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

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.