Convex, n = 10 record
A =
the smallest positive root of
= 0.0519930715229432002545036182708711660295373243…
solved here from the 3-parameter symmetry-reduced tie system (16 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
Classes
Symmetry
Symmetry
2-fold dihedral symmetry (group D2, order 4).
10 points in 3 orbits (4 + 4 + 2) — hover a point to see its orbit.
Minimal triangles
16 triangles tied (within relative 10−9) at the minimal area, in 4 congruence classes:
| count | side lengths | triangles (point indices) | |
|---|---|---|---|
| 4 | 0.5104 · 0.9724 · 1.4023 |
(0,1,2) (0,2,6) (3,7,9) (7,8,9) | |
| 4 | 0.6858 · 0.8513 · 1.4741 |
(1,4,5) (3,4,5) (4,5,6) (4,5,8) | |
| 4 | 0.6858 · 1.2098 · 1.8617 |
(0,5,8) (1,4,9) (2,3,4) (5,6,7) | |
| 4 | 0.6858 · 1.4741 · 2.1359 |
(1,4,8) (1,5,8) (3,4,6) (3,5,6) |
Provenance
- Found by David Cantrell, June 2007.
- Coordinates by an external contributor, re-verified here:
David Cantrell's Mathematica notebook UltimateHeilbronn.nb (2007), received September 2026— Original author's arrangement, previously shown only as a reconstruction; given exactly in the notebook.. - Verified in exact arithmetic: all 120 triples enumerated, 16 tied at the minimum.
Record history
- 2026-09-15 value identified as the smallest positive root of 9088A⁴ − 4928A³ + 708A² − 44A + 1 = 0.051993071522943… (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
