Square, n = 8 proven
A =
= 0.0723764243184441470310894796519582207292026826…
Classes
Symmetry
Symmetry
180° Rotationally symmetric (group C2, order 2).
Minimal triangles
12 triangles tied (within relative 10−9) at the minimal area, in 6 congruence classes:
| count | side lengths | triangles (point indices) | |
|---|---|---|---|
| 2 | 0.2993 · 0.4925 · 0.6228 |
(1,2,7) (4,5,6) | |
| 2 | 0.4430 · 0.5889 · 0.9885 |
(0,6,7) (3,6,7) | |
| 2 | 0.2993 · 0.7676 · 1.0176 |
(0,1,2) (3,4,5) | |
| 2 | 0.4430 · 0.6228 · 1.0267 |
(0,5,6) (1,3,7) | |
| 2 | 0.4925 · 0.6547 · 1.1170 |
(1,4,6) (2,5,7) | |
| 2 | 0.4430 · 0.9885 · 1.4142 |
(0,3,6) (0,3,7) |
Provenance
- Found by F. Comellas and J. Yebra, December 2001.
- Proved optimal by L. Dehbi and Z. Zeng, 2022.
- Coordinates from spiralulam/heilbronn (MIT):
spiralulam/heilbronn config_n08.json— Dehbi, Zeng (2022). - Verified in exact arithmetic: all 56 triples enumerated, 12 tied at the minimum.
- Friedman's page lists:
A = (√13-1) / 36 = .07237+.
Downloads
- points.txt coordinates, tab-separated
- points.csv coordinates, CSV
- points.json full record: value, provenance, verification
- figure.svg this figure
