On regular and equivelar Leonardo polyhedra

Abstract

A Leonardo polyhedron is a 2-manifold without boundary, embedded in Euclidean 3-space E³, built up of convex polygons and with the geometric symmetry (or rotation) group of a Platonic solid and of genus g ≥ 2. The polyhedra are named in honour of Leonardo's famous illustrations in [1] (cf. also [2]). Only six combinatorially regular Leonardo polyhedra are known: Coxeter's four regular skew polyhedra, and the polyhedral realizations of the regular maps by Klein of genus 3 and by Fricke and Klein of genus 5. In this paper we construct infinite series of equivelar (i.e. locally regular) Leonardo polyhedra, which share some properties with the regular ones, namely the same Schläfli symbols and related topological structure. So the weaker condition of local regularity allows a much greater variety of symmetric polyhedra.

[1] L. Pacioli, De Divina Proportione (Disegni di Leonardo da Vinci 1500-1503), Faksimile Dominiani, Como, 1967.
[2] D. Huylebrouk, Lost in triangulation: Leonardo da Vinci's mathematical slip-up, Scientific American, March 29, 2011.