From the Icosahedron to E8

Here’s a draft of a little thing I’m writing for the Newsletter of the London Mathematical Society. The regular icosahedron is connected to many ‘exceptional objects’ in mathematics, and here I describe two ways of using it to construct $\mathrm{E}_8.$ One uses a subring of the quaternions called the ‘icosians’, while the other uses Patrick du Val’s work on the resolution of Kleinian singularities. I leave it as a challenge to find the connection between these two constructions!

From the Icosahedron to E8...