Triangle, n = 18 record

A = the second-smallest positive root of

43200 x^{12} - 310177539 x^{11} + 381486159 x^{10} + 10547986913 x^{9} - 5356562854 x^{8} + 3623073265 x^{7} + 565010257 x^{6} - 356137529 x^{5} - 40275162 x^{4} + 2253891 x^{3} + 6619 x^{2} - 717 x + 4

= 0.0148943429879598621067985008479439749710999240…

solved here as the exact critical point of the 5-parameter symmetry-reduced tie system (33 tied minimal triangles); minimal polynomial found by lattice reduction on 1650 digits and verified against a 3000-digit solution

33 minimal triangles in 6 congruence classes, colored by class.

Symmetry

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

18 points in 4 orbits (6 + 6 + 3 + 3) — hover a point to see its orbit.

Minimal triangles

33 triangles tied (within relative 10⁻⁹) at the minimal area, in 6 congruence classes:

count	side lengths	triangles (point indices)
6	`0.1439 · 0.2858 · 0.3988`	(0,5,9) (0,6,9) (1,4,15) (2,8,16) (4,7,15) (8,16,17)
6	`0.2421 · 0.3031 · 0.5335`	(1,13,14) (2,10,12) (3,6,11) (3,14,17) (5,12,13) (7,10,11)
6	`0.2858 · 0.4438 · 0.7247`	(0,6,7) (1,15,17) (1,16,17) (2,5,8) (2,5,9) (4,6,7)
6	`0.3210 · 0.5335 · 0.8515`	(1,2,13) (1,3,6) (2,6,10) (3,7,17) (5,7,10) (5,13,17)
6	`0.3210 · 0.7485 · 1.0678`	(0,1,3) (2,13,15) (3,7,16) (4,5,10) (6,8,10) (9,13,17)
3	`0.5006 · 0.5006 · 0.9993`	(0,4,11) (8,9,12) (14,15,16)

Provenance

Found by Nathan Sudermann-Merx, August 2026.
Coordinates reconstructed by local optimization: heilbronn-site exact tie-system solve — exact solution of the symmetry-reduced tie system (5 parameters, 33 tied minimal triangles); literals are the exact values rounded to 30 decimals.
Verified in exact arithmetic: all 816 triples enumerated, 33 tied at the minimum.

Record history

2026-09-15 coordinates replaced by the exact tie-system solution (30 decimals); value identified as the second-smallest positive root of 43200A¹² − 310177539A¹¹ + 381486159A¹⁰ + 10547986913A⁹ − 5356562854A⁸ + 3623073265A⁷ + 565010257A⁶ − 356137529A⁵ − 40275162A⁴ + 2253891A³ + 6619A² − 717A + 4 = 0.014894342987959… (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