Eitan’s Nebula, a 3D-printed puzzle by Eitan Cher. The puzzle features icosahedral geometry with cuts at two different depths and has a whopping 743 parts (including internal mechanism) and 920 stickers. Because of the nature of the face-turning icosahedral geometry, the puzzle can jumble. Jumbling is a somewhat difficult term to define, but it’s presence can be seen in the shape-shifting moves in the last picture (notice the corner that is not where it “should” be). “Normal” moves are rational rotations. For example, image 3 shows a 120-degree counterclockwise rotation of the top face (and clockwise on the smaller layer below it), or simply 1/3 of a full turn. At this position (”stop”), all moves that were available previously are still available. The geometry has not changed. With a jumbling move, a particular stop allows some moves but not others. Cuts line up in mysterious ways and unexpected moves are possible. In the 5th image, the pink face was turned less than 120 degrees before a move on the black face became possible (6th image). Some pieces were “not ready yet” and got moved along “upside down,” resulting in the shape-shifted appearance. In fact, the pink face was rotated by exactly arccos((-1+3√5)/8) = 44.47751219... degrees. Across all types of puzzles with all sorts of geometries, it seems that jumbling angles are always irrational. Curious. To see in action, check out a video of Eitan showing off his creation here.