# 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.