From Pentagons to Pentagrams

I recently showed you that if you take the regular icosahedron: considered in a coordinate system based on the golden ratio, and then replace √5 by -√5 in all your formulas, you get the great icosahedron: But this fact isn’t an isolated one-off! If we do the same for the regular dodecahedron: we get the great stellated dodecahedron: There’s also a star polyhedron called the great dodecahedron: and if we play the same game, replacing by in all the formulas, we get the small stellated dodecahedron: These six polyhedra form a family; the four nonconvex ones are called the Kepler–Poinsot polyhedra. I never understood what was so great about them, though of course they look ravishingly attractive. So it was nice to learn that if we include the convex ones, they come in three pairs related by the operation of replacing by which is called Galois conjugation. This is mentioned near the end of this book: • John Horton Conway, Heidi Burgiel and Chaim Goodman-Strauss, The Symmetries of Things, A…