Message #1393

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Interesting object
Date: Tue, 08 Feb 2011 11:05:12 -0600

Wow, I pose a question before falling asleep and wake up to two great
answers. Thanks! I’m grateful for both. Andrey’s modular arithmetic
approach seems like it can be very helpful with the code calculations that
will need to be done, and Melinda’s explanation provides a great deal of
insight into the connections between faces.

With Melinda’s description, I think I know why my existing coloring rule
could not generate this tiling. It is because the pattern arises from these
hyperbolic translations of 6
ultraparallel<http://en.wikipedia.org/wiki/Ultraparallel#Non-intersecting_lines>lines
(with 4 faces on each line), whereas my tiling rule is based on
patterns that arise in a rotationally symmetric way. I need to study
further, but if I’m right, one can not do a 1/8th, face-centered rotation of
a 24 colored {8,3} - it does not appear to be a symmetry of the object.

A coincident aside related to these hyperbolic translations is that I found
out yesterday I missed a class of checkerboards on KQ which are due to
symmetries of this kind. You can have six translational 4-cycles of KQ,
much like what we are discussing here (KQ is a more symmetric object though,
having the rotational symmetries as well). There is no analogous
checkerboard of Megaminx, since there are no parallel lines in spherical
geometry. I’ll likely make the actual checkerboard this weekend, and will
email if I do. It will indeed be the last possible type, because I found
this out reading part of this
paper<http://library.msri.org/books/Book35/files/karcher.pdf>(see
p32), which enumerates all the possible KQ symmetries.

And on the mystery puzzle, that’s exactly it Melinda. The image turned up
unexpectedly when I was playing with some code this weekend, and I was so
pleased that it looked very similar to the stereo graphically projected
Megaminx. You can do the same inversion on KQ to get a similar looking
picture of that as well.

seeya,
Roice

P.S. I hope this email came across ok, as the formatting of my last
one seems to be all messed up on the yahoo group site. Gmail and Yahoo
groups appear to be a little incongruous lately.