Contents

group theory

# Contents

## Idea

…one of the regular polytopes in dimension 4…

…hence a higher dimensional analog of the Platonic solids

## Definition

The 24-cell is the regular polyhedron in the Cartesian space/Euclidean space $\mathbb{R}^4$ whose vertices are, under the identification $\mathbb{R}^4 \simeq_{\mathbb{R}} \mathbb{Q}$ with the space of quaternions, the 8 unit quaternions $\pm 1$, $\pm i$, $\pm j$, $\pm k$ and the 16 unit quaternions given by $\frac1{2}(\varepsilon_0 1 + \varepsilon_1 i + \varepsilon_2 j + \varepsilon_3 k)$ where $(\varepsilon_0, \ldots, \varepsilon_3) \in \{-1, 1\}^4$.

(These 24 quaternions form a group under quaternion multiplication, and this group is isomorphic to the binary tetrahedral group.)

## Properties

### Symmetry group

The finite rotation group inside O(4) which is the symmetry group of the 24-cell is the Coxeter group F4.