Title: From the Icosahedron to E8,
Abstract: The regular icosahedron is connected to many exceptional objects in
mathematics. Here we describe two constructions of the $\mathrm{E}_8$ lattice
from the icosahedron. One uses a subring of the quaternions called the
"icosians", while the other uses du Val's work on the resolution of Kleinian
singularities. Together they link the golden ratio, the quaternions, the
quintic equation, the 600-cell, and the Poincare homology 3-sphere. We leave it
as a challenge to the reader to find the connection between these two
constructions.