# Message #1317

From: Andrey <andreyastrelin@yahoo.com>

Subject: Re: [MC4D] 8Colors solved

Date: Fri, 31 Dec 2010 05:54:55 -0000

Roice,

do you know any Klein bottle-like periodic painting of square or hexagonal tiling of plane that has cells or two kinds (left- and right-handed) for every color? I try to find such thing, but results have very small group of movements: for example, classic model of Klein bottle (square with connected opposite sides) enable all shifts in one direction, half-side shifts in other directions and two families of 180-deg rotations - that is the factor of movement group of infinite rectangular prizm. There are no 90-deg rotations in this model. I wonder if we can design "douprizm" based on this type of connections.

Another way to make non-orientable puzzle is to take projective-plane-like faces (3-colored cube of 6-colored dodecahedron). 3-colored cube generates puzzle that looks like 3^4 with the same color of opposite sides and with special movements (when you twist some cube, opposite cube also is twisted somehow). Puzzle has 4 faces, 6 2Cs, 4 3Cs and one 4C cubie. I don’t know if there is painting of 120cell that gives regular puzzle made of 6 color half-dodecahedra.

And it will be interesting to look for non-orientable paintings of "bitruncated cubic honeycomb" (made of trucated octahedra) :)

Andrey