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Rubik's Klein Quartic

Originally shared by Layra Idarani

Rubik's Klein Quartic
And other shapes

The usual Rubik's Cube is a cube cut up into smaller cubes, called "cubies", identical to each other. The basic move on a Rubik's cube is to rotate an nxnxn1 layer of cubies around its central point on an axis parallel to the short side of the layer. The interesting behavior comes from the fact that the layers intersect.
Suppose we inflate the cube like a balloon. The overall shape becomes a sphere, and the cuts that separate piece from piece and layer from layer become rounded. In fact, just assume that the cuts are circular, like latitude lines not centered at the North or South pole. Solving this sphere is just the same as solving a regular Rubik's cube; we didn't change the coloring or the allowable moves or how the layers intersect, we just changed the shapes of the individual pieces.
Now we've got the setup for a fairly general "twisting puzzle", just a sphere with a …