There are 4 compact regular honeycombs in 3D hyperbolic space: a honeycomb of dodecahedra which is its own self-dual, a honeycomb of icosahedra which is its own self-dual, a honeycomb of dodecahedra and a honeycomb of cubes which are duals of each other. Investigate these for puzzles and games, almost certainly in a virtual setting.
Cubes connecting in weird ways could be a basis for a 3D maze or collection of rooms. Incidentally this would also work for 2D hyperbolic tiling of squares.
Multiple dodecahedra or multiple icosahedra glued to each other could be a basis for monstrously complicated put-together puzzles. Pieces based on the 3D Euclidean cubic honeycomb can be rotated in (only) 24 different ways. Icosahedral symmetry allows 60 different orientations.
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