Forty-seven

Here’s a nice one, 47S_3 from Stewart’s Chapter VII ‘Exploration of (R)(A) toroids’. Take an octahedron and attach a pair of octahedrons to each face:(That’s 17 octahedra, one in the center and two in each of the eight arms.) That makes something roughly cube shaped. Now make a kind of pyramid of five octahedra:and you can attach one of these to each face of the cube. That’s 30 more octahedra for a total of 47, and it looks like this:If you stare at it long enough you can see each of the resulting twelve holes is in the center of a rhombus — there’s a rhombic dodecahedron underlying all this:Spin her up.

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