Methods & resources

This site is an open, fully reproducible archive of best-known configurations for the Heilbronn problem. Every figure, table, and data download is compiled directly from exact coordinate records. All data, search tools, and exact verification pipelines are open source in tejstead/heilbronn-site, meaning any result on this site can be verified independently on your own machine.

Where the coordinates come from

• Community contributions — new configurations and exact algebraic proofs are submitted via pull requests (see CONTRIBUTING). Continuous integration computes all triangle areas in exact rational arithmetic and posts an automated verification report on the PR. Recent record submissions include contributions from Nathan Sudermann-Merx, Rhys Chappell, and Chouaieb Nemri.
• This site's own search campaigns — records for n = 17…36 across the variants, found with the toolkit under search/ and submitted through the same lane as everyone else's.
• cnemri/heilbronn-alphaevolve — Chouaieb Nemri's AlphaEvolve-evolved search program (square n = 17, 21, 22 records); the evolved program's architecture also powers several of the later batch records.
• spiralulam/heilbronn (MIT) — companion repository to Nathan Sudermann-Merx's certified-optimality papers: square n = 3…16, including the previously unpublished configurations of Peter Karpov and Mark Beyleveld. His original coordinates for later entries were also submitted here directly.
• rhyschappell/heilbronn-n14-exact — Rhys Chappell's exact algebraic realization of the n = 14 square configuration; he has since contributed further records and the degree-15 exact value for square n = 24.
• google-deepmind/alphaevolve_results — the original AlphaEvolve constructions (triangle n = 11, convex n = 13, 14).
• Published exact constructions from the proofs (see the bibliography).
• Local reconstruction: for configurations whose coordinates were never published (mostly David Cantrell's), we re-derive them by numerical optimization seeded from the record figures, and accept a reconstruction only if its exact value and symmetry match the record entry. These are labeled reconstructed and never claim to be the original author's exact arrangement.

How New Records Are Discovered

Finding candidate configurations requires navigating high-dimensional, non-convex landscapes where local optima proliferate rapidly. The repository includes an optimization toolkit (under search/) employing several complementary strategies:

• Basin-hopping & Local Search (attack.py): Alternates random coordinate perturbations with local minimization to escape shallow basins.
• Successive Linear Programming (refine.py): Uses trust-region SLP polishing with Karush-Kuhn-Tucker (KKT) tightening to drive candidate coordinates to machine precision.
• Symmetry Restriction (sym.py): Restricts point placements to candidate point groups (such as dihedral and cyclic symmetries), drastically reducing the dimension of the search space.
• Laddering & Seeding: Bootstraps an n-point configuration by strategically inserting an additional point into the (n-1) optimum and re-optimizing.
• Evolved Heuristics: Several recent records (e.g. square n = 17, 21, and 22) originated from heuristic search algorithms generated by DeepMind's AlphaEvolve system, executed and refined by community contributors.

Exact Algebraic Values

Whenever a configuration has an exact closed form, we determine its minimal polynomial through one of three paths: formal optimality proofs from the literature, polynomials provided by contributors (validated to 45 digits at build time), or direct symbolic derivation. For derived values, we identify the tightest triangle constraints, reduce them by the configuration's symmetry group, and solve the polynomial tie system using Gröbner bases followed by linear programming for any interior slack points.

This process yielded the degree-5 quintic for triangle n = 15 and the cubics for square n = 10 and 12. In cases where a polynomial's Galois group is not solvable by radicals, algebraic theory prohibits any closed form in terms of nested roots; for those entries, we report the root of the minimal polynomial directly.

Verification

Every configuration on this site is checked with exact rational arithmetic: the coordinates' decimal literals are taken exactly, all C(n,3) triangle areas are enumerated, and the reported value is the exact minimum, normalized to a unit-area container. Two independent verifiers (search/verify_a.py, search/verify_b.py) agree on every entry; the build runs a library form of the first, submissions are re-verified by CI, and the in-browser verifier runs the same computation.

Normalization conventions

The container has unit area. Beware when comparing with papers: work in the unit right triangle (area ½) quotes triangle values half as large, and the retired circle variant used a unit-radius disk (area π). The triangle problem is affine-invariant, so coordinates are stored in the right frame (0,0),(1,0),(0,1) and displayed equilateral.

Historical Origin & Attribution

These record tables build on the work of Erich Friedman, whose Packing Center cataloged Heilbronn configurations for decades before going offline in 2026. We preserved his historical records, attribution notes, and symmetry classifications in data/curated/records.json. All figures on this site are generated anew from verified coordinate sets. Individual record holders and proof credits are tracked on the leaderboard.

Bibliography

• M. Goldberg, Maximizing the smallest triangle made by N points in a square, Mathematics Magazine 45(3) (1972) 135–144
• L. Yang, J. Z. Zhang, Z. B. Zeng, A conjecture on the first several Heilbronn numbers and a computation, Chinese Ann. Math. Ser. A 13 (1992) 503–515
• A. Dress, L. Yang, Z. Zeng, Heilbronn problem for six points in a planar convex body, in Minimax and Applications, Springer (1995) 173–190
• F. Comellas, J. L. A. Yebra, New lower bounds for Heilbronn numbers, Electron. J. Combin. 9 (2002) #R6
• Z. Chen, Z. Zeng, On the Heilbronn optimal configuration of seven points in the square, Automated Deduction in Geometry, LNAI 6301, Springer (2011) 196–224
• L. Chen, Z. Zeng, W. Zhou, An upper bound of Heilbronn number for eight points in triangles, J. Comb. Optim. 28(4) (2014) 854–874
• L. Chen et al., Searching approximate global optimal Heilbronn configurations of nine points in the unit square via GPGPU computing, J. Global Optim. (2016)
• L. Dehbi, Z. Zeng, Heilbronn's problem of eight points in the square, J. Syst. Sci. Complex. (2022)
• AlphaEvolve (Google DeepMind), 2025 — new configurations for triangle n=11 and convex n=13, 14
• N. Sudermann-Merx, Certified global optimality and exact coordinates for Heilbronn configurations in the square (2026)
• N. Sudermann-Merx, The Heilbronn problem in triangles: boundary structure and certified optima for n ≤ 8 (2026)
• A. Cohen, C. Pohoata, D. Zakharov, A new upper bound for the Heilbronn triangle problem (2023)
• Monji, Modir, Kocuk, Solving the Heilbronn triangle problem using global optimization methods (2025)
• OEIS A343851 — decimal expansion of the n=7 Heilbronn constant for the square
• OEIS A379533 — (√13−1)/36, the n=8 Heilbronn constant for the square
• OEIS A379534 — (9√65−55)/320, the n=9 Heilbronn constant for the square
• E. Friedman, The Heilbronn problem for squares — Packing Center record table (offline since 2026)
• E. Friedman, The Heilbronn problem for triangles — Packing Center record table (offline since 2026)
• E. Friedman, The Heilbronn problem for convex regions — Packing Center record table (offline since 2026)
• MathWorld — Heilbronn Triangle Problem
• Wikipedia — Heilbronn triangle problem
• spiralulam/heilbronn — square configurations n ≤ 16, exact coordinates, optimization models (MIT)
• google-deepmind/alphaevolve_results — published AlphaEvolve constructions
• L. Yang, Z. Zeng, Heilbronn problem for seven points in a planar convex body, in Minimax and Applications, Springer (1995) 191–218
• Z. Zeng, L. Chen, Determining the Heilbronn configuration of seven points in triangles via symbolic computation, CASC 2019, LNCS 11661, Springer (2019) 458–477
• R. Chappell, Exact realization of the best-known n=14 Heilbronn configuration (2026), github.com/rhyschappell/heilbronn-n14-exact
• T. Stead, Convex Heilbronn values and optimizers for n = 3–8, Lean 4 source and proof guide (2026), github.com/tejstead/heilbronn-site
• T. Stead, heilbronn-convex-3-8, Palomar registry PALOMAR-2026-09-02-000012 v1 (Lean 4 formalization, machine-checked), 2 September 2026