Coloring the 600 Cell

TL;DR

This paper explores the structure of the 600-cell's colorings, revealing its connection to Latin squares and constructing a complex with specific symmetries and quotient spaces.

Contribution

It establishes a detailed relationship between the 600-cell's colorings and Latin squares, and constructs the associated coloring space with symmetry considerations.

Findings

The 600-cell has exactly 10 5-colorings.

The coloring space has 1344 colorings and is isomorphic to the space of 5x5 Latin squares.

The complex decomposes into four quotient spaces and two spaces of even Latin squares.

Abstract

The 600 cell S has exactly 10 5-colorings. From these colorings we can construct the space of colorings $B(S)$. This complex has 1344 colorings, and is isomorphic to the space of 5 by 5 Latin Squares. These simplices split into 4 copies of a quotient of S by an involution, and two copies of a space made up of even Latin Squares.