Octonion multiplication and Heawood's map

TL;DR

This paper demonstrates how the octonion multiplication table can be derived from the geometric structure of Heawood's map, a regular tessellation of a torus by seven hexagons.

Contribution

It introduces a novel geometric approach to recover octonion multiplication from a specific tessellation of the torus.

Findings

Octonion multiplication can be derived from Heawood's map.

The geometric structure encodes algebraic properties of octonions.

Provides a new perspective linking geometry and algebra in higher dimensions.

Abstract

In this note, the octonion multiplication table is recovered from a regular tesselation of the "equilateral" two dimensional torus by seven hexagons, also known as Heawood's map.