Square, n = 27 record
A = 0.007102404979886…
full precision
Exact value of the published coordinate literals:
0.00710240497988603473385057358523
Symmetry
Symmetry
Mirror symmetric (group D1, order 2).
27 points in 14 orbits (2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 1) — hover a point to see its orbit.
Minimal triangles
47 triangles tied (within relative 10−9) at the minimal area, in 24 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 2 | 0.1289 · 0.1669 · 0.2771 |
(10,16,26) (11,17,26) |
| 2 | 0.0807 · 0.3072 · 0.3761 |
(18,21,24) (19,20,25) |
| 2 | 0.1562 · 0.3304 · 0.4825 |
(8,23,25) (9,22,24) |
| 2 | 0.1669 · 0.3689 · 0.5326 |
(8,17,26) (9,16,26) |
| 2 | 0.2389 · 0.3561 · 0.5930 |
(0,6,20) (1,7,21) |
| 2 | 0.1289 · 0.4907 · 0.6170 |
(6,10,16) (7,11,17) |
| 2 | 0.1587 · 0.4689 · 0.6254 |
(0,9,21) (1,8,20) |
| 2 | 0.3233 · 0.3583 · 0.6803 |
(0,1,24) (0,1,25) |
| 2 | 0.3417 · 0.3581 · 0.6987 |
(0,8,12) (1,9,13) |
| 2 | 0.2373 · 0.4689 · 0.7049 |
(0,9,15) (1,8,14) |
| 2 | 0.1289 · 0.5811 · 0.7081 |
(0,10,16) (1,11,17) |
| 2 | 0.2621 · 0.5262 · 0.7873 |
(2,8,11) (3,9,10) |
| 2 | 0.2115 · 0.6142 · 0.8248 |
(0,13,26) (1,12,26) |
| 2 | 0.2655 · 0.5655 · 0.8302 |
(6,13,23) (7,12,22) |
| 2 | 0.1640 · 0.6673 · 0.8302 |
(4,14,25) (5,15,24) |
| 2 | 0.3831 · 0.5057 · 0.8883 |
(4,19,23) (5,18,22) |
| 2 | 0.4275 · 0.4825 · 0.9095 |
(2,9,24) (3,8,25) |
| 2 | 0.2098 · 0.7146 · 0.9237 |
(2,16,23) (3,17,22) |
| 2 | 0.4275 · 0.5282 · 0.9552 |
(2,9,18) (3,8,19) |
| 2 | 0.1587 · 0.8532 · 1.0112 |
(0,2,21) (1,3,20) |
| 2 | 0.2947 · 0.7957 · 1.0900 |
(2,10,20) (3,11,21) |
| 2 | 0.4533 · 0.7492 · 1.2022 |
(12,15,25) (13,14,24) |
| 2 | 0.4434 · 0.8369 · 1.2801 |
(4,13,18) (5,12,19) |
| 1 | 0.4789 · 0.4789 · 0.9574 |
(4,5,26) |
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, ruin-and-recreate, soft-min L-BFGS warm-up, trust-region SLP polish); tie system refined to 80 digits by damped Gauss-Newton; verified in exact rational arithmetic— Mirror symmetric about x = 1/2 (one point on the axis); 47 tied minimal triangles. Found by mirror-restricted basin hopping.. - Verified in exact arithmetic: all 2925 triples enumerated, 47 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
