Message #1008

From: Melinda Green <melinda@superliminal.com>
Subject: Re: [MC4D] Magic Tiles
Date: Thu, 15 Jul 2010 17:23:58 -0700

Andrey wrote:
> I’ve started to play with Magic Tiles from octagon, 6 colors and nonagons, 4 colors. I can says that it’s a very nice joke! Thank you for it, Roice!
> For heptagons, 24 colors I used the method that I developed for 120 cell first: to select some set of centers, put in place all pieces between them (call these centers "solved"), then select some center adjacent to others and put in place all pieces between it and solved centers. Just don’t rotate solved centers in this method (only in "macros") and be careful to keep area of unsolved ceneters connected. When one face remained I just used operations from 3^3 (it’s not most effective, but I didn’t need to think).
>

Your method of solving the {7,3} (AKA the Klein quartic
<http://math.ucr.edu/home/baez/klein.html>) is not guaranteed to work.
Nelson Garcia was the first to point out the possible "two bottoms"
<http://games.groups.yahoo.com/group/4D_Cubing/message/837> problem that
can result which lead to a nice discussion of topology and genus. In
short, the top-down method is only guaranteed to work on surfaces
topologically equivalent to the sphere (I.E. genus == 0). All of the
puzzles that we’ve dealt with up till then had this property, but the
Klein quartic has genus == 3.

> Nice puzzle!
> What do we know about finite factor-lattices of dodecahedral honeycomb?

I’m not sure what you mean by "finite factor-lattices of dodecahedral
honeycomb". That sounds to me like simply the 120-cell. There are
certainly twisty puzzles that can be defined on honeycomb lattices. The
duel of the {7,3} is the {3,7}
<http://www.superliminal.com/geometry/infinite/3_7a.htm> which is
naturally defined as a finite tiling in a repeating space. These tile
puzzles of Roice’s can have such 3D polygonal equivalents and are
therefore mathematically quite interesting.

-Melinda