Happy Holidays

What is this?

Besides possibly a festive tree ornament, it is a sneak preview of a new Rubik analogue program I've been playing with this past year but have yet to publish (update: I put a version of this online in late January). One can bring to light a huge set of puzzles by considering the original cube puzzle as a special case where the colored faces are a regular tiling of squares, and then abstracting by asking "What is the most precise analogue for other polygonal tilings?". For me, this simple question lead to all kinds of engrossing pathways, discoveries, and of course... many more questions.

We live on an island surrounded by a sea of ignorance. As our island of knowledge grows, so does the shore of our ignorance.

- John Wheeler

I saw that quote this week on John Baez's site, and ground zero on my personal island appears to be the Rubik's Cube.

Anyway, the picture above is a checkerboard pattern made by twisting up a puzzle based on a regular tiling of octagons, which requires hyperbolic geometry to fit together, naturally! The octagonal faces are delineated by yellow edges, and the black circles slice them up into stickers. This particular puzzle has 6 unique face colors which repeat in a certain pattern, and that pattern is interesting to study in itself. All the faces having the same center colors are actually identical and twist together. Kaleidoscopical to watch :D

Best wishes to all my readers in 2010. I love all three-or-so of you!