Folklore, the Borromean rings, the icosahedron, and three dimensions

There is a relationship between the Borromean rings, the icosahedron and something called the Poincar\'e homology sphere. This relationship is explored in a wandering path that introduces fundamental ideas from topology and a geometric construction of an icosahedral compound of octahedra. This exploration results in proofs that the orientation-preserving symmetry group of an icosahedron is the alternating group of five symbols, the fact that the Borromean rings are linked, and background related to the Poincar\'e conjecture. This is an exposition of known results aimed at undergraduates.