Square, n = 21 record
A = 0.011473760785701…
full precision
Exact value of the published coordinate literals:
0.0114737607857012102683491831776
Symmetry
Not symmetric (group C1, order 1).
Minimal triangles
37 triangles tied (within relative 10−9) at the minimal area, in 37 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 1 | 0.1282 · 0.2115 · 0.3000 |
(3,5,10) |
| 1 | 0.1255 · 0.2459 · 0.3427 |
(1,9,16) |
| 1 | 0.1384 · 0.2369 · 0.3494 |
(2,11,19) |
| 1 | 0.2301 · 0.2480 · 0.4679 |
(12,17,19) |
| 1 | 0.2393 · 0.2626 · 0.4932 |
(0,15,18) |
| 1 | 0.1550 · 0.3741 · 0.5201 |
(4,6,17) |
| 1 | 0.1282 · 0.4137 · 0.5321 |
(3,5,18) |
| 1 | 0.2184 · 0.3486 · 0.5607 |
(9,14,17) |
| 1 | 0.1995 · 0.4078 · 0.6019 |
(8,18,20) |
| 1 | 0.2369 · 0.3874 · 0.6196 |
(2,9,11) |
| 1 | 0.1255 · 0.5021 · 0.6207 |
(1,10,16) |
| 1 | 0.2423 · 0.3898 · 0.6276 |
(2,8,12) |
| 1 | 0.2459 · 0.3874 · 0.6288 |
(2,9,16) |
| 1 | 0.2916 · 0.3432 · 0.6306 |
(4,10,14) |
| 1 | 0.2057 · 0.4382 · 0.6393 |
(7,9,12) |
| 1 | 0.2916 · 0.3823 · 0.6704 |
(2,4,14) |
| 1 | 0.2184 · 0.4719 · 0.6865 |
(14,16,17) |
| 1 | 0.2115 · 0.5021 · 0.7100 |
(1,5,10) |
| 1 | 0.2240 · 0.5112 · 0.7321 |
(7,14,15) |
| 1 | 0.2569 · 0.5060 · 0.7602 |
(0,14,20) |
| 1 | 0.3694 · 0.4137 · 0.7808 |
(3,13,18) |
| 1 | 0.3189 · 0.4695 · 0.7861 |
(11,12,20) |
| 1 | 0.2626 · 0.5321 · 0.7923 |
(0,5,18) |
| 1 | 0.2240 · 0.6329 · 0.8548 |
(7,14,18) |
| 1 | 0.4263 · 0.4307 · 0.8553 |
(7,11,16) |
| 1 | 0.3817 · 0.4932 · 0.8732 |
(0,10,15) |
| 1 | 0.1995 · 0.7115 · 0.9090 |
(18,19,20) |
| 1 | 0.3946 · 0.5187 · 0.9119 |
(6,11,14) |
| 1 | 0.4127 · 0.5030 · 0.9143 |
(1,4,18) |
| 1 | 0.2301 · 0.7036 · 0.9319 |
(10,12,17) |
| 1 | 0.2645 · 0.6756 · 0.9385 |
(3,12,15) |
| 1 | 0.3983 · 0.5661 · 0.9632 |
(0,7,17) |
| 1 | 0.2301 · 0.7731 · 1.0017 |
(5,12,17) |
| 1 | 0.4719 · 0.5624 · 1.0334 |
(8,14,16) |
| 1 | 0.3490 · 0.6983 · 1.0463 |
(5,8,15) |
| 1 | 0.4914 · 0.5607 · 1.0512 |
(9,13,17) |
| 1 | 0.4679 · 0.7154 · 1.1827 |
(3,17,19) |
Provenance
- Found by Rob Gardiner, September 2026.
- Coordinates by an external contributor, re-verified here:
Rob Gardiner, Heilbronn square configurations, September 2026.— Search from random starting configurations. - Verified in exact arithmetic: all 1330 triples enumerated, 37 tied at the minimum.
Record history
- 2026-07 0.00987+ — Tej Stead
- 2026-08 0.010814+ — Tej Stead
- 2026-08-03 0.0110002677 — cnemri via AlphaEvolve (verified here; record)
- 2026-08-21 .01117+ — Tej Stead (coordinates never published)
Downloads
- points.txt coordinates, tab-separated
- points.csv coordinates, CSV
- points.json full record: value, provenance, verification
- figure.svg this figure
