# Sums and Products of Regular Polytopes' Squared Chord Lengths

## TL;DR

This paper investigates whether known relationships involving squared chord lengths of regular polygons extend to all regular polytopes, finding that some do, with varying degrees of generality, and discovering related corollaries and theorems.

## Contribution

It generalizes certain chord length relationships from regular polygons to higher-dimensional regular polytopes, identifying which facts hold universally and which are limited to specific families.

## Key findings

- One fact generalizes to all regular polytopes of dimension ≥ 2.
- Another fact generalizes to all regular polytopes except most simplices.
- A third fact generalizes only to crosspolytopes and the 24-cell.

## Abstract

Although previous research has found several facts concerning chord lengths of regular polytopes, none of these investigations has considered whether any of these facts define relationships that might generalize to the chord lengths of all regular polytopes. Consequently, this paper explores whether four findings of previous studies-- viz., the four facts relating to the sums and products of squared chord lengths of regular polygons inscribed in unit circles-- can be generalized to all regular ($n$-dimensional) polytopes (inscribed in unit $n$-spheres). We show that (a) one of these four facts actually does generalize to all regular polytopes (of dimension $n \geq 2$), (b) one generalizes to all regular polytopes except most simplices, (c) one generalizes only to the family of crosspolytopes, and (d) one generalizes only to the crosspolytopes and 24-cell. We also discover several corollaries (due to reciprocation) and some theorems specific to the three-dimensional regular polytopes along the way.