Message #1524

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Re: new hemi-cube and hemi-dodecahedron puzzles
Date: Mon, 07 Mar 2011 21:43:38 -0600

>
> On Mon, Mar 7, 2011 at 3:09 PM, Andrey wrote:
>
>> If you twist opposite sides in same direction (relative to sphee surface),
>> you’ll get orientable puzzle. I don’t know, if there are sphere paintings
>> that are invariant to this transformation (i.e. order of adjacent colors of
>> all intances of one face is the same). There is 5-color painting if
>> icosahedron (all faces of outscribed tetrahedra have the same color), but
>> I’m not sure that it will work.
>
>
Here is something else interesting I realized about the
"orientable hemi-cube" we’ve been talking about… In order for twists to
move in the same direction relative to the surface, and for it to still be
interpreted as a hemi-cube puzzle, the identification of 2C/3C piece pairs
must change after every twist! Without allowing this kind of wackiness,
which honestly doesn’t feel legit at all, I think the answer to Andrey’s
question about the existence of orientable periodic paintings for the sphere
is provably ‘no’ for any tiling on the sphere (regular, uniform, whatever).

Here is how the logic showing that could go… If an orientable painting was
possible, we are talking about choosing a subset of tiles, which when
repeated would cover the sphere (the "universal cover" of the chosen subset
would be the sphere). The topology of the surface represented by this
subset of tiles must admit elliptical geometry, but there are only two
topologies which do so, the sphere and the projective plane (see chapter 12
of Jeff Weeks’ "The Shape of
Space<http://www.amazon.com/gp/product/0824707095?ie=UTF8&tag=gravit-20&linkCode=as2&camp=1789&creative=390957&creativeASIN=0824707095>").
The latter is non-orientable, so the topology of our subset must be the
sphere. However, the universal cover of the sphere is the sphere itself
(see here <http://mathworld.wolfram.com/UniversalCover.html>), so the the
only set of tiles which will work is the full set of tiles, which already
covers the entire sphere. So there is no periodic painting possible.

Also, I have fixed the issue Nan found with the even-length hemi-cube
puzzles, and uploaded a new version. Thank you again Nan!

Roice