Triangle, n = 12 record
A =
the smallest positive root of
= 0.0310047817435254466386619953378577300655973679…
solved here as the exact critical point of the 3-parameter symmetry-reduced tie system (21 tied minimal triangles); minimal polynomial found by lattice reduction on 1650 digits and verified against a 3000-digit solution
Classes
Symmetry
Symmetry
3-fold dihedral symmetry (group D3, order 6).
12 points in 3 orbits (6 + 3 + 3) — hover a point to see its orbit.
Minimal triangles
21 triangles tied (within relative 10−9) at the minimal area, in 4 congruence classes:
| count | side lengths | triangles (point indices) | |
|---|---|---|---|
| 6 | 0.3082 · 0.4632 · 0.7528 |
(0,6,10) (1,2,10) (2,4,10) (2,5,6) (2,6,8) (6,9,10) | |
| 6 | 0.2502 · 0.7658 · 1.0057 |
(0,5,7) (0,5,11) (1,3,8) (1,7,8) (3,4,9) (4,9,11) | |
| 6 | 0.5751 · 0.6451 · 1.2160 |
(0,2,3) (1,6,11) (2,7,9) (3,5,10) (4,6,7) (8,10,11) | |
| 3 | 0.2502 · 0.2502 · 0.3770 |
(0,1,7) (3,8,9) (4,5,11) |
Provenance
- Found by David Cantrell, July 2006.
- Coordinates reconstructed by local optimization:
heilbronn-site exact tie-system solve— exact solution of the symmetry-reduced tie system (3 parameters, 21 tied minimal triangles); literals are the exact values rounded to 30 decimals. - Verified in exact arithmetic: all 220 triples enumerated, 21 tied at the minimum.
Record history
- 2026-09-15 coordinates replaced by the exact tie-system solution (30 decimals); value identified as the smallest positive root of 729A⁶ − 291168A⁵ + 390241A⁴ − 43317A³ − 4386A² + 5A + 5 = 0.031004781743525… (this site, from the symmetry-reduced tie system)
Downloads
- points.txt coordinates, tab-separated
- points.csv coordinates, CSV
- points.json full record: value, provenance, verification
- figure.svg this figure
