undented

Enumerate, solve, and prove all prime neoplatonic solids.

A neoplatonic solid is an undented polyhedron with equilateral triangle faces, meeting at most six to a vertex. It is prime if it has no degree-3 vertex.

Quick start

make all VMAX=12       # everything through v=12 (~1 minute)

For large v, clone to a compute server:

make all VMAX=50 JOBS=96

Requires: C compiler, python3, numpy, scipy. GNU parallel is optional (used for faster sharded enumeration if present).

Pipeline

make seed-one  V=N      generate one seed file
make seeds     VMAX=N   generate seed files through N
make primes    VMAX=N   grow primes by recurrence: prime(v) = grow(prime(v-1)) ∪ seed(v)
make solve     VMAX=N   Euclidean embeddings via homotopy from ideal
make prove     VMAX=N   existence proofs (IEEE 754 + error bounds)
make check     VMAX=N   independent validation (dent, length, defect, embed)
make checkers           build the standalone check tools
make all       VMAX=N   everything
make status             show results

Seed and prime semantics

• seed(6) = the octahedron (base case)
• seed(v) = empty for v < 6 and 7 ≤ v < 12
• seed(v) = fullerene duals via buckygen for v ≥ 12
• prime(4) = the tetrahedron (exceptional, does not seed the recurrence)
• prime(5) = empty
• prime(v) = grow(prime(v-1)) ∪ seed(v) for v ≥ 6

What the prover checks

0. Undented — every vertex turning sum > 0
1. Embedded — no triangle-triangle intersections
2. 3|V| ≥ |E|
3. Collision distance > 0 — non-adjacent simplices separated
4. σ_min > 0 and ρ < σ_min²/(16√E) — rigidity + near solution
5. Perturbation bound < CD/√V — solution stays embedded

See PROOF_METHOD.txt for details.

Layout

src/                    source code
  clers.c                 CLERS tool: decode, encode, name, canonical
  clers.py                CLERS decoder (Python, for scripting)
  grow_step.c             vertex insertion → offspring names
  neoeuc_c.c              homotopy solver (ideal → Euclidean, with polish)
  prove.c                 existence prover (C, LAPACK SVD, IEEE 754 error bounds)
  prove_float.py          existence prover (Python, legacy)
  plantri_to_poly         planar_code → face-list converter
  dent_check.c            min link turning (undentedness)
  length_check.c          max edge length deviation
  defect_check.c          min angle defect
  embed_check.cpp         self-intersection test (requires CGAL, optional)
  check_all.c             runs dent + length + defect in one pass
third_party/buckygen/   Brinkmann's fullerene generator (GPL)
data/                   generated data (gitignored)
run/                    scratch and logs (gitignored)
PROOF_METHOD.txt        why the float prover is rigorous