Convex, n = 16 record
A =
the smallest positive root of
= 0.0222728771289812680876159711627856993758033607…
solved here as the exact critical point of the 4-parameter symmetry-reduced tie system (28 tied minimal triangles); minimal polynomial found by lattice reduction on 1650 digits and verified against a 3000-digit solution
Classes
Symmetry
Symmetry
4-fold dihedral symmetry (group D4, order 8).
16 points in 3 orbits (8 + 4 + 4) — hover a point to see its orbit.
Minimal triangles
28 triangles tied (within relative 10−9) at the minimal area, in 4 congruence classes:
| count | side lengths | triangles (point indices) | |
|---|---|---|---|
| 8 | 0.2493 · 0.2735 · 0.5038 |
(1,3,6) (1,6,7) (2,3,13) (2,13,15) (5,7,10) (5,10,11) (9,11,14) (9,14,15) | |
| 8 | 0.2797 · 0.3269 · 0.5949 |
(0,1,4) (0,4,10) (0,6,12) (0,12,13) (2,8,12) (4,5,8) (4,8,14) (8,9,12) | |
| 8 | 0.3937 · 0.5949 · 0.9859 |
(0,3,10) (0,7,13) (1,4,11) (2,8,11) (3,9,12) (4,7,14) (5,8,15) (6,12,15) | |
| 4 | 0.2735 · 0.2735 · 0.5308 |
(1,2,3) (5,6,7) (9,10,11) (13,14,15) |
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 560 triples enumerated, 28 tied at the minimum.
Record history
- 2026-09-15 value identified as the smallest positive root of 4608A⁵ − 10752A⁴ + 9824A³ − 3712A² + 482A − 9 = 0.022272877128981… (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
