Sunday, March 04, 2018

[omvgtkqp] Dodecahedral edge cover

Can one partition the faces of an icosahedron into 10 non-overlapping pairs of adjacent faces?  Some simple investigation suggests yes.  If so, one can set up a grid on each nonplanar rhombus by connecting opposite edges.  If distortion is OK (e.g., in a game), this can be a map with 10 square shaped regions connected to each other.

Are there any nicely symmetric face pairings?  Zonohedra, polyhedra with all faces parallelograms (the "more restrictive definition" according to George Hart) exist, including the nicely symmetric rhombic dodecahedron and rhombic triacontahedron.

Equivalently, this partitions the surface of a sphere into 10 congruent rhombus shaped patches.  Equivalently, it is an edge cover of the regular dodecahedral graph.

What happens when this is done on a octahedron?  There seem to be two ways of doing it: 4 wedges from pole to pole, or 2 sideways with respect to the other 2 reminiscent of a sphericon.

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