Triangle, n = 11 record
A =
the smallest positive root of
= 0.0365298898800302164248471279615801122384728660…
solved here as the exact critical point of the 8-parameter symmetry-reduced tie system (17 tied minimal triangles); minimal polynomial found by lattice reduction on 1650 digits and verified against a 3000-digit solution
Symmetry
Symmetry
Mirror symmetric (group D1, order 2).
11 points in 6 orbits (2 + 2 + 2 + 2 + 2 + 1) — hover a point to see its orbit.
Minimal triangles
17 triangles tied (within relative 10−9) at the minimal area, in 9 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 2 | 0.3731 · 0.4598 · 0.8131 |
(0,3,9) (3,4,10) |
| 2 | 0.3731 · 0.4662 · 0.8199 |
(3,4,5) (3,8,9) |
| 2 | 0.2817 · 0.6764 · 0.9428 |
(2,5,8) (5,6,8) |
| 2 | 0.2881 · 0.8515 · 1.1298 |
(0,1,6) (2,7,10) |
| 2 | 0.4598 · 0.6970 · 1.1495 |
(0,5,9) (4,8,10) |
| 2 | 0.5395 · 0.6351 · 1.1679 |
(1,2,3) (3,6,7) |
| 2 | 0.5213 · 0.6970 · 1.2122 |
(0,4,8) (5,9,10) |
| 2 | 0.5213 · 0.7248 · 1.2403 |
(0,2,4) (6,9,10) |
| 1 | 0.5395 · 0.5395 · 1.0704 |
(1,3,7) |
Provenance
- Found by AlphaEvolve, June 2025.
- Coordinates reconstructed by local optimization:
heilbronn-site exact tie-system solve— exact solution of the symmetry-reduced tie system (8 parameters, 17 tied minimal triangles); literals are the exact values rounded to 30 decimals. - Verified in exact arithmetic: all 165 triples enumerated, 17 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 386726293821797392786129A¹⁶ + 5390497410699904416698271A¹⁵ + 9150784972225868037074965A¹⁴ + 29882099349975917905402053A¹³ + 96907959773628435631853994A¹² + 8630833231832902477977498A¹¹ + 163485900461635219362256358A¹⁰ − 73728098063632444097933977A⁹ − 62591326820175894704348590A⁸ + 19409148088019531147406410A⁷ + 8189217351544212941245001A⁶ − 440725435064635936977209A⁵ − 172988796750685677368738A⁴ + 6949815954004975270119A³ + 907432336598652742572A² − 61466944226230595232A + 1011466570038909840 = 0.036529889880030… (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
