Message #1013

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Magic Tiles
Date: Fri, 16 Jul 2010 11:02:18 -0500

> > Melinda wrote:
> >
> > > I’m not sure what you mean by "finite factor-lattices of dodecahedral
> > > honeycomb". That sounds to me like simply the 120-cell.
> >
> >
> > I was wondering what Andrey meant by this as well. I thought maybe he
> was
> > asking about a hyperbolic dodecahedral
> > honeycomb<http://en.wikipedia.org/wiki/Order-4_dodecahedral_honeycomb>,
> > and whether you could cover the entire infinite space by coloring the
> cells
> > in a repeating manner using a finite number of colors. I haven’t seen
> any
> > info on whether this is possible (and what the corresponding topologies
> > might be), but I’d love to be pointed to it.
> >
> Yes, you are right. Question is about the periodic (in some sense)
> paintings of dodecahedral honeycomb. I can easily imagine one (in 2 colors -
> we have 4 dodecahedra meeting at each edge so checkerboard painting is
> possible) but is there something more?
> Another question is about splitting of dodecahedron in such puzzle, but we
> always have "non-geometric" variant based on megaminx splitting: when you
> twist a cell you catch 3 stickers from the edge of the cell that is
> connected by edge to yours and one corner sticker from the face that is
> connected by vertex, and don’t create extra sub-edge and sub-corner 1C
> stickers. Animation will be with intersections of stickers, but dodecahedra
> are round enough :)
>

Ah, yeah :) Maybe your checkerboard coloring thought easily leads to a few
more, namely a 4 color and 8 color puzzle. I’m not sure either works, but
it seems that as long as the number of colors is less than or equal to than
the number of cells meeting at a vertex, you have a chance of repeating the
patten in a vertex transitive way. In the 2D world for instance, I found
that one could always make a 3 color puzzle for even-sided polygons (colors
were equal to the number of cells meeting at a vertex). It feels like the
more interesting question would be if there is a painting in this 3D
hyperbolic space with more than 8 colors that would work, since that would
produce a puzzle where colors were not always adjacent to all the other
colors. My intuition on this is that it is not possible, but hopefully
we’ll see!

As far as the splitting, I feel the best analogue is to put a "slicing
sphere" at the center of each dodecahedron. These spheres are slightly
larger than the dodecahedra, and slice up the adjacent cells into stickers.
When you twist a cell, you move all the material inside its sphere, and *no
material overlaps* (which I think is a worthy design goal). However, since
the honeycomb doesn’t have a simplex vertex figure, we don’t get the simple
1C, 2C, 3C, 4C piece types - the stickers are not as neat. You could do it
like you described as well of course, leading to a distant relative of
"Impossiball", but using slicing spheres feels more elegant (especially
since the spherical puzzles supported by MC4D can be cast this way without
changing their nature).

Take Care,
Roice