Triangle, n = 25 record
A = 0.007802638741054…
full precision
Exact value of the published coordinate literals:
0.00780263874105424296579400583256
Symmetry
Symmetry
120° Rotationally symmetric (group C3, order 3).
25 points in 9 orbits (3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 1) — hover a point to see its orbit.
Minimal triangles
42 triangles tied (within relative 10−9) at the minimal area, in 14 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 3 | 0.1718 · 0.2701 · 0.4357 |
(3,13,21) (4,14,22) (5,12,23) |
| 3 | 0.1228 · 0.4249 · 0.5433 |
(15,22,24) (16,23,24) (17,21,24) |
| 3 | 0.1228 · 0.4481 · 0.5669 |
(6,17,21) (7,15,22) (8,16,23) |
| 3 | 0.2526 · 0.4207 · 0.6716 |
(3,7,9) (4,8,10) (5,6,11) |
| 3 | 0.3217 · 0.3552 · 0.6753 |
(9,19,23) (10,20,21) (11,18,22) |
| 3 | 0.2628 · 0.4672 · 0.7285 |
(0,9,12) (1,10,13) (2,11,14) |
| 3 | 0.3502 · 0.4297 · 0.7788 |
(9,13,16) (10,14,17) (11,12,15) |
| 3 | 0.3768 · 0.4481 · 0.8240 |
(6,15,21) (7,16,22) (8,17,23) |
| 3 | 0.2376 · 0.6358 · 0.8725 |
(0,5,15) (1,3,16) (2,4,17) |
| 3 | 0.2526 · 0.6839 · 0.9357 |
(3,4,10) (3,5,9) (4,5,11) |
| 3 | 0.4171 · 0.5479 · 0.9645 |
(0,3,23) (1,4,21) (2,5,22) |
| 3 | 0.2923 · 0.7859 · 1.0777 |
(3,8,15) (4,6,16) (5,7,17) |
| 3 | 0.3552 · 0.7804 · 1.1352 |
(0,18,22) (1,19,23) (2,20,21) |
| 3 | 0.3552 · 0.8252 · 1.1800 |
(18,19,22) (18,20,21) (19,20,23) |
Provenance
- Found by Fable 5.1 with Shengtong Zhang, September 2026.
- Coordinates by an external contributor, re-verified here:
Fable51 search campaign, September 2026: symmetry-reduced basin hopping (orbit parametrization, greedy ruin-and-recreate, soft-min warm-up, trust-region SLP) and insertion ladders from neighbouring records; tie system refined to 80 digits by damped Gauss-Newton; verified in exact rational arithmetic— 3-fold rotationally symmetric; 42 tied minimal triangles. - Verified in exact arithmetic: all 2300 triples enumerated, 42 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
