colors than those which require more.



-Melinda



On 5/11/2012 10:01 AM, Roice Nelson wrote:
cite="mid:CAEMuGXrbQY=+ymm+Bqtj8E_psqQZ-BHbfTnzoPZUG5WNkfAvnQ@mail.gmail.com"
type="cite">

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---

From: "schuma" <mananself@gmail.com>
Date: Sat, 12 May 2012 08:28:31 -0000
Subject: Re: Edge turning {3,7} IRP solved

I like the feature of "randomly generating colors". It would be useful in s=
olving. An example is the situation that Melinda met. She knew two colors a=
long an orbit are identical or very similar, but to find out which two colo=
rs took some time. If there's a "randomize colors" button, those two colors=
will pop out immediately.=20

I have also solved this puzzle (IRP {3,7} 56-color E0:1:0). I solved it in =
the IRP view.=20

There are 168 small pieces that actually move. I THOUGHT they were divided =
into 7 orbits, each of which has 24 pieces. But I was wrong. There are four=
shorter orbits, each with 6 pieces. Sum of them =3D 24. So it seems there =
are 6 longer orbits with length 24, and 4 shorter orbits with length 6.=20

The four shorter orbits are as follows (I'm writing only the first two colo=
rs because the rest are determined)
1. triangles with colors 8 - 18 - ...
2. triangles with colors 20 - 9 - ...
3. triangles with colors 46 - 53 - ...
4. triangles with colors 29 - 25 - ...

In the image of this page:
http://www.superliminal.com/geometry/infinite/3_7a.htm
The four short orbits are around the "necks" colored by blue, red, green an=
d a blocked (thus unknown) color. So the faces and edges are actually not e=
quivalent. I consider it a generalized "deltahedron"
[http://en.wikipedia.org/wiki/Deltahedron]

After I read more about the infinite polyhedra/skew polyhedra on wikipedia,=
I found Coxeter and Petrie considered the most stringent regularity, just =
like "regular" polyhedra. And there are only three "regular infinite skew p=
olyhedra".=20

Gott relaxed it to include more shapes, including {3,7}. "Where Coxeter and=
Petrie had required that the vertices be symmetrical, Gott required only t=
hat they be congruent" [http://en.wikipedia.org/wiki/Infinite_skew_polyhedr=
on].=20

On this page:
http://www.superliminal.com/geometry/infinite/infinite.htm
"Regular polyhedra are those that are composed of only one type of regular =
polygon", therefore no requirement on vertices.=20

There are more IRP puzzles, e.g., {4,5} 30C, with non-equivalent faces and =
maybe orbits with different lengths, waiting for us to discover.

Nan


--- In 4D_Cubing@yahoogroups.com, Melinda Green wrote:
>
> Yes, that was it. I also adjusted another pair that was distinct but=20
> very close. Maybe orange and 255,128,0?
>=20
> You may like to look at the generateVisuallyDistinctColors method I=20
> wrote for MC4D which I'm rather proud of. It finds N colors as different=
=20
> from each other as possible in the YUV color space and with the=20
> restriction that within each color, at least one component of its RGB=20
> equivalent is greater than a given minimum and one that is less than a=20
> given maximum. One of the best things about using a dynamically=20
> generated list is that those cases that need fewer colors (the common=20
> cases) will have more freedom to choose more contrasting colors than=20
> those which require more.
>=20
> -Melinda
>=20
> On 5/11/2012 10:01 AM, Roice Nelson wrote:
> >
> >
> > Nan found some double colors early on in the defaults, which were=20
> > corrected. Looks like this latest one must be color 7, "Cyan" and=20
> > color 27, "Aqua". Different names, but visually they look identical=20
> > to me. I'll try to pick a better default for color 27, and please let=
=20
> > me know if these weren't the ones that caused the confusion.
> > Thanks!
> > Roice
> >
> > On Fri, May 11, 2012 at 2:16 AM, Melinda Green=20
> > > wrote:
> >
> >
> >
> > Oh, so that's what you meant! Roice had reminded me about your
> > "aha" but we thought you might have meant something else. So it's
> > just a bug in the default colors? It sure drove me batty for a
> > while but I'm glad that I thought to check the colors.
> >
> > Your puzzle was the edge turning {4,4|3}, one of the "Harlequin"
> > puzzles which are really the direct cousin of this one. By the
> > way, toggle the "show as skew" setting for this one. It's really
> > quite amazing.
> >
> > -Melinda
> >
> >
> > On 5/10/2012 11:47 PM, Eduard Baumann wrote:
> >> The doubled cyan color! This was my "AHA" I mentioned enigmaticly
> >> in the comment of a smaller case.
> >
> >
> >
> >
> >
> >
>

---

From: Melinda Green <melinda@superliminal.com>
Date: Sat, 12 May 2012 02:51:44 -0700
Subject: Re: [MC4D] Re: Edge turning {3,7} IRP solved

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On 5/12/2012 1:28 AM, schuma wrote:
> I like the feature of "randomly generating colors". It would be useful in solving. An example is the situation that Melinda met. She knew two colors along an orbit are identical or very similar, but to find out which two colors took some time. If there's a "randomize colors" button, those two colors will pop out immediately.

What a great idea, Nan! I was only suggesting that Roice consider just
using my random function once per puzzle but I really like your idea to
let the user randomize at any time!

Roice: I've also noticed that the settings sometimes all reset as a
result of switching puzzles which seems like a small bug. It may have
something to do with using two instances of MT at the same time. I'm not
sure.

>
> I have also solved this puzzle (IRP {3,7} 56-color E0:1:0). I solved it in the IRP view.

Now how the heck did you do that, Nan?! And using less than at third of
the twists that I used! I love the 3D geometry of this one probably more
than any other IRP but it just seemed impossible to solve it in that
mode. Your result seems like you did it one-handed and blindfolded. How
did you go about that and was it like to solve it that way?

>
> There are 168 small pieces that actually move. I THOUGHT they were divided into 7 orbits, each of which has 24 pieces. But I was wrong. There are four shorter orbits, each with 6 pieces. Sum of them = 24. So it seems there are 6 longer orbits with length 24, and 4 shorter orbits with length 6.

That's pretty cool, isn't it? This was one of the features that I want
to emphasize in that sculpture I mentioned earlier.

>
> The four shorter orbits are as follows (I'm writing only the first two colors because the rest are determined)
> 1. triangles with colors 8 - 18 - ...
> 2. triangles with colors 20 - 9 - ...
> 3. triangles with colors 46 - 53 - ...
> 4. triangles with colors 29 - 25 - ...
>
> In the image of this page:
> http://www.superliminal.com/geometry/infinite/3_7a.htm
> The four short orbits are around the "necks" colored by blue, red, green and a blocked (thus unknown) color. So the faces and edges are actually not equivalent. I consider it a generalized "deltahedron"
> [http://en.wikipedia.org/wiki/Deltahedron]
>
> After I read more about the infinite polyhedra/skew polyhedra on wikipedia, I found Coxeter and Petrie considered the most stringent regularity, just like "regular" polyhedra. And there are only three "regular infinite skew polyhedra".
>
> Gott relaxed it to include more shapes, including {3,7}. "Where Coxeter and Petrie had required that the vertices be symmetrical, Gott required only that they be congruent" [http://en.wikipedia.org/wiki/Infinite_skew_polyhedron].
>
> On this page:
> http://www.superliminal.com/geometry/infinite/infinite.htm
> "Regular polyhedra are those that are composed of only one type of regular polygon", therefore no requirement on vertices.

That's my page and I'm surprised that I didn't really mention vertices.
For the definition that I prefer, all faces must be the same regular
polygon and all vertices must be transitive. The one constraint that I
relax from true regularity is that the edges do not need to be identical
which implies that the faces aren't necessarily transitive. I should
clarify that.

>
> There are more IRP puzzles, e.g., {4,5} 30C, with non-equivalent faces and maybe orbits with different lengths, waiting for us to discover.

I was in the middle of solving the edge turning version of that one when
your message came in! :-)

There is also at least one puzzle in which paths loop around and crosses
itself kind of like the shape of the number 6. Because of that you can
move a petal from one corner of a triangle, sending it out around its
orbit only to come back in such a way that you can place it into a
different corner of the triangle that it came from. The orbit must loop
around three times like that like a Celtic triangle
. I'll
let you figure out which puzzle that is. :-)

-Melinda


>
> --- In 4D_Cubing@yahoogroups.com, Melinda Green wrote:
>> Yes, that was it. I also adjusted another pair that was distinct but
>> very close. Maybe orange and 255,128,0?
>>
>> You may like to look at the generateVisuallyDistinctColors method I
>> wrote for MC4D which I'm rather proud of. It finds N colors as different
>> from each other as possible in the YUV color space and with the
>> restriction that within each color, at least one component of its RGB
>> equivalent is greater than a given minimum and one that is less than a
>> given maximum. One of the best things about using a dynamically
>> generated list is that those cases that need fewer colors (the common
>> cases) will have more freedom to choose more contrasting colors than
>> those which require more.
>>
>> -Melinda
>>
>> On 5/11/2012 10:01 AM, Roice Nelson wrote:
>>>
>>> Nan found some double colors early on in the defaults, which were
>>> corrected. Looks like this latest one must be color 7, "Cyan" and
>>> color 27, "Aqua". Different names, but visually they look identical
>>> to me. I'll try to pick a better default for color 27, and please let
>>> me know if these weren't the ones that caused the confusion.
>>> Thanks!
>>> Roice
>>>
>>> On Fri, May 11, 2012 at 2:16 AM, Melinda Green
>>> > wrote:
>>>
>>>
>>>
>>> Oh, so that's what you meant! Roice had reminded me about your
>>> "aha" but we thought you might have meant something else. So it's
>>> just a bug in the default colors? It sure drove me batty for a
>>> while but I'm glad that I thought to check the colors.
>>>
>>> Your puzzle was the edge turning {4,4|3}, one of the "Harlequin"
>>> puzzles which are really the direct cousin of this one. By the
>>> way, toggle the "show as skew" setting for this one. It's really
>>> quite amazing.
>>>
>>> -Melinda
>>>
>>>
>>> On 5/10/2012 11:47 PM, Eduard Baumann wrote:
>>>> The doubled cyan color! This was my "AHA" I mentioned enigmaticly
>>>> in the comment of a smaller case.

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On 5/12/2012 1:28 AM, schuma wrote:

I like the feature of "randomly generating colors". It would be useful in solving. An example is the situation that Melinda met. She knew two colors along an orbit are identical or very similar, but to find out which two colors took some time. If there's a "randomize colors" button, those two colors will pop out immediately. 

What a great idea, Nan! I was only suggesting that Roice consider
just using my random function once per puzzle but I really like your
idea to let the user randomize at any time!



Roice: I've also noticed that the settings sometimes all reset as a
result of switching puzzles which seems like a small bug. It may
have something to do with using two instances of MT at the same
time. I'm not sure.

I have also solved this puzzle (IRP {3,7} 56-color E0:1:0). I solved it in the IRP view. 

Now how the heck did you do that, Nan?! And using less than at third
of the twists that I used! I love the 3D geometry of this one
probably more than any other IRP but it just seemed impossible to
solve it in that mode. Your result seems like you did it one-handed
and blindfolded. How did you go about that and was it like to solve
it that way?

There are 168 small pieces that actually move. I THOUGHT they were divided into 7 orbits, each of which has 24 pieces. But I was wrong. There are four shorter orbits, each with 6 pieces. Sum of them = 24. So it seems there are 6 longer orbits with length 24, and 4 shorter orbits with length 6. 

That's pretty cool, isn't it? This was one of the features that I
want to emphasize in that sculpture I mentioned earlier.

The four shorter orbits are as follows (I'm writing only the first two colors because the rest are determined)
1. triangles with colors 8 - 18 - ...
2. triangles with colors 20 - 9 - ...
3. triangles with colors 46 - 53 - ...
4. triangles with colors 29 - 25 - ...

In the image of this page:
http://www.superliminal.com/geometry/infinite/3_7a.htm
The four short orbits are around the "necks" colored by blue, red, green and a blocked (thus unknown) color. So the faces and edges are actually not equivalent. I consider it a generalized "deltahedron"
[http://en.wikipedia.org/wiki/Deltahedron]

After I read more about the infinite polyhedra/skew polyhedra on wikipedia, I found Coxeter and Petrie considered the most stringent regularity, just like "regular" polyhedra. And there are only three "regular infinite skew polyhedra". 

Gott relaxed it to include more shapes, including {3,7}. "Where Coxeter and Petrie had required that the vertices be symmetrical, Gott required only that they be congruent" [http://en.wikipedia.org/wiki/Infinite_skew_polyhedron]. 

On this page:
http://www.superliminal.com/geometry/infinite/infinite.htm
"Regular polyhedra are those that are composed of only one type of regular polygon", therefore no requirement on vertices. 

That's my page and I'm surprised that I didn't really mention
vertices. For the definition that I prefer, all faces must be the
same regular polygon and all vertices must be transitive. The one
constraint that I relax from true regularity is that the edges do
not need to be identical which implies that the faces aren't
necessarily transitive. I should clarify that.

There are more IRP puzzles, e.g., {4,5} 30C, with non-equivalent faces and maybe orbits with different lengths, waiting for us to discover.

I was in the middle of solving the edge turning version of that one
when your message came in! :-)



There is also at least one puzzle in which paths loop around and
crosses itself kind of like the shape of the number 6. Because of
that you can move a petal from one corner of a triangle, sending it
out around its orbit only to come back in such a way that you can
place it into a different corner of the triangle that it came from.
The orbit must loop around three times like that like a href="http://www.clker.com/cliparts/z/d/7/8/q/C/celtic-triangle-md.png">Celtic
triangle. I'll let you figure out which puzzle that is.  :-)



-Melinda

--- In 4D_Cubing@yahoogroups.com, Melinda Green <melinda@...> wrote:

Yes, that was it. I also adjusted another pair that was distinct but 
very close. Maybe orange and 255,128,0?

You may like to look at the generateVisuallyDistinctColors method I 
wrote for MC4D which I'm rather proud of. It finds N colors as different 
from each other as possible in the YUV color space and with the 
restriction that within each color, at least one component of its RGB 
equivalent is greater than a given minimum and one that is less than a 
given maximum. One of the best things about using a dynamically 
generated list is that those cases that need fewer colors (the common 
cases) will have more freedom to choose more contrasting colors than 
those which require more.

-Melinda

On 5/11/2012 10:01 AM, Roice Nelson wrote:

Nan found some double colors early on in the defaults, which were 
corrected.  Looks like this latest one must be color 7, "Cyan" and 
color 27, "Aqua".  Different names, but visually they look identical 
to me.  I'll try to pick a better default for color 27, and please let 
me know if these weren't the ones that caused the confusion.
Thanks!
Roice

On Fri, May 11, 2012 at 2:16 AM, Melinda Green 
<melinda@... <mailto:melinda@...>> wrote:



    Oh, so that's what you meant! Roice had reminded me about your
    "aha" but we thought you might have meant something else. So it's
    just a bug in the default colors? It sure drove me batty for a
    while but I'm glad that I thought to check the colors.

    Your puzzle was the edge turning {4,4|3}, one of the "Harlequin"
    puzzles which are really the direct cousin of this one. By the
    way, toggle the "show as skew" setting for this one. It's really
    quite amazing.

    -Melinda


    On 5/10/2012 11:47 PM, Eduard Baumann wrote:

    The doubled cyan color! This was my "AHA" I mentioned enigmaticly
    in the comment of a smaller case.

--------------060507040204090400020009--

---

From: "schuma" <mananself@gmail.com>
Date: Sat, 12 May 2012 18:08:44 -0000
Subject: Re: Edge turning {3,7} IRP solved

Hi Melinda,

I made a short video explaining my strategy:

http://www.youtube.com/watch?v=3D5DrPDEyKMBg


Nan

--- In 4D_Cubing@yahoogroups.com, Melinda Green wrote:
> Now how the heck did you do that, Nan?! And using less than at third of=20
> the twists that I used! I love the 3D geometry of this one probably more=
=20
> than any other IRP but it just seemed impossible to solve it in that=20
> mode. Your result seems like you did it one-handed and blindfolded. How=20
> did you go about that and was it like to solve it that way?

---

From: Melinda Green <melinda@superliminal.com>
Date: Sat, 12 May 2012 14:06:06 -0700
Subject: Re: [MC4D] Re: Edge turning {3,7} IRP solved

Hello Nan,

I'm now very familiar with solving edge piece orbits now that I've
solved a few of these puzzles. The part that I do not understand is how
you did all that in the 3D view where it is difficult to follow a long
orbit.

I love how can solve these puzzles through pure intuition, but there
are enough things that I need to pause and think about that it is not
completely trivial. I don't play Sudoku but I imagine that the mental
effort is similar.

-Melinda

On 5/12/2012 11:08 AM, schuma wrote:
> Hi Melinda,
>
> I made a short video explaining my strategy:
>
> http://www.youtube.com/watch?v=5DrPDEyKMBg
>
>
> Nan
>
> --- In 4D_Cubing@yahoogroups.com, Melinda Green wrote:
>> Now how the heck did you do that, Nan?! And using less than at third of
>> the twists that I used! I love the 3D geometry of this one probably more
>> than any other IRP but it just seemed impossible to solve it in that
>> mode. Your result seems like you did it one-handed and blindfolded. How
>> did you go about that and was it like to solve it that way?
>
>
> ------------------------------------
>
> Yahoo! Groups Links
>
>
>
>

---

From: "Eduard Baumann" <baumann@mcnet.ch>
Date: Sun, 13 May 2012 00:08:13 +0200
Subject: Re: Edge turning {3,7} IRP solved

------=_NextPart_000_0018_01CD309C.7D2D88E0
Content-Type: text/plain;
charset="iso-8859-1"
Content-Transfer-Encoding: quoted-printable

Superbe this communication by video ! :-)
------=_NextPart_000_0018_01CD309C.7D2D88E0
Content-Type: text/html;
charset="iso-8859-1"
Content-Transfer-Encoding: quoted-printable



>

Superbe this communication by video ! =
;=20
:-)

------=_NextPart_000_0018_01CD309C.7D2D88E0--

---

From: "schuma" <mananself@gmail.com>
Date: Mon, 14 May 2012 05:08:23 -0000
Subject: Re: Edge turning {3,7} IRP solved

Hi Melinda,

I can see {3,9}a 24-Color E0:1:0 is close, but not exact. There, an orbit c=
an visit a triangle twice, from two directions, but not three times. I have=
n't found a similar property on other {3,n} puzzles. But, for those puzzles=
where the faces are not equivalent, I cannot exclude them unless I have tr=
ied all the faces.=20

Can you reveal the answer now?

Nan

--- In 4D_Cubing@yahoogroups.com, Melinda Green wrote:
> There is also at least one puzzle in which paths loop around and crosses=
=20
> itself kind of like the shape of the number 6. Because of that you can=20
> move a petal from one corner of a triangle, sending it out around its=20
> orbit only to come back in such a way that you can place it into a=20
> different corner of the triangle that it came from. The orbit must loop=20
> around three times like that like a Celtic triangle=20
> . I'll=
=20
> let you figure out which puzzle that is. :-)

---

From: Melinda Green <melinda@superliminal.com>
Date: Mon, 14 May 2012 00:55:32 -0700
Subject: Re: [MC4D] Re: Edge turning {3,7} IRP solved

I think you mean the {3,9}a 36-Color (there is no 24-color version), and
yes, that is the right answer. You are also right that each orbit only
intersects itself once, not three times. That happens in a figure-8
pattern. This gives it the strange property of allowing you to swap a
single edge in isolation. When I saw that state during my solve I
naturally thought that I had another duplicate color hiding somewhere
but searching for it turned up nothing. I then figured that I would
probably have to use the fact that its orbit self-intersects in order to
solve it and sure enough that worked.

This puzzle is an especially lovely and symmetric object and closely
related to the {3,7}. I think you can really see it as a kind of union
of two {3,7}'s. You can build a {3,7} by starting with an icosahedron
and attaching 4 octahedra to 4 of its faces along tetrahedral axes in a
pattern of carbon bonds in a diamond lattice. That uses up 4 of each
icosahedron's 20 faces. Well you can make this {3,9} by adding yet
another 4 octrahedra in another diamond lattice. It is therefore much
denser and difficult to see well in MT 3D mode but rotating it around or
viewing in stereo will show you what I mean.

So here is a new challenge: Find the shortest way to flip a single edge
of this lovely IRP.
Enjoy!
-Melinda

On 5/13/2012 10:08 PM, schuma wrote:
> Hi Melinda,
>
> I can see {3,9}a 24-Color E0:1:0 is close, but not exact. There, an orbit can visit a triangle twice, from two directions, but not three times. I haven't found a similar property on other {3,n} puzzles. But, for those puzzles where the faces are not equivalent, I cannot exclude them unless I have tried all the faces.
>
> Can you reveal the answer now?
>
> Nan
>
> --- In 4D_Cubing@yahoogroups.com, Melinda Green wrote:
>> There is also at least one puzzle in which paths loop around and crosses
>> itself kind of like the shape of the number 6. Because of that you can
>> move a petal from one corner of a triangle, sending it out around its
>> orbit only to come back in such a way that you can place it into a
>> different corner of the triangle that it came from. The orbit must loop
>> around three times like that like a Celtic triangle
>> . I'll
>> let you figure out which puzzle that is. :-)
>
>
>
> ------------------------------------
>
> Yahoo! Groups Links
>
>
>
>

---

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