The Icosidodecahedron

TL;DR

This paper explores the geometric and algebraic properties of the icosidodecahedron, linking it to higher-dimensional polytopes and symmetries, including connections to the E8 root polytope and quaternionic structures.

Contribution

It provides a detailed explanation of the icosidodecahedron's relation to higher-dimensional polytopes and their symmetries, highlighting connections to the E8 root system and quaternionic algebra.

Findings

The icosidodecahedron is related to higher-dimensional polytopes through projections.

Connections between the icosidodecahedron and the E8 root polytope are established.

The paper links geometric structures to quaternionic and icosian algebra.

Abstract

The icosidodecahedron has 30 vertices, one at the center of each edge of a regular icosahedron -- or equivalently, one at the center of each edge of a regular dodecahedron. It is a beautiful, highly symmetrical shape. But it is just a projection down to 3d space of a more symmetrical 6-dimensional polytope with 60 vertices. It is also a slice of a more symmetrical 4d polytope with 120 vertices, which in turn is the projection down to 4d space of an even more symmetrical 8-dimensional polytope with 240 vertices: the E8 root polytope. Here we explain all these constructions, and their connection to the quaternions and icosians.