Square, n = 12 record
A =
the unique real root of
= 0.0325988586918196982187640066235154080492419884…
computed here from the Comellas–Yebra closed-form configuration (arXiv:2603.11107 appendix)
Classes
Symmetry
Symmetry
4-fold dihedral symmetry (group D4, order 8).
12 points in 2 orbits (8 + 4) — hover a point to see its orbit.
Minimal triangles
20 triangles tied (within relative 10−9) at the minimal area, in 3 congruence classes:
| count | side lengths | triangles (point indices) | |
|---|---|---|---|
| 8 | 0.1631 · 0.4249 · 0.5042 |
(0,2,4) (0,2,5) (1,3,4) (1,3,6) (5,8,10) (6,9,11) (7,8,10) (7,9,11) | |
| 8 | 0.4249 · 0.4518 · 0.8636 |
(0,4,6) (1,4,5) (2,5,7) (3,6,7) (4,5,8) (4,6,9) (5,7,11) (6,7,10) | |
| 4 | 0.5042 · 0.5042 · 1.0000 |
(0,5,10) (1,6,11) (2,3,4) (7,8,9) |
Provenance
- Found by F. Comellas and J. Yebra, December 2001.
- Coordinates exact, as published in:
arXiv:2603.11107— Comellas-Yebra closed form (appendix). - Verified in exact arithmetic: all 220 triples enumerated, 20 tied at the minimum.
Record history
- 2026-08-21 value identified as the unique real root of 64A³ + 80A² + 28A − 1 = 0 (this site, from the published closed-form configuration)
Downloads
- points.txt coordinates, tab-separated
- points.csv coordinates, CSV
- points.json full record: value, provenance, verification
- figure.svg this figure
