Mean distance in polyhedra

TL;DR

This paper derives exact formulas for the moments of the distance between two uniformly chosen points in any polyhedron, including explicit results for all Platonic solids, using elementary functions.

Contribution

It provides a general method to compute all moments of point-to-point distances in polyhedra, with explicit formulas for Platonic solids, advancing geometric probability calculations.

Findings

All moments of the distance can be expressed with elementary functions.

Exact mean distances are derived for all Platonic solids.

The approach simplifies distance calculations in polyhedral geometries.

Abstract

Given any polyhedron from which we select two random points uniformly and independently, we show that all the moments of the distance between those points can be always written in terms of elementary functions. As an illustration, the mean distance is found in the exact form for all Platonic solids.