The Universe of Discourse

Sat, 17 Nov 2018

Yesterday Katara asked me out of nowhere “When you make a stella octangula, do you build it up from an octahedron or a tetrahedron?” Stella octangula was her favorite polyhedron eight years ago.

“Uh,” I said. “Both?”

Then she had to make one to see what I meant. You can start with a regular octahedron:

$a regular octahedron made of six steel ball bearings and twelve blue and green magnetic struts$

Then you erect spikes onto four of the octahedron's faces; this produces a regular tetrahedron:

Then you erect spikes onto the other four of the octahedron's faces. Now you have a stella octangula.

$The octahedron from before, but with red tripods attached to its other four faces, making eight tripods in all. The final result looks like interpenetrating red and orange tetrahedra, with the original octahedron in their intersection$

So yeah, both. Or instead of starting with a unit octahedron and erecting eight spikes of size 1, you can start with a unit tetrahedron and erect four spikes of size ½. It's both at once.

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