The perimeter and volume of a Reuleaux polyhedron

TL;DR

This paper investigates the geometric properties of Reuleaux polyhedra, focusing on calculating their perimeter and volume using advanced mathematical tools, and extends these methods to approximate the volume of Meissner polyhedra, which are bodies of constant width.

Contribution

It introduces a method to compute perimeter and volume of Reuleaux polyhedra using Gauss-Bonnet theorem and integral formulas, and adapts this approach for Meissner polyhedra.

Findings

Derived formulas for perimeter and volume of Reuleaux polyhedra.

Extended the method to approximate volumes of Meissner polyhedra.

Provided computational techniques for bodies of constant width.

Abstract

A ball polyhedron is the intersection of a finite number of closed balls in $\mathbb{R}^3$ with the same radius. In this note, we study ball polyhedra in which the set of centers defining the balls have the maximum possible number of diametric pairs. We explain how to compute the perimeter and volume of these shapes by employing the Gauss-Bonnet theorem and another integral formula. In addition, we show how to adapt this method to approximate the volume of Meissner polyhedra, which are constant width bodies constructed from ball polyhedra.