Square, n = 14 record
A =
the smallest positive root of
= 0.0243039796209924867482019578100155839585389904…
exact form computed by Rhys Chappell
Symmetry
Symmetry
2-fold dihedral symmetry (group D2, order 4).
14 points in 4 orbits (4 + 4 + 4 + 2) — hover a point to see its orbit.
Minimal triangles
26 triangles tied (within relative 10−9) at the minimal area, in 7 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 4 | 0.2768 · 0.4444 · 0.7073 |
(0,10,12) (1,11,13) (2,8,12) (3,9,13) |
| 4 | 0.2024 · 0.6262 · 0.8168 |
(0,4,7) (1,5,6) (2,5,6) (3,4,7) |
| 4 | 0.2768 · 0.5721 · 0.8399 |
(4,11,13) (5,10,12) (6,9,13) (7,8,12) |
| 4 | 0.3218 · 0.5479 · 0.8618 |
(4,8,9) (5,8,9) (6,10,11) (7,10,11) |
| 4 | 0.4444 · 0.5024 · 0.9411 |
(0,6,12) (1,7,13) (2,4,12) (3,5,13) |
| 4 | 0.3218 · 0.8618 · 1.1799 |
(4,5,8) (4,5,9) (6,7,10) (6,7,11) |
| 2 | 0.5024 · 0.5024 · 1.0000 |
(4,6,12) (5,7,13) |
Provenance
- Found by Mark Beyleveld, August 2006.
- Exact value and coordinates computed by Rhys Chappell, August 2026 — an exact algebraic realization of Beyleveld's configuration (coordinates in Q(a), 8a³ − 12a² − 12a + 1 = 0), re-verified here.
- Coordinates by an external contributor, re-verified here:
github.com/rhyschappell/heilbronn-n14-exact— exact algebraic realization of Beyleveld's configuration: all coordinates in Q(a) with 8a³−12a²−12a+1=0; re-verified here (26 exact ties at the minimum). - Verified in exact arithmetic: all 364 triples enumerated, 26 tied at the minimum.
Record history
- 2026-08-21 coordinates replaced by Rhys Chappell's exact realization of Beyleveld's configuration; value identified as the smallest positive root of 320A³ + 768A² − 60A + 1 = 0.024303979620992… (github.com/rhyschappell/heilbronn-n14-exact, re-verified here)
Downloads
- points.txt coordinates, tab-separated
- points.csv coordinates, CSV
- points.json full record: value, provenance, verification
- figure.svg this figure
