Tuesday, June 06, 2017

[kcxvhsoi] 3D puzzles in hyperbolic space

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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