Message #2457

From: Melinda Green <melinda@superliminal.com>
Subject: Re: [MC4D] MagicTile Solving
Date: Sat, 03 Nov 2012 11:55:31 -0700

The face adjacency graphs are the duels of the vertex graphs that you
can decode from the WRL files. Note also that for all vertices in each
IRPs are geometrically identical, and not just topologically identical.
That means that using only affine transformations (rotate and translate
only) you can take any vertex to any other vertex and end up with the
same structure. That is not the case for most or all of Roice’s
hyperbolic puzzles.

To properly examine them in 3D, you don’t want to just load a cell file
into Cortona, instead you should use MT for the IRPs that Roice supports
so far and toggle Settings > Skew Polyhedra > Show as Skew. For the rest
of the IRPs you should load them from my table
<http://superliminal.com/geometry/infinite/infinite.htm> because in both
programs you can interactively add and remove tilings in the X, Y, and Z
directions. In MT the keys are x, y, and z for removing layers and X, Y,
and Z to add them. Stereo viewing is very helpful once you learn to view
them that way.

Regarding your graphs, various programs can help you to relax them but
you may need to interact with them to find nicely symmetrical views. I
wrote some code to do something like that a long time ago when studying
flexible polyhedra but it is currently hard-coded to deal with just one
model. You can try it here
<http://superliminal.com/geometry/flexible/applet/applet.htm>. These are
hard problems.

-Melinda

On 11/3/2012 10:35 AM, Eduard Baumann wrote:
>
>
> Results of my _color graph study for MT irp {4,5} 30_.
> First the _adjacency list of b30_:
> 1 2 3 4 5
> 2 1 6 7 8
> 3 1 7 9 10
> 4 1 7 11 12
> 5 1 7 13 14
> 6 2 9 9 15
> 7 2 3 4 5
> 8 2 13 13 16
> 9 3 6 6 17
> 10 3 11 11 18
> 11 4 10 10 19
> 12 4 14 14 20
> 13 5 8 8 21
> 14 5 12 12 22
> 15 6 24 23 25
> 16 8 26 27 28
> 17 9 23 25 29
> 18 10 26 27 29
> 19 11 24 26 27
> 20 12 23 25 28
> 21 13 26 27 30
> 22 14 23 25 30
> 23 15 17 20 22
> 24 15 19 30 30
> 25 15 17 20 22
> 26 16 18 19 21
> 27 16 18 19&n bsp;21
> 28 16 20 29 29
> 29 17 18 28 28
> 30 21 22 24 24
> It is interesting that here we have _12_ vertices which have _doubled
> neighbours_ (6, 8-14,24 and 28-30). Then also we have two _pairs_ of
> vertices which have _same_ neighbours (1+7) and (26+27).
> The _adjacency list of a30_ had none of these specialities.
> My try to embed a30 gave the following. I hope the uploaded pictures
> to wiki are linkable.
> http://wiki.superliminal.com/wiki/File:Color_graph_a30_manual.PNG
> And now Mathematica helping me:
> http://wiki.superliminal.com/wiki/File:Color_graph_ab30_Mtica.PNG
> The spring embedding procedure is certainly performant but the graphs
> to be shown are complex and not very regular.
> Regards
> Ed
>
> —– Original Message —–
> *From:* Melinda Green <mailto:melinda@superliminal.com>
> *To:* 4D_Cubing@yahoogroups.com <mailto:4D_Cubing@yahoogroups.com>
> *Sent:* Friday, November 02, 2012 11:52 PM
> *Subject:* Re: [MC4D] MagicTile Solving
>
> Ah, I missed the ‘6’, thank you for the correction. This is one of
> the 3 IRPs that are as perfectly symmetric as the Platonic solids
> in every way. It is also the IRP twin of the original Rubik’s
> cube. I would still like to know why Nan’s solution is so much
> shorter.
>
> I also do not understand why you see the IRP 4-5 b30 f001 as a
> warm-up exercise to the IRP {4,5} a30 F 0:0:1. True they both have
> 30 colors and genus 4, but they have different symmetries which I
> would guess would make the ‘a’ puzzle the simpler of the two.
>
> -Melinda
>
> On 11/2/2012 2:05 PM, Eduard Baumann wrote:
>> Wait.
>> The similar puzzle I mentioned is
>> NOT
>> MT irp {4,5} a30 F 0:0:1
>> BUT
>> MT irp {4,6} 12 F 0:0:1
>> I will attack
>> MT irp {4,5} a30 F 0:0:1
>> next time but I wanted study before he color topology of a30 and b30.
>> Ed
>>
>> —– Original Message —–
>> *From:* Melinda Green <mailto:melinda@superliminal.com>
>> *To:* 4D_Cubing@yahoogroups.com
>> <mailto:4D_Cubing@yahoogroups.com>
>> *Sent:* Friday, November 02, 2012 9:53 PM
>> *Subject:* Re: [MC4D] MagicTile Solving
>>
>> {4,5} a30 is one of my favorite IRPs. I find it to be quite beautiful and symmetric. It is the one that I showcase on themain geometry page <http://superliminal.com/geometry/geometry.htm> to introduce the subject. (Third image down.) The ‘b’ puzzle that surprised you is less symmetric but is still a fascinating structure. It looks very much like an apartment complex. I would like to know why Nan was able to solve it with such a smaller number of twists. Unless your macros are extremely long, it doesn’t seem like that can be the only difference. What do you think, Nan?
>>
>> -Melinda
>>
>> On 11/2/2012 11:17 AM, Eduard wrote:
>>> Solving of MT irp {4,5} b30 F 0:0:1 —– || 11/02/2012 || 2393
>>>
>>> Remark:
>>> Over 2000 twists. I worked without macros this time. Not low hanging fruit. Here 30 colors. In the similar puzzle "irp 4-6 12 f001" with 12 colors I worked with macros and needed 21’000 twists (Nan only 400 !!).
>>
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