Square, n = 13 record
A = 0.027019916669111…
full precision
Exact value of the published coordinate literals:
0.027019916669111101000446039172
Symmetry
Not symmetric (group C1, order 1).
Minimal triangles
20 triangles tied (within relative 10−9) at the minimal area, in 20 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 1 | 0.2333 · 0.2898 · 0.4687 |
(0,5,12) |
| 1 | 0.1326 · 0.4384 · 0.5026 |
(6,11,12) |
| 1 | 0.2606 · 0.3255 · 0.5519 |
(2,4,10) |
| 1 | 0.1326 · 0.5226 · 0.6144 |
(6,9,11) |
| 1 | 0.2394 · 0.4732 · 0.6931 |
(3,7,8) |
| 1 | 0.2687 · 0.5136 · 0.7681 |
(8,10,12) |
| 1 | 0.3754 · 0.4530 · 0.8177 |
(0,2,7) |
| 1 | 0.2413 · 0.6290 · 0.8588 |
(2,8,9) |
| 1 | 0.2898 · 0.6215 · 0.9022 |
(5,9,12) |
| 1 | 0.4422 · 0.4801 · 0.9147 |
(2,3,5) |
| 1 | 0.2413 · 0.6931 · 0.9248 |
(7,8,9) |
| 1 | 0.2687 · 0.6775 · 0.9375 |
(8,10,11) |
| 1 | 0.4687 · 0.5026 · 0.9648 |
(0,6,12) |
| 1 | 0.3955 · 0.6144 · 1.0039 |
(1,9,11) |
| 1 | 0.4422 · 0.5680 · 1.0043 |
(3,4,5) |
| 1 | 0.4384 · 0.5893 · 1.0221 |
(7,11,12) |
| 1 | 0.4086 · 0.6201 · 1.0229 |
(0,3,10) |
| 1 | 0.4912 · 0.6270 · 1.1139 |
(4,6,8) |
| 1 | 0.4923 · 0.7599 · 1.2491 |
(1,5,8) |
| 1 | 0.4923 · 0.8382 · 1.3279 |
(0,1,8) |
Provenance
- Found by Peter Karpov, August 2011.
- Coordinates reconstructed by local optimization:
heilbronn-site reconstruct/reconstruct.py— in-basin tightening of Peter Karpov's configuration (published at inversed.ru/Ascension, score 0.0270192722391139); same arrangement, fully converged coordinates. - Verified in exact arithmetic: all 286 triples enumerated, 20 tied at the minimum.
Record history
- 2011-08 0.0270192722 — Peter Karpov (coordinates published 2018 at inversed.ru/Ascension)
- 2026-08 0.0270199167 — in-basin tightening of Karpov's coordinates (this site)
Downloads
- points.txt coordinates, tab-separated
- points.csv coordinates, CSV
- points.json full record: value, provenance, verification
- figure.svg this figure
