Several hours later, I have now found the difference between an octahedron and an icosahedron. I had been stuck on generating the coordinates of the octahedron. A little reading and experimentation directed my attention to the cube circumscribing the icosahedron. The way I’ve set things up, its vertices are all [±u, ±u, ±u], where u is t + 1. (And curiously, since t is the golden ratio, t + 1 is the same as t * t.)

An octahedron is a dual of a cube, meaning its six vertices are just the six centroids of the circumscribing cube. With those 6 vertices stitched together to form eight faces, I had an octahedron, from which I subtracted the icosahedron.