## Icosahedron, Part 2

*December 31, 2017 by Chris Johnson. Filed under madeup, public.*

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.

As Dr. Ward used to say, “Viola!” (sic):

## Leave a Reply