Square, n = 5 proven
A =
= 0.192450089729875254836382926833985818549200584…
Classes
Symmetry
Symmetry
Mirror symmetric (group D1, order 2).
5 points in 3 orbits (2 + 2 + 1) — hover a point to see its orbit.
Minimal triangles
4 triangles tied (within relative 10−9) at the minimal area, in 2 congruence classes:
| count | side lengths | triangles (point indices) | |
|---|---|---|---|
| 2 | 0.5977 · 0.6667 · 1.0040 |
(0,1,2) (1,2,3) | |
| 2 | 0.6667 · 0.6667 · 1.1547 |
(0,1,4) (2,3,4) |
Provenance
- Proved optimal by Yang Lu, Zhang Jingzhong, and Zeng Zhenbing, 1991.
- Coordinates exact, as published in:
arXiv:2603.11107. - Verified in exact arithmetic: all 10 triples enumerated, 4 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
- L. Yang, J. Z. Zhang, Z. B. Zeng, A conjecture on the first several Heilbronn numbers and a computation, Chinese Ann. Math. Ser. A 13 (1992) 503–515