Message #1318

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] 8Colors solved
Date: Fri, 31 Dec 2010 13:36:31 -0600

inlines below :)

On Thu, Dec 30, 2010 at 11:54 PM, Andrey <andreyastrelin@yahoo.com> wrote:

> 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.
>

I think there are many possibilities, but I just made a pictorial example of
one in the hexagonal case and uploaded it
here<http://www.gravitation3d.com/magictile/pics/klein_bottle_example.png>.
That would be a 9-color, non-orientable puzzle if it were sliced up. For
any cell, the copies above/below would twist in the same direction. The
copies that are three columns to the left/right would twist in an opposite
direction. Rotations of a face would be 60 degrees, just as in the other
MagicTile hexagonal puzzles.

Is this what you were looking for? (I didn’t understand the phrasing
"shifts in one direction, half-side shifts in other directions…")

> 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.
>

Regarding a "puzzle made of 6 color half-dodecahedra", it sounds like what
you are heading towards is the 57-cell<http://en.wikipedia.org/wiki/57-cell>,
which is a four dimensional "abstract" regular polytope composed of
hemi-dodecahedra. I’m unsure if this could be visualized as a painting of
the 120cell. If a puzzle were made based on the 57-cell, each
hemi-dodecahedron would be a solid color though (not 6 colors). The only
other polotype I’m aware of which is built up of hemi-objects is the
11-cell<http://en.wikipedia.org/wiki/11-cell>, made
from hemi-icosahedra, so if there is the possibility of a polytope composed
of hemi-cubes like you’re describing, that is news to me.

You can paint the 120cell into a non-orientable hemi-120cell (by simply
identifying antipodal cells), which could easily be presented using the
Magic120Cell interface. And you could create a hemi-8cell puzzle as well
(which is what I thought you were describing at first). But in both these
cases, cells would be full polyhedra, not hemi-polyhedra.

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

I agree :)

Happy New Year,
Roice