Krystal Guo on Nostr: Sorry, one of the messages didn't send. So, the squished-looking ones are because I ...

Sorry, one of the messages didn't send. So, the squished-looking ones are because I computed the eigenvalues using SageMath's default, which tries very hard to output an integer eigenvector, which is why the entries are nice. But they're not orthogonal or anything. Now, I found the eigenvalues with linalg from scipy. Now, the eigenvectors for sqrt(5) produce the icosahedron and those for -sqrt(5) produce the great icosahedron. Each of these eigenspaces happens to have dimension 3, so they produce coordinates in 3D.

The map swapping sqrt(5) and -sqrt(5) is the (only) non-trivial Galois automorphism of the splitting field of the eigenvalues, so this map takes an eigenvector for sqrt(5) to an eigenvector of -sqrt(5), because the adjacency matrix has rational (in fact, integer) entries, which are invariant under all Galois automorphisms.