A 6-net is a simplicial triangulation of the 2-sphere with maximum degree at most 6. Experiments suggest that every 6-net has a realization, unique up to isometry, as an undented Euclidean polyhedron built from equilateral triangles of side length 1, and as an ideal equilateral hyperbolic polyhedron. We call such realizations neoplatonic solids and ideal neoplatonics.
A net is prime if every triangular circuit bounds a face. Non-prime neoplatonics are obtained from prime ones by elementary gluing operations. A computer-assisted proof shows that every prime 6-net with v ≤ 50 (8,239,684 in all) has a realization as a convex ideal neoplatonic, unique among convex realizations. These realizations continue via numerical homotopy to approximate realizations as Euclidean neoplatonics, and a computer-assisted proof shows that a true realization lies nearby.
The canonical name of a neoplatonic is its lex-first CLERS encoding (CBEAD variant, with L→B, R→A, S→D).