Triangle, n = 31 record
A = 0.005161284315624…
full precision
Exact value of the published coordinate literals:
0.00516128431562437454682520846843
Symmetry
Symmetry
120° Rotationally symmetric (group C3, order 3).
31 points in 11 orbits (3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 1) — hover a point to see its orbit.
Minimal triangles
57 triangles tied (within relative 10−9) at the minimal area, in 19 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 3 | 0.1509 · 0.2305 · 0.3773 |
(15,18,28) (16,19,29) (17,20,27) |
| 3 | 0.0797 · 0.3233 · 0.3977 |
(0,6,26) (1,7,24) (2,8,25) |
| 3 | 0.1585 · 0.3152 · 0.4715 |
(6,12,28) (7,13,29) (8,14,27) |
| 3 | 0.1897 · 0.2935 · 0.4812 |
(12,18,22) (13,19,23) (14,20,21) |
| 3 | 0.2436 · 0.3626 · 0.6051 |
(3,4,11) (3,5,10) (4,5,9) |
| 3 | 0.1685 · 0.4810 · 0.6485 |
(9,14,30) (10,12,30) (11,13,30) |
| 3 | 0.1976 · 0.5130 · 0.7098 |
(15,26,30) (16,24,30) (17,25,30) |
| 3 | 0.2594 · 0.4660 · 0.7248 |
(3,16,18) (4,17,19) (5,15,20) |
| 3 | 0.2157 · 0.5122 · 0.7272 |
(0,3,22) (1,4,23) (2,5,21) |
| 3 | 0.1685 · 0.5809 · 0.7487 |
(6,10,30) (7,11,30) (8,9,30) |
| 3 | 0.3152 · 0.4373 · 0.7520 |
(0,14,27) (1,12,28) (2,13,29) |
| 3 | 0.3233 · 0.4373 · 0.7601 |
(0,26,27) (1,24,28) (2,25,29) |
| 3 | 0.2935 · 0.6014 · 0.8945 |
(18,22,29) (19,23,27) (20,21,28) |
| 3 | 0.3497 · 0.6304 · 0.9799 |
(9,17,28) (10,15,29) (11,16,27) |
| 3 | 0.1897 · 0.8351 · 1.0244 |
(12,18,19) (13,19,20) (14,18,20) |
| 3 | 0.2594 · 0.7960 · 1.0551 |
(3,16,26) (4,17,24) (5,15,25) |
| 3 | 0.2594 · 0.8005 · 1.0596 |
(3,6,16) (4,7,17) (5,8,15) |
| 3 | 0.4389 · 0.6332 · 1.0719 |
(3,15,24) (4,16,25) (5,17,26) |
| 3 | 0.3351 · 0.9618 · 1.2968 |
(6,7,18) (6,8,20) (7,8,19) |
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; 57 tied minimal triangles. - Verified in exact arithmetic: all 4495 triples enumerated, 57 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
