Square, n = 24 record
A = 0.009089008077248…
full precision
Exact value of the published coordinate literals:
0.009089008077248714764430035994
Symmetry
Symmetry
2-fold dihedral symmetry (group D2, order 4).
24 points in 6 orbits (4 + 4 + 4 + 4 + 4 + 4) — hover a point to see its orbit.
Minimal triangles
48 triangles tied (within relative 10−9) at the minimal area, in 12 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 4 | 0.2501 · 0.3206 · 0.5670 |
(3,9,17) (3,9,23) (5,15,21) (11,15,21) |
| 4 | 0.1735 · 0.4180 · 0.5876 |
(0,10,22) (4,6,16) (4,16,18) (10,12,22) |
| 4 | 0.1735 · 0.4193 · 0.5889 |
(0,1,22) (4,18,19) (6,7,16) (10,12,13) |
| 4 | 0.3832 · 0.4027 · 0.7846 |
(0,14,19) (1,8,18) (2,7,12) (6,13,20) |
| 4 | 0.1548 · 0.6368 · 0.7895 |
(2,3,20) (2,20,21) (8,9,14) (8,14,15) |
| 4 | 0.2957 · 0.5876 · 0.8822 |
(0,10,23) (4,6,17) (5,16,18) (11,12,22) |
| 4 | 0.3461 · 0.5670 · 0.9122 |
(0,5,15) (3,12,17) (6,11,21) (9,18,23) |
| 4 | 0.1548 · 0.7895 · 0.9428 |
(2,3,21) (3,20,21) (8,9,15) (9,14,15) |
| 4 | 0.3832 · 0.5887 · 0.9711 |
(0,14,16) (2,4,12) (6,20,22) (8,10,18) |
| 4 | 0.3003 · 0.6748 · 0.9742 |
(1,3,11) (5,7,9) (13,15,23) (17,19,21) |
| 4 | 0.3468 · 0.7687 · 1.1149 |
(2,16,23) (4,11,14) (5,10,20) (8,17,22) |
| 4 | 0.3300 · 0.9298 · 1.2594 |
(0,13,17) (1,5,12) (6,19,23) (7,11,18) |
Provenance
- Found by Ilkyoo Choi, May 2026.
- Coordinates by an external contributor, re-verified here:
AlphaEvolve Early Access experiment, run 2026-05-14; re-polished in heilbronn-site (search/refine.py)— Coordinates generated during Ilkyoo Choi's AlphaEvolve experiment, re-polished to convergence by the site maintainer: 0.0090890081 against 0.0090890061 for the submitted coordinates (+0.00002%), 48 tied minimal triples (submitted: 28).. - Verified in exact arithmetic: all 2024 triples enumerated, 48 tied at the minimum.
Downloads
- points.txt coordinates, tab-separated
- points.csv coordinates, CSV
- points.json full record: value, provenance, verification
- figure.svg this figure
