On 3 and 4 dimensional regular solids, Part 1: The 4-simplex generates the free group

TL;DR

This paper demonstrates that the vertices of a 4-simplex, represented as unit quaternions, generate a free group, introducing a new algebraic length concept and linking it to number theory through trace analysis.

Contribution

It introduces a novel alternating length for words in free groups and connects geometric properties of the 4-simplex to algebraic and number-theoretic concepts.

Findings

Vertices of the 4-simplex generate a free group.

The new alternating length can be computed via the trace's algebraic denominator.

A number-theoretic interpretation of the alternating length is established.

Abstract

The 4-simplex has vertices 5 unit quaternions, which we arrange so that one of them is the unit. We show that the remaining 4 vertices are the generators of a free group. For the proof, we introduce a new alternating length on words in free groups. We show that for words in simplex vertices the necklace form of the alternating length can be read number theoretically, as the logarithm of the algebraic denominator of their trace.