Convex, n = 20 record
A =
the smallest positive root of
= 0.0144642050306691662611456823958236465772564393…
solved here as the exact critical point of the 4-parameter symmetry-reduced tie system (40 tied minimal triangles); minimal polynomial found by lattice reduction on 1650 digits and verified against a 3000-digit solution
Classes
Symmetry
Symmetry
5-fold dihedral symmetry (group D5, order 10).
20 points in 3 orbits (10 + 5 + 5) — hover a point to see its orbit.
Minimal triangles
40 triangles tied (within relative 10−9) at the minimal area, in 4 congruence classes:
| count | side lengths | triangles (point indices) | |
|---|---|---|---|
| 10 | 0.2129 · 0.3819 · 0.5901 |
(0,1,13) (0,4,12) (1,2,14) (2,3,10) (3,4,11) (5,6,11) (5,9,12) (6,7,10) (7,8,14) (8,9,13) | |
| 10 | 0.3691 · 0.4026 · 0.7698 |
(0,11,17) (1,12,18) (2,13,19) (3,14,15) (4,10,16) (5,13,17) (6,12,16) (7,11,15) (8,10,19) (9,14,18) | |
| 10 | 0.1874 · 0.6910 · 0.8764 |
(0,10,15) (1,11,16) (2,12,17) (3,13,18) (4,14,19) (5,14,19) (6,13,18) (7,12,17) (8,11,16) (9,10,15) | |
| 10 | 0.4026 · 0.5532 · 0.9548 |
(0,2,19) (0,3,17) (1,3,15) (1,4,18) (2,4,16) (5,7,15) (5,8,17) (6,8,19) (6,9,16) (7,9,18) |
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 (4 parameters, 40 tied minimal triangles); literals are the exact values rounded to 30 decimals. - Verified in exact arithmetic: all 1140 triples enumerated, 40 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 38125A⁶ + 304000A⁵ + 238650A⁴ − 101700A³ − 31260A² + 4690A − 61 = 0.014464205030669… (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
