Square, n = 7 proven
A =
the smallest positive root of
= 0.0838590090075134066379667435447605568443247682…
three real roots (casus irreducibilis): no real nested-root form exists
Symmetry
Not symmetric (group C1, order 1).
Minimal triangles
8 triangles tied (within relative 10−9) at the minimal area, in 8 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 1 | 0.3394 · 0.4949 · 0.5839 |
(1,2,6) |
| 1 | 0.4066 · 0.5268 · 0.8422 |
(0,5,6) |
| 1 | 0.4066 · 0.5432 · 0.8648 |
(0,4,5) |
| 1 | 0.3394 · 0.7255 · 1.0010 |
(0,1,2) |
| 1 | 0.5057 · 0.5839 · 1.0404 |
(1,3,6) |
| 1 | 0.5432 · 0.7154 · 1.2280 |
(1,4,5) |
| 1 | 0.4949 · 0.8253 · 1.2927 |
(2,4,6) |
| 1 | 0.5057 · 0.8422 · 1.3221 |
(0,3,6) |
Provenance
- Found by F. Comellas and J. Yebra, December 2001.
- Proved optimal by Zhenbing Chen and Liangyu Chen, 2011.
- Coordinates exact, as published in:
arXiv:2603.11107— in terms of the middle root of 19f^3 - 27f^2 + 11f - 1. - Verified in exact arithmetic: all 35 triples enumerated, 8 tied at the minimum.
Downloads
- points.txt coordinates, tab-separated
- points.csv coordinates, CSV
- points.json full record: value, provenance, verification
- figure.svg this figure
References
- 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
- F. Comellas, J. L. A. Yebra, New lower bounds for Heilbronn numbers, Electron. J. Combin. 9 (2002) #R6
- OEIS A343851 — decimal expansion of the n=7 Heilbronn constant for the square