Equal volumes of regular polytopes with commensurable edges

Abstract

An equilateral triangle and a square can be rescaled to have the same area, but their side lengths cannot both be rational. We prove the corresponding statement for every pair of regular convex polytopes in a fixed dimension: equal volume and a rational ratio of edge lengths force congruence. For polygons, the proof uses a fractional linear expression for the cotangent that generates the full cyclotomic field. In higher dimensions, a prime valuation argument treats the simplex, cube and cross-polytope together, leaving only the exceptional three- and four-dimensional types. We also give a direct proof of a known rational-angle classification, apply it to algebraic areas of regular star polygons, and record an arithmetic criterion for equal numerical volumes in different dimensions.