Triangle, n = 21 record
A = 0.011395976979638…
full precision
Exact value of the published coordinate literals:
0.0113959769796381381312837019684
Symmetry
Symmetry
120° Rotationally symmetric (group C3, order 3).
21 points in 7 orbits (3 + 3 + 3 + 3 + 3 + 3 + 3) — hover a point to see its orbit.
Minimal triangles
39 triangles tied (within relative 10−9) at the minimal area, in 13 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 3 | 0.1614 · 0.2116 · 0.3488 |
(0,14,16) (1,12,17) (2,13,15) |
| 3 | 0.2082 · 0.2780 · 0.4765 |
(3,7,19) (4,8,20) (5,6,18) |
| 3 | 0.3414 · 0.3586 · 0.6970 |
(12,18,20) (13,18,19) (14,19,20) |
| 3 | 0.2402 · 0.4664 · 0.7033 |
(6,13,17) (7,14,15) (8,12,16) |
| 3 | 0.4203 · 0.4679 · 0.8867 |
(0,3,15) (1,4,16) (2,5,17) |
| 3 | 0.4664 · 0.4742 · 0.9393 |
(6,10,13) (7,11,14) (8,9,12) |
| 3 | 0.3177 · 0.6471 · 0.9634 |
(9,17,18) (10,15,19) (11,16,20) |
| 3 | 0.3193 · 0.6471 · 0.9651 |
(6,16,20) (7,17,18) (8,15,19) |
| 3 | 0.3538 · 0.6970 · 1.0498 |
(12,15,18) (13,16,19) (14,17,20) |
| 3 | 0.5297 · 0.6467 · 1.1757 |
(3,9,20) (4,10,18) (5,11,19) |
| 3 | 0.5389 · 0.6582 · 1.1965 |
(0,7,10) (1,8,11) (2,6,9) |
| 3 | 0.2939 · 1.0299 · 1.3232 |
(0,11,13) (1,9,14) (2,10,12) |
| 3 | 0.5389 · 0.9393 · 1.4778 |
(6,9,10) (7,10,11) (8,9,11) |
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; 39 tied minimal triangles. - Verified in exact arithmetic: all 1330 triples enumerated, 39 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
