Neoplatonic solids
arXiv:2607.26363
The paper studies special triangulations of the sphere called 6-nets and shows they can be uniquely realized as Euclidean and hyperbolic polyhedra built from unit equilateral triangles, providing computational proofs for many cases.
Abstract
A \emph{6-net} is a simplicial triangulation of the -sphere with maximum degree . Experiments suggest that every -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 -net with 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 -nets with , counted up to combinatorial isomorphism.