Neoplatonic solids

Abstract

A \emph{6-net} is a simplicial triangulation of the $2$-sphere with maximum degree $\leq 6$. Experiments suggest that every $6$-net admits a unique realization as an undented Euclidean polyhedron built from unit equilateral triangles, and a unique realization as an ideal equilateral hyperbolic polyhedron. We call these \emph{neoplatonic solids} and \emph{ideal neoplatonics}. A net is \emph{prime} if every 3-cycle bounds a face. A computer-assisted proof shows that every prime $6$-net with $v \leq 50$ has a unique realization as a convex ideal neoplatonic. Numerical homotopy from this realization yields an approximate Euclidean neoplatonic, and a computer-assisted proof shows that a true Euclidean neoplatonic lies nearby, though we do not prove uniqueness. Using the separating-triangle decomposition, we extend Euclidean existence to all $10{,}412{,}340$ $6$-nets with $v\leq50$, counted up to combinatorial isomorphism.