Triangle, n = 22 record

A = the smallest positive root of

4113894419667609830181603 x^{22} + 29807468238423455008126578 x^{21} - 39085790603220143290920177 x^{20} + 24488515905115680120305898 x^{19} - 10736525499024455231632668 x^{18} + 3768816750540027903994113 x^{17} - 1158442379073845833748328 x^{16} + 324799853969658361401171 x^{15} - 71937195458392144391355 x^{14} + 11920737923277897236508 x^{13} - 1491784289359000650099 x^{12} + 155491081480323732357 x^{11} - 18192256172188373745 x^{10} + 1431857137555076508 x^{9} - 104692732480960029 x^{8} + 11430606627963063 x^{7} - 705604556304666 x^{6} + 16028440098039 x^{5} + 15064749231 x^{4} - 2374399435 x^{3} - 25225143 x^{2} + 119664 x + 2704

= 0.0102661276072629636732082937422183376425520208…

solved here as the exact critical point of the 6-parameter symmetry-reduced tie system (39 tied minimal triangles); minimal polynomial found by lattice reduction on 1650 digits and verified against a 3000-digit solution

39 triangles tie for the minimal area.

Symmetry

3-fold dihedral symmetry (group D3, order 6).

22 points in 5 orbits (6 + 6 + 6 + 3 + 1) — hover a point to see its orbit.

Minimal triangles

39 triangles tied (within relative 10⁻⁹) at the minimal area, in 7 congruence classes:

count	side lengths	triangles (point indices)
6	`0.1581 · 0.1620 · 0.2861`	(0,9,13) (1,10,12) (2,11,15) (3,6,14) (4,7,17) (5,8,16)
6	`0.1620 · 0.4062 · 0.5624`	(0,13,21) (1,12,21) (2,15,21) (3,14,21) (4,17,21) (5,16,21)
6	`0.3853 · 0.5851 · 0.9694`	(0,8,12) (1,11,13) (2,10,14) (3,7,15) (4,6,16) (5,9,17)
6	`0.3600 · 0.6652 · 1.0244`	(0,6,18) (1,7,18) (2,8,19) (3,9,19) (4,10,20) (5,11,20)
6	`0.2861 · 0.8273 · 1.1126`	(0,9,11) (1,8,10) (2,7,11) (3,6,10) (4,7,9) (5,6,8)
6	`0.3122 · 0.8510 · 1.1625`	(6,13,19) (7,12,20) (8,15,20) (9,14,18) (10,17,18) (11,16,19)
3	`0.4062 · 0.4062 · 0.8108`	(0,3,21) (1,4,21) (2,5,21)

Provenance

Found by Tej Stead, August 2026.
Coordinates reconstructed by local optimization: heilbronn-site exact tie-system solve — exact solution of the symmetry-reduced tie system (6 parameters, 39 tied minimal triangles); literals are the exact values rounded to 30 decimals.
Verified in exact arithmetic: all 1540 triples enumerated, 39 tied at the minimum.

Record history

2026-09-15 coordinates replaced by the exact tie-system solution (30 decimals); value identified as the smallest positive root of 4113894419667609830181603A²² + 29807468238423455008126578A²¹ − 39085790603220143290920177A²⁰ + 24488515905115680120305898A¹⁹ − 10736525499024455231632668A¹⁸ + 3768816750540027903994113A¹⁷ − 1158442379073845833748328A¹⁶ + 324799853969658361401171A¹⁵ − 71937195458392144391355A¹⁴ + 11920737923277897236508A¹³ − 1491784289359000650099A¹² + 155491081480323732357A¹¹ − 18192256172188373745A¹⁰ + 1431857137555076508A⁹ − 104692732480960029A⁸ + 11430606627963063A⁷ − 705604556304666A⁶ + 16028440098039A⁵ + 15064749231A⁴ − 2374399435A³ − 25225143A² + 119664A + 2704 = 0.010266127607262… (this site, from the symmetry-reduced tie system)

Downloads

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