Message #2460

From: Melinda Green <melinda@superliminal.com>
Subject: Re: [MC4D] MagicTile Solving
Date: Sun, 04 Nov 2012 20:28:35 -0800

Well yes, all graphs are 2D, and it is far simpler to do these sorts of
things in 2D when possible. Even so it is a very difficult problem to
find any hidden symmetries and structure within large graphs unless you
already know something about them. I recently corresponded with Michael
Anttila who caught my eye with a nice page describing the "Devil’s
Algorithm <http://anttila.ca/michael/devilsalgorithm/>" which he
describes as sort of the opposite of God’s Algorithm. He had tried
various things to find structure within the 2^3 state graph but didn’t
get very far.

-Melinda

On 11/3/2012 12:45 PM, Eduard Baumann wrote:
>
>
> Interesting, but be aware that the Mathematica embeddings are
> two-dimensional !
> Ed
>
> —– Original Message —–
> *From:* Melinda Green <mailto:melinda@superliminal.com>
> *To:* 4D_Cubing@yahoogroups.com <mailto:4D_Cubing@yahoogroups.com>
> *Sent:* Saturday, November 03, 2012 7:55 PM
> *Subject:* Re: [MC4D] MagicTile Solving
>
> 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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