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 nxnx1 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 bunch of cuts in it forming it into layers, where the basic move is to twist a layer so that the cuts line up.  Most twisting puzzles on the market are equivalent to this, with some funny shaping. For example, the Megaminx, essentially a Rubik's dodecahedron, is a sphere with twelve cuts, one for each face.
The interesting structure is not the fact that we've got a sphere, but that we've got all of these cuts making these intersecting layers. So forget the sphere. Consider a flat, infinite plane with a square grid on it. Color all of the squares different colors. For each square, draw a circle centered at the center of the square; the circle should be a little bit but only a little bit larger than the square it contains and all the circles should be the same size. Cut along the circles (but not the squares!). Consider the contents of each circle to be a "layer". We again have a twisting puzzle.
Okay, but this object is too big, i.e. infinite. So take our square grid and color it with six colors, arranged so that for any given square, the squares it shares edges with are all different colors from the given square and from each other. If we spin a layer whose center piece is a given color, we have to spin every layer whose center piece is that color in the same way. Now we really only have six independent layers, just like a regular Rubik's cube, only now they're arranged differently. Instead of six circles cut into a sphere, we have a six circles cut into what would be a torus if we were allowed to glue each square of a given color to all other squares of that color. A Rubik's torus?

Puzzle 1: How are the six colors arranged?

Puzzle 2: There's a way to tile a torus with five square "faces". On the plane this translates into coloring a square grid with five colors so that so that for any given square, the squares it shares edges with are all different colors from the given square and from each other. How are the colors arranged on the grid?

Puzzle 3: How many colors are needed for the square grid so that for any square, the squares it shares edges /and corners/ with are all different colors from the given square and from each other?

Similarly we can take a tiling of the hyperbolic plane by heptagons, pick 24 colors, and arrange the colors so that gluing heptagons of the same color gives us a tiling of Klein quartic surface by heptagons. Cutting the appropriate circles then gives us a Rubik's Klein quartic.

We can also make nonorientable Rubik's things, and Rubik's things in higher dimensions (with n-spherical cuts in n+1-dimensional shapes of constant curvature). The important bit seems to be that the curvature should be constant so that rotating a layer preserves the overall shape.

Puzzle 4:  Are there extra necessary conditions for Rubik's orbifolds?

Linkies:

MagicTile by +Roice Nelson, the wonderful program that taught me all of this. Includes a Rubik's Klein Quartic.
Video: https://www.youtube.com/watch?x-yt-ts=1422040409&x-yt-cl=84637285&v=kN5_1vJbW9M
Site: http://www.gravitation3d.com/magictile/  
I really suggest playing with it, even if you can't solve a regular Rubik's cube.

Wikipedia on the Klein Quartic:
http://en.wikipedia.org/wiki/Klein_quartic

Twisty Puzzles. Check out their museum of twisty puzzles:
http://www.twistypuzzles.com/

#LayraExplains