Methods & resources

Everything on this site is generated ahead of time from exact coordinate data; the pages you're reading are static files. The generator, data and deployment live in the site repository.

Where the coordinates come from

• TejSteadQC/heilbronn-configurations — coordinates for n = 17…35 in all three variants (and several records below that), each verified by two independent exact verifiers; also the search toolkit used to find and polish them (trust-region successive-LP polishing, KKT tightening, symmetry-restricted search).
• spiralulam/heilbronn (MIT) — square configurations for n = 3…16, the companion repository to Nathan Sudermann-Merx's certified-optimality work, including the previously unpublished configurations of Peter Karpov (n = 13, 15) and Mark Beyleveld (n = 14, 16).
• google-deepmind/alphaevolve_results — the 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 published figures, and accept a reconstruction only if its exact value and symmetry match the published entry. These are labeled reconstructed and never claim to be the original author's exact arrangement.

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. The in-browser verifier runs the same computation.

Normalization conventions

Values here follow Friedman's pages: 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.

Attribution

This site is an enhanced presentation of the record tables curated for decades by Erich Friedman; values, credits and symmetry labels are recorded from his pages, and each configuration page links back to its row. His images are not reproduced — every figure is regenerated from coordinates.

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)
• E. Friedman, The Heilbronn problem for triangles (Packing Center)
• E. Friedman, The Heilbronn problem for convex regions (Packing Center)
• MathWorld — Heilbronn Triangle Problem
• Wikipedia — Heilbronn triangle problem
• TejSteadQC/heilbronn-configurations — coordinates, dual verifiers, search toolkit
• spiralulam/heilbronn — square configurations n ≤ 16, exact coordinates, optimization models (MIT)
• google-deepmind/alphaevolve_results — published AlphaEvolve constructions