Convex, n = 31 record
A = 0.006309670708089…
full precision
Exact value of the published coordinate literals:
0.00630967070808958091075187665278
Symmetry
Not symmetric (group C1, order 1).
Minimal triangles
55 triangles tied (within relative 10−9) at the minimal area, in 55 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 1 | 0.1970 · 0.2325 · 0.4253 |
(8,11,30) |
| 1 | 0.1807 · 0.2492 · 0.4256 |
(5,15,20) |
| 1 | 0.1807 · 0.2540 · 0.4306 |
(5,20,22) |
| 1 | 0.2009 · 0.2371 · 0.4340 |
(17,27,30) |
| 1 | 0.2068 · 0.2430 · 0.4462 |
(0,2,18) |
| 1 | 0.1582 · 0.3036 · 0.4581 |
(11,17,23) |
| 1 | 0.1621 · 0.3045 · 0.4631 |
(2,13,29) |
| 1 | 0.1801 · 0.3034 · 0.4804 |
(24,28,30) |
| 1 | 0.1709 · 0.3153 · 0.4830 |
(9,14,18) |
| 1 | 0.1685 · 0.3182 · 0.4836 |
(7,21,30) |
| 1 | 0.1771 · 0.3099 · 0.4840 |
(2,20,25) |
| 1 | 0.1889 · 0.2984 · 0.4844 |
(4,16,22) |
| 1 | 0.1692 · 0.3420 · 0.5085 |
(0,5,25) |
| 1 | 0.1542 · 0.3700 · 0.5215 |
(6,23,27) |
| 1 | 0.1692 · 0.3748 · 0.5416 |
(5,25,29) |
| 1 | 0.2794 · 0.2815 · 0.5591 |
(11,24,27) |
| 1 | 0.2815 · 0.2905 · 0.5704 |
(11,21,27) |
| 1 | 0.1789 · 0.3954 · 0.5722 |
(6,8,29) |
| 1 | 0.3065 · 0.3467 · 0.6520 |
(1,3,29) |
| 1 | 0.3153 · 0.3384 · 0.6525 |
(9,14,19) |
| 1 | 0.3182 · 0.3530 · 0.6702 |
(3,7,21) |
| 1 | 0.2524 · 0.4581 · 0.7096 |
(11,17,22) |
| 1 | 0.2517 · 0.4631 · 0.7138 |
(2,13,15) |
| 1 | 0.2360 · 0.4830 · 0.7181 |
(3,9,18) |
| 1 | 0.2492 · 0.4784 · 0.7267 |
(5,15,17) |
| 1 | 0.2513 · 0.4836 · 0.7340 |
(19,21,30) |
| 1 | 0.2540 · 0.4840 · 0.7371 |
(2,20,22) |
| 1 | 0.1582 · 0.6182 · 0.7753 |
(4,11,23) |
| 1 | 0.1621 · 0.6281 · 0.7892 |
(13,16,29) |
| 1 | 0.3859 · 0.4306 · 0.8159 |
(5,22,26) |
| 1 | 0.2515 · 0.5722 · 0.8230 |
(8,12,29) |
| 1 | 0.3284 · 0.5215 · 0.8494 |
(6,9,27) |
| 1 | 0.2346 · 0.6226 · 0.8565 |
(13,18,21) |
| 1 | 0.4216 · 0.4525 · 0.8736 |
(4,12,25) |
| 1 | 0.1295 · 0.7676 · 0.8962 |
(5,11,16) |
| 1 | 0.1521 · 0.7557 · 0.9070 |
(8,20,21) |
| 1 | 0.1287 · 0.7868 · 0.9146 |
(4,13,20) |
| 1 | 0.1499 · 0.7690 · 0.9182 |
(1,5,9) |
| 1 | 0.1295 · 0.7956 · 0.9243 |
(5,8,16) |
| 1 | 0.1268 · 0.8011 · 0.9270 |
(2,27,28) |
| 1 | 0.4477 · 0.4873 · 0.9346 |
(6,7,14) |
| 1 | 0.1289 · 0.8123 · 0.9404 |
(0,12,17) |
| 1 | 0.1287 · 0.8153 · 0.9433 |
(1,4,20) |
| 1 | 0.3467 · 0.6010 · 0.9474 |
(1,2,3) |
| 1 | 0.3639 · 0.5972 · 0.9608 |
(8,17,19) |
| 1 | 0.2532 · 0.7092 · 0.9620 |
(8,15,25) |
| 1 | 0.1324 · 0.8314 · 0.9630 |
(2,7,11) |
| 1 | 0.1386 · 0.8314 · 0.9693 |
(2,11,26) |
| 1 | 0.1268 · 0.8568 · 0.9829 |
(0,27,28) |
| 1 | 0.1339 · 0.8508 · 0.9839 |
(13,14,17) |
| 1 | 0.1289 · 0.8568 · 0.9850 |
(0,12,27) |
| 1 | 0.1383 · 0.8508 · 0.9884 |
(13,17,24) |
| 1 | 0.3467 · 0.7189 · 1.0653 |
(1,3,26) |
| 1 | 0.3639 · 0.7167 · 1.0804 |
(8,19,24) |
| 1 | 0.4052 · 0.6943 · 1.0992 |
(5,6,19) |
Provenance
- Found by Alexandar Lackovic with help of Opus 5.5, September 2026.
- Coordinates by an external contributor, re-verified here:
Basin hopping search (with best spot reinsertion and shake moves, elite pool, laddered/symmetric seeds) with SLSQP polishing of the max-min-area problem. Originally implemented by Alexandar Lackovic in Python, then optimized and ported to C++ by Opus 5.5. https://github.com/codrAlex/heilbronn-search— Not symmetric. Found by basin hopping with best spot reinsertion moves and SLSQP polishing of the max-min-area problem. - Verified in exact arithmetic: all 4495 triples enumerated, 55 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
