Tuesday, September 26, 2017

[kdjykabg] Go on a sphere

Playing go 囲碁 on the surface of a cube is the easiest way to transport the game to a topologically spherical manifold while preserving the square-lattice feel of the original game.  Other ways are possible: subdivide the rhombic faces of a rhombic dodecahedron, or similarly any polyhedron whose faces are all quadrilaterals (preferably parallelograms).

Inspired by HEALpix, which subdivides a sphere into 12 squarish regions, then subdivides those "squares".

Given a polyhedron with all triangular faces, join pairs of adjacent faces forming non-planar quadrilaterals, which can then be subdivided into topological square lattices.  Slightly tricky in preferring pairs forming convex quadrilaterals, though nonconvex can be subdivided with curved lines.  There might be a few triangles left over representing topological defects where the game might be more interesting (similar to fights for a corner or edge in the original game).  Of course, if a face has more than three sides, it can be triangulated.

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