2026/07/29 by Peter Doyle, Matthew Ellison · 1 voice
Mathematics · #math.CO #math.MG
arxiv created 2026/07/29 · arxiv updated 2026/07/30
A 6-net is a simplicial triangulation of the 2-sphere with maximum degree ≤ 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 neoplatonic solids and ideal neoplatonics. A net is prime if every 3-cycle bounds a face. A computer-assisted proof shows that every prime 6-net with v ≤ 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≤50, counted up to combinatorial isomorphism.