Square, n = 22 record

A = 0.010672853853126…

full precision

Exact value of the published coordinate literals: 0.0106728538531263737530373516021

42 triangles tie for the minimal area.

Symmetry

180° Rotationally symmetric (group C2, order 2).

22 points in 11 orbits (2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2) — hover a point to see its orbit.

Minimal triangles

42 triangles tied (within relative 10⁻⁹) at the minimal area, in 21 congruence classes:

count	side lengths	triangles (point indices)
2	`0.2070 · 0.2149 · 0.4087`	(0,9,12) (1,8,13)
2	`0.2077 · 0.3005 · 0.5008`	(4,6,9) (5,7,8)
2	`0.2705 · 0.2860 · 0.5510`	(0,15,21) (1,14,20)
2	`0.2581 · 0.3364 · 0.5900`	(0,6,11) (1,7,10)
2	`0.3014 · 0.3095 · 0.6069`	(8,12,14) (9,13,15)
2	`0.2059 · 0.4657 · 0.6680`	(6,14,15) (7,14,15)
2	`0.2149 · 0.4983 · 0.7102`	(0,7,12) (1,6,13)
2	`0.2581 · 0.4706 · 0.7261`	(6,11,12) (7,10,13)
2	`0.2276 · 0.5053 · 0.7301`	(2,5,16) (3,4,17)
2	`0.2070 · 0.5503 · 0.7546`	(0,9,18) (1,8,19)
2	`0.3422 · 0.4228 · 0.7630`	(8,10,20) (9,11,21)
2	`0.3512 · 0.5135 · 0.8633`	(10,16,18) (11,17,19)
2	`0.2059 · 0.6680 · 0.8720`	(6,7,14) (6,7,15)
2	`0.2149 · 0.6606 · 0.8737`	(0,10,12) (1,11,13)
2	`0.2059 · 0.6808 · 0.8848`	(6,8,15) (7,9,14)
2	`0.3005 · 0.6585 · 0.9578`	(4,6,16) (5,7,17)
2	`0.2581 · 0.7429 · 0.9998`	(2,6,11) (3,7,10)
2	`0.3512 · 0.6549 · 1.0052`	(10,18,21) (11,19,20)
2	`0.4173 · 0.6069 · 1.0233`	(8,12,21) (9,13,20)
2	`0.2705 · 0.7786 · 1.0481`	(0,3,21) (1,2,20)
2	`0.4313 · 0.8418 · 1.2726`	(4,10,14) (5,11,15)

Provenance

Found by Fable 5.1 with Shengtong Zhang, September 2026.
Found with AlphaEvolve (Google Cloud) and submitted in August 2026, superseding the previous .009569+ record (Nathan Sudermann-Merx).
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 — C2 (180° rotational) symmetric, 42 tied minimal triangles; found by C2-restricted basin hopping.
Verified in exact arithmetic: all 1540 triples enumerated, 42 tied at the minimum.

Record history

2026-07 0.00886+ → 0.009458+ — Tej Stead
2026-08 0.009569+ — Nathan Sudermann-Merx
2026-08-03 0.0104725344 — cnemri via AlphaEvolve (verified here; record)

Downloads

points.txt coordinates, tab-separated
points.csv coordinates, CSV
points.json full record: value, provenance, verification
figure.svg this figure