Heilbronn's triangle problem in the disk: a certified n = 14 construction

An explicit 14-point configuration in exact integer coordinates, improving the best 14-point value we could locate by about 1.075%, with every triangle area decided in exact rational arithmetic.

The bound

Place n points in the closed disk of radius 1. Among the C(n,3) triangles they form, look at the smallest — how large can that smallest triangle be forced to be? Call that quantity α_disk(n). For almost every n, nobody knows its exact value. This deposit settles one thing about n = 14:

It exceeds by about +1.075% the best earlier 14-point value we could locate — A = .0758+, attributed to David Cantrell (June 2007) on Erich Friedman's page The Heilbronn Problem for Circles, offline since around 2024 and surviving only in web archives. It is not a rediscovery: Friedman's page describes the 2007 configuration as horizontally symmetric, while this one has no reflection symmetry at all — 8 points on the boundary circle, 6 in the interior.

Why you need not trust the search

A lower bound is settled by a single configuration, so nothing about the search that found it needs to be believed.

• All 14 points are stored as integers over a common scale of 10³³, so disk containment and all 364 triangle areas are decided in exact rational arithmetic. No floating point appears anywhere in the certificate.
• A 157-line verifier using nothing but the Python standard library re-derives the bound from the deposited files — short enough to read in full before believing it.

git clone https://github.com/SeverinVisionary/heilbronn-records
cd heilbronn-records/disk && python3 verify.py configs   # 12 of 12 verified

The basin is hard to hit: in a seed-recorded study an independent restart landed in it about once in 1800 tries (9 hits in 16384). Reproducible, and easy to miss.

What is claimed, and what is not

• This is a certified lower bound on an unknown quantity, improving the best previously documented construction we could locate. It is not a proved record and not an optimality result.
• No published coordinates for Cantrell's construction were found, and no improvement in the nineteen years since — but the deposit states plainly which sources were not checked. “The best value we could locate”, not “the first improvement since 2007”.
• The companion unit-square lane in the repository broke no records, and none is claimed.
• Two retractions are kept in the repository rather than deleted: six configurations once claimed as best known were later beaten by a naive annealer, and a claimed +3.79% at n = 11 turned out to be Cantrell's own 2006 configuration reached through a wrong baseline row.

The square lane: structure, not records

• A rigidity audit of the record landscape, 7 ≤ n ≤ 12, each best-known square configuration certified in its own exact field. Five are infinitesimally rigid; n = 10 is the unique exception.
• n = 10 is prestress stable — first-order flexible (rank 19 in dimension 20) but second-order rigid: an exact strictly positive stress collapses the feasible cone onto the flex line and the stress-weighted Hessian is negative (Q = −109.241), so the configuration is a strict, isolated local maximum.
• An n = 12 minimal-core theorem: all 6196 subsets of the active triangles classified with exact two-sided certificates — exactly three D₄-classes of inclusion-minimal rigid cores, of sizes 17/18/18, and no rigid 16-subset.

Cite

Yang, Hanyu (2026). Heilbronn's triangle problem in the disk: a certified
n = 14 construction, with a census for 7 <= n <= 16 [Data set].
Zenodo. https://doi.org/10.5281/zenodo.22091169

Read further

The deposit note as a typeset PDF (6 pages) — a rendering of disk/NOTE.md as it stands on main at commit c06bdb5, the same file that is in the Zenodo record, with only a title block added. The Markdown file is the version of record.