Square, n = 19 record
A = 0.013947862081484…
full precision
Exact value of the published coordinate literals:
0.0139478620814845012856999805677
Symmetry
Not symmetric (group C1, order 1).
Minimal triangles
31 triangles tied (within relative 10−9) at the minimal area, in 31 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 1 | 0.1655 · 0.2053 · 0.3291 |
(0,3,8) |
| 1 | 0.2928 · 0.3178 · 0.6036 |
(8,11,16) |
| 1 | 0.2648 · 0.3476 · 0.6053 |
(0,6,11) |
| 1 | 0.2593 · 0.3864 · 0.6396 |
(10,15,16) |
| 1 | 0.2033 · 0.4517 · 0.6484 |
(9,11,17) |
| 1 | 0.2280 · 0.4386 · 0.6606 |
(2,14,15) |
| 1 | 0.2981 · 0.3702 · 0.6629 |
(2,17,18) |
| 1 | 0.2305 · 0.4386 · 0.6633 |
(2,4,15) |
| 1 | 0.2981 · 0.3768 · 0.6696 |
(1,17,18) |
| 1 | 0.2053 · 0.4815 · 0.6809 |
(0,3,13) |
| 1 | 0.2137 · 0.4815 · 0.6896 |
(0,4,13) |
| 1 | 0.1655 · 0.5335 · 0.6925 |
(3,5,8) |
| 1 | 0.1936 · 0.5335 · 0.7217 |
(5,8,12) |
| 1 | 0.2593 · 0.5116 · 0.7670 |
(10,13,15) |
| 1 | 0.2447 · 0.5328 · 0.7736 |
(6,15,17) |
| 1 | 0.2716 · 0.5063 · 0.7742 |
(5,7,9) |
| 1 | 0.2447 · 0.5354 · 0.7763 |
(6,14,17) |
| 1 | 0.2453 · 0.5352 · 0.7767 |
(1,9,15) |
| 1 | 0.2611 · 0.5364 · 0.7940 |
(14,16,18) |
| 1 | 0.2611 · 0.5366 · 0.7942 |
(7,16,18) |
| 1 | 0.2648 · 0.6484 · 0.9107 |
(0,9,11) |
| 1 | 0.4546 · 0.4818 · 0.9344 |
(4,9,18) |
| 1 | 0.4473 · 0.4903 · 0.9357 |
(10,12,17) |
| 1 | 0.4014 · 0.5736 · 0.9733 |
(2,3,16) |
| 1 | 0.4818 · 0.6696 · 1.1503 |
(1,4,18) |
| 1 | 0.4903 · 0.6629 · 1.1522 |
(2,12,17) |
| 1 | 0.4074 · 0.7487 · 1.1550 |
(3,15,18) |
| 1 | 0.2509 · 0.9057 · 1.1550 |
(12,13,16) |
| 1 | 0.4173 · 0.7455 · 1.1618 |
(3,7,17) |
| 1 | 0.2750 · 0.8899 · 1.1635 |
(4,5,6) |
| 1 | 0.3124 · 0.9355 · 1.2468 |
(3,11,14) |
Provenance
- Found by Tej Stead, August 2026.
- Coordinates by an external contributor, re-verified here:
Original construction: Tej Stead, August 2026, https://math.tejstead.com/heilbronn/square/19/. Refined coordinates: Rob Gardiner, September 2026, https://github.com/renidrag/heilbronn-solutions.— Original construction: found with the search program evolved by cnemri (heilbronn-alphaevolve), finished with a trust-region SLP polisher; part of the 2026-08-21 batch. Independently rediscovered from random starting configurations and numerically refined by Rob Gardiner, September 2026.. - Verified in exact arithmetic: all 969 triples enumerated, 31 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
