The Heilbronn Problem

The Heilbronn problem asks how to place n points in a unit-area region so that the smallest triangle formed by any three points is as large as possible; the maximal smallest-triangle area is denoted A(n). The compilation presents the best known point configurations for three classic containers - the unit square, an equilateral triangle, and optimal convex regions - providing exact coordinates, symmetry and congruence analysis, references to published proofs, and an in-browser rational-arithmetic verifier that regenerates every figure from exactly verified coordinates. Methods and attributions are documented alongside each entry so configurations can be independently checked.

The record log shows active improvements, with many recent updates in late 2026: for example, Triangle n=20 improved to 0.01260939 (+2.37%) and Square n=23 to 0.00975728 (+1.18%), with other gains by contributors including Rob Gardiner, Marc-Emmanuel Coupvent des Graviers, and Alexandar Lackovic. A table of best-known values (truncated to eight decimals, with proven-optimal entries flagged) runs from n=3 - where areas are large (square 0.5, triangle 1.0) - down to n=36, where square values fall to about 0.00418; values decline predictably with n. The resource emphasizes exact, reproducible configurations and maintains an Atom feed for new records.