Convex, n = 9 record

A = the smallest positive root of

4636072741330 x^{14} + 1774549666005 x^{13} - 1233481443979 x^{12} - 1367692325626 x^{11} - 694825007492 x^{10} + 38268929375 x^{9} + 102061390459 x^{8} + 123696372 x^{7} - 5255868622 x^{6} + 66844595 x^{5} + 131075035 x^{4} - 8755802 x^{3} - 171936 x^{2} - 24711 x + 2645

= 0.0640647523079416997628889229773599724292772168…

solved here from the 6-parameter symmetry-reduced tie system (11 tied minimal triangles), whose solutions form a curve, together with stationarity of the value along that curve; minimal polynomial found by lattice reduction on 825 digits and verified against a 1500-digit solution

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

Symmetry

Mirror symmetric (group D1, order 2).

9 points in 5 orbits (2 + 2 + 2 + 2 + 1) — hover a point to see its orbit.

Minimal triangles

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

count	side lengths	triangles (point indices)
2	`0.2469 · 0.4673 · 0.6544`	(3,4,7) (3,7,8)
2	`0.3611 · 0.4673 · 0.7978`	(2,3,4) (6,7,8)
2	`0.3967 · 0.5009 · 0.8744`	(0,1,8) (0,4,5)
2	`0.4884 · 0.5009 · 0.9726`	(0,2,8) (0,4,6)
2	`0.3967 · 0.6341 · 1.0153`	(0,1,7) (0,3,5)
1	`0.4884 · 0.4884 · 0.9595`	(0,2,6)

Provenance

Found by David Cantrell, June 2007.
David Cantrell's original mirror-symmetric coordinates (2007), tightened to 15 decimals; this site's earlier asymmetric re-derivation of the same optimum differed only at rounding level.
Coordinates reconstructed by local optimization: heilbronn-site exact tie-system solve — exact solution of the symmetry-reduced tie system (6 parameters, 11 tied minimal triangles) with the value stationary along its one-parameter solution curve; literals are the exact values rounded to 30 decimals.
Verified in exact arithmetic: all 84 triples enumerated, 11 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 4636072741330A¹⁴ + 1774549666005A¹³ − 1233481443979A¹² − 1367692325626A¹¹ − 694825007492A¹⁰ + 38268929375A⁹ + 102061390459A⁸ + 123696372A⁷ − 5255868622A⁶ + 66844595A⁵ + 131075035A⁴ − 8755802A³ − 171936A² − 24711A + 2645 = 0.064064752307941… (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