Convex, n = 21 record
A = 0.013191087454153…
full precision
Exact value of the published coordinate literals:
0.0131910874541530346697632405654
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 8 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 6 | 0.0855 · 0.1875 · 0.2312 |
(3,15,18) (4,16,19) (5,17,20) (6,16,19) (7,17,20) (8,15,18) |
| 6 | 0.1971 · 0.2138 · 0.4040 |
(0,4,6) (1,5,7) (2,3,8) (3,8,11) (4,6,9) (5,7,10) |
| 6 | 0.1645 · 0.3499 · 0.5105 |
(0,10,16) (0,10,20) (1,11,17) (1,11,18) (2,9,15) (2,9,19) |
| 6 | 0.2312 · 0.3409 · 0.5696 |
(0,7,17) (1,8,15) (2,6,16) (3,9,18) (4,10,19) (5,11,20) |
| 6 | 0.1875 · 0.6992 · 0.8858 |
(3,15,16) (4,16,17) (5,15,17) (6,18,19) (7,19,20) (8,18,20) |
| 3 | 0.2583 · 0.3693 · 0.6257 |
(12,13,16) (12,14,15) (13,14,17) |
| 3 | 0.2292 · 0.4567 · 0.6843 |
(3,7,12) (4,8,13) (5,6,14) |
| 3 | 0.2292 · 0.5894 · 0.8176 |
(3,12,20) (4,13,18) (5,14,19) |
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; 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
