The Geometry of $H_4$ Polytopes

TL;DR

This paper explores the geometric structure of 24-cell arrangements within the 600-cell and applies these findings to relate the $E_8$ lattice to a 4-dimensional space over $f{F}_4$, revealing new geometric correspondences.

Contribution

It introduces a detailed geometric analysis of $H_4$ polytopes and demonstrates their application to transforming the $E_8$ lattice into a 4-space over $f{F}_4$.

Findings

Describes the arrangement of 24-cells in the 600-cell.

Shows how the 600-cell relates to the $E_8$ lattice.

Establishes a correspondence between geometric objects and algebraic structures.

Abstract

We describe the geometry of an arrangement of 24-cells inscribed in the 600-cell. In $\S$7 we apply our results to the even unimodular lattice $E_8$ and show how the 600-cell transforms $E_8$/2$E_8$, an 8-space over the field $\bf{F}$$_2$, into a 4-space over $\bf{F}$$_4$ whose points, lines and planes are labeled by the geometric objects of the 600-cell.