Triangle, n = 15 record
A =
the unique real root of
= 0.0210907694602666879593204076637693494055410966…
leading coefficient 3⁷·7⁴; Galois group S₅, so no radical (nested-root) form exists
Classes
Symmetry
Symmetry
120° Rotationally symmetric (group C3, order 3).
15 points in 5 orbits (3 + 3 + 3 + 3 + 3) — hover a point to see its orbit.
Minimal triangles
15 triangles tied (within relative 10−9) at the minimal area, in 3 congruence classes:
| count | side lengths | triangles (point indices) | |
|---|---|---|---|
| 6 | 0.3987 · 0.6208 · 1.0160 |
(3,4,12) (3,5,14) (4,5,13) (6,7,12) (6,8,14) (7,8,13) | |
| 6 | 0.2819 · 0.7788 · 1.0568 |
(0,3,7) (0,5,6) (1,3,7) (1,4,8) (2,4,8) (2,5,6) | |
| 3 | 0.6668 · 0.6668 · 1.3320 |
(0,1,12) (0,2,14) (1,2,13) |
Provenance
- Found by David Cantrell, June 2007.
- The tight structure splits 12 + 3: twelve points form a mirror-symmetric frame carrying all 15 minimal triples — the exact critical point of its tie system, with coordinates algebraic of degree 5 — while the other three are floaters whose every triple stays ≥ 1.0178× the minimum.
- Each floater is confined to a triangular cell of diameter ≈ 0.003; the six such cells form two mirror-image orbits and the floaters must occupy alternating cells, so the configuration is chiral — the frame's mirror symmetry cannot extend to all 15 points. Floaters are placed at their cell's maximal-clearance point, which pins the whole configuration algebraically.
- David Cantrell's own coordinates for this entry (received September 2026) are on file in the repository: a mirror-symmetric arrangement with 24 tied minimal triples and value 0.02109025…, about 5.2 × 10⁻⁷ below the chiral configuration shown. Local polishing of his coordinates moves directly into the chiral configuration, so the two are the same basin.
- Coordinates reconstructed by local optimization:
heilbronn-site reconstruct/reconstruct.py— exact reconstruction: the twelve points carrying all 15 minimal triples form a mirror-symmetric frame solved as the exact critical point of its tie system (degree-5 algebraic); the three remaining points are floaters (all their triples ≥ 1.0178× the minimum), each confined to a triangular cell of diameter ≈ 0.003 and placed at its cell's maximal-clearance point. Literals: exact values rounded to 30 decimals, rotation images taken of the rounded generators. - Verified in exact arithmetic: all 455 triples enumerated, 15 tied at the minimum.
Record history
- 2007-06 0.0210+ — David Cantrell
- 2026-08-21 coordinates replaced by the exact tie-system solution; value identified as the unique real root of 5250987A⁵ + 1609650A⁴ + 161469A³ + 5438A² − 13A − 4 = 0.021090769460266… (this site)
Downloads
- points.txt coordinates, tab-separated
- points.csv coordinates, CSV
- points.json full record: value, provenance, verification
- figure.svg this figure
