Message #1319

From: Andrey <andreyastrelin@yahoo.com>
Subject: Re: [MC4D] 8Colors solved
Date: Fri, 31 Dec 2010 20:55:58 -0000

>
> Is this what you were looking for? (I didn’t understand the phrasing
> "shifts in one direction, half-side shifts in other directions…")
>
Not exactly… If you turn this picture around some cell by 90 degrees, then picture will be changed (but some colors remain same) - so this painting is not invariant wrt this rotation. It can be used for non-oriented puzzle, but not as non-oriented face of the puzzle
> 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.

Yes, 57-cell and 11-cell are polytopes with non-oriented faces, so they can be used for the puzzle. And they don’t mention 4-cell because it has only one vertex (like your 3-color paintings in Magic Tile) that is not good for the polytope :)
>
> > And it will be interesting to look for non-orientable paintings of
> > "bitruncated cubic honeycomb" (made of trucated octahedra) :)
>
>
> I agree :)
>
but faces will be oriented :(

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Happy New Year!