Geometrical and Topological Analogues of Rubik's Cube

MagicTile is a program that abstracts the Rubik's Cube as a colored regular tiling of squares on the sphere. The familiar "cubeness" is lost, but all the important properties of the original puzzle remain. Once we have a nice analogue of a twist from this perspective, we can consider other colored regular tilings and a huge number of new twisty puzzles become possible, some living in the world of hyperbolic geometry!

Mathologer has made a wonderful introduction to MagicTile on YouTube, which includes a special challenge to solve THE Klein Bottle Rubik's Cube. If you are one of the first 100 to do so, your name will be listed in the Mathologer Challenge Hall of Fame.

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A quick overview of what MagicTile supports:
MagicTile is currently approaching one thousand configured twisty puzzles, with thousands more possible. The homebase for discussion of this program is the 4D_Cubing Yahoo group, and we track solutions on the Superliminal wiki.

The MagicTile code is available on GitHub.