Message #2462
From: Eduard Baumann <baumann@mcnet.ch>
Subject: Re: [MC4D] MagicTile Solving
Date: Mon, 05 Nov 2012 09:23:02 +0100
Yes, very nice describing !
Thanks.
Ed
—– Original Message —–
From: Melinda Green
To: 4D_Cubing@yahoogroups.com
Sent: Monday, November 05, 2012 5:28 AM
Subject: Re: [MC4D] MagicTile Solving
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" 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 ----- <br>
From: Melinda Green <br>
To: 4D_Cubing@yahoogroups.com <br>
Sent: Saturday, November 03, 2012 7:55 PM<br>
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 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. 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:<br>
1 2 3 4 5<br>
2 1 6 7 8<br>
3 1 7 9 10<br>
4 1 7 11 12<br>
5 1 7 13 14<br>
6 2 9 9 15<br>
7 2 3 4 5<br>
8 2 13 13 16<br>
9 3 6 6 17<br>
10 3 11 11 18<br>
11 4 10 10 19<br>
12 4 14 14 20<br>
13 5 8 8 21<br>
14 5 12 12 22<br>
15 6 24 23 25<br>
16 8 26 27 28<br>
17 9 23 25 29<br>
18 10 26 27 29<br>
19 11 24 26 27<br>
20 12 23 25 28<br>
21 13 26 27 30<br>
22 14 23 25 30<br>
23 15 17 20 22<br>
24 15 19 30 30<br>
25 15 17 20 22<br>
26 16 18 19 21<br>
27 16 18 19&n bsp;21<br>
28 16 20 29 29<br>
29 17 18 28 28<br>
30 21 22 24 24<br>
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.<br>
http://wiki.superliminal.com/wiki/File:Color_graph_a30_manual.PNG
And now Mathematica helping me:<br>
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<br>
Ed
—– Original Message —–
From: Melinda Green
To: 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 <br>
NOT<br>
MT irp {4,5} a30 F 0:0:1<br>
BUT<br>
MT irp {4,6} 12 F 0:0:1
I will attack <br>
MT irp {4,5} a30 F 0:0:1<br>
next time but I wanted study before he color topology of a30 and b30.
Ed
----- Original Message ----- <br>
From: Melinda Green <br>
To: 4D_Cubing@yahoogroups.com <br>
Sent: Friday, November 02, 2012 9:53 PM<br>
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 the main geometry page 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 !!).