Message #2588
From: Eduard Baumann <ed.baumann@bluewin.ch>
Subject: Re: [MC4D] MagicTile Coloring
Date: Wed, 02 Jan 2013 00:17:07 +0100
Thanks very much.
Yes, I was aware that simply turning the prismes has no impact.
I didn’t read the articles about skew polyhedra before. Im glad that my desriptions of the "organisation" is correct (but evidently not complete).
For new colorings: it is now evident for me that finding such colorings is a very deep and delicate process.
I have finished now the deep cut runcinated 5-cell v020 for which the "theorem of Baumann" applies (no pieces stay at home). Counting more carefully the orbits (not an easy job) I had to reformulate the theorem.
The "theorem of Astrelin" applies to bitruncated 5-cell v200 where no pieces stay at home.
Theorem of Baumann old, erroneous
A 90-deg rotation of the whole puzzle "MT skew {4,6|3} 30 v020" (Roice Nelson) around some face gives odd permutation of the edges (15 4-loops) and even permutation of centers (6 4-loops + 1 fixed points). This is reducible to a single edge swap.
Theorem of Baumann new, corrected
A 90-deg rotation of the whole puzzle "MT skew {4,6|3} 30 v020" (Roice Nelson) around some face gives even permutation of the edges (14 4-loops and 2 2-loops) and odd permutation of centers (7 4-loops + 2 fixed points). This is reducible to a single center swap.
The edge 2-loops are about the two opposite colors of the rotation axis (here white and (192,192,0)).
Conclusion of the theorems: after scrambling avoid turning of the whole before you have fixed the home positions.
Best regards
Ed
—– Original Message —–
From: Roice Nelson
To: 4D_Cubing@yahoogroups.com
Sent: Tuesday, January 01, 2013 10:47 PM
Subject: Re: [MC4D] MagicTile Coloring
Hi Ed,
Yep, I’ve been following your posts and progress. Nice job on all your solves btw! Getting to the half-way mark would be a big milestone, and I hope you make it.
I may be missing something, but it seems that if you recolor one of the triangular prisms by cycling the 3 colors on it, the puzzle hasn’t really changed. So it seems to me that all of the colorings you are describing are equivalent. It is still a 30-faced puzzle with 30 colors, connected up with the same global topology. The edge sets in this puzzle had to be made to fit the topology of the {4,6|3} skew polyhedron, and changing the edge sets would change the topology (resulting in some other shape).
But maybe you are thinking something else. Are you talking about twisting up one of the triangular prisms and re-gluing, such that one triangle base remains unchanged and the other is rotated 60 degrees? If so, that would indeed be different, but the resulting shape wouldn’t be this skew polyhedron, and MagicTile can’t currently support something like this.
Here’s some links I used when making these two puzzles. They might be helpful for further study.
a.. For the {4,6|3}, the wikipedia page on the runcinated 5-cell.
b.. For the {6,4|3}, the wikipedia page on the bitruncated 5-cell.
c.. Also, see the section ‘Finite regular skew polyhedra of 4-space’ for other topology possibilities (unfortunately, most would have too many faces to make good puzzles).
Let me know if I’m on track with my understanding.
seeya,
Roice
On Mon, Dec 31, 2012 at 1:03 PM, Eduard <ed.baumann@bluewin.ch> wrote:
Hi Roice,
Have you seen my description of the organisation of "MT skew {4,6|3} 30 v020" ?<br>
Each 3-prismatique edge can be untwisted or twisted by +60° or -60°<br>
(separated from tetrahedron-vertex and reglued). So 10^3 different colorings can be constructed. That's a lot. Are some of them equivalent?<br>
Is it difficult to find the corresponding "edge-sets"?
Kind regards<br>
Ed
------------------------------------
Yahoo! Groups Links