Square, n = 9 proven
A =
= 0.0548759991708967089728109971022935630631548961…
Classes
Symmetry
Symmetry
Mirror symmetric (group D1, order 2). (Friedman's page says: symmetric about a diagonal.)
Minimal triangles
11 triangles tied (within relative 10−9) at the minimal area, in 6 congruence classes:
| count | side lengths | triangles (point indices) | |
|---|---|---|---|
| 2 | 0.2601 · 0.5664 · 0.7648 |
(0,1,6) (2,7,8) | |
| 2 | 0.2453 · 0.6328 · 0.8247 |
(0,3,4) (3,4,7) | |
| 2 | 0.2601 · 0.6328 · 0.8446 |
(0,1,4) (3,7,8) | |
| 2 | 0.3576 · 0.6708 · 1.0022 |
(0,2,5) (5,6,7) | |
| 2 | 0.3576 · 0.8103 · 1.1493 |
(2,4,5) (3,5,6) | |
| 1 | 0.5920 · 0.5920 · 1.1689 |
(1,5,8) |
Provenance
- Found by F. Comellas and J. Yebra, December 2001.
- Proved optimal by Nathan Sudermann-Merx, March 2026.
- Coordinates from spiralulam/heilbronn (MIT):
spiralulam/heilbronn config_n09.json— Sudermann-Merx (2026). - Verified in exact arithmetic: all 84 triples enumerated, 11 tied at the minimum.
- Friedman's page lists:
A = (9√65-55) / 320 = .05487+.
Downloads
- points.txt coordinates, tab-separated
- points.csv coordinates, CSV
- points.json full record: value, provenance, verification
- figure.svg this figure
References
- N. Sudermann-Merx, Certified global optimality and exact coordinates for Heilbronn configurations in the square (2026)
- L. Chen et al., Searching approximate global optimal Heilbronn configurations of nine points in the unit square via GPGPU computing, J. Global Optim. (2016)
- F. Comellas, J. L. A. Yebra, New lower bounds for Heilbronn numbers, Electron. J. Combin. 9 (2002) #R6
- OEIS A379534 — (9√65−55)/320, the n=9 Heilbronn constant for the square