Square, n = 20 record
A = 0.012938196714727…
full precision
Exact value of the published coordinate literals:
0.0129381967147270176121146278205
Symmetry
Symmetry
2-fold dihedral symmetry (group D2, order 4).
20 points in 6 orbits (4 + 4 + 4 + 4 + 2 + 2) — hover a point to see its orbit.
Minimal triangles
42 triangles tied (within relative 10−9) at the minimal area, in 11 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 4 | 0.2086 · 0.2616 · 0.4561 |
(0,10,19) (1,11,18) (2,8,19) (3,9,18) |
| 4 | 0.2856 · 0.3951 · 0.6763 |
(8,12,15) (9,13,14) (10,13,14) (11,12,15) |
| 4 | 0.2616 · 0.5007 · 0.7589 |
(4,10,19) (5,11,18) (6,8,19) (7,9,18) |
| 4 | 0.3951 · 0.5107 · 0.9040 |
(4,12,15) (5,13,14) (6,13,14) (7,12,15) |
| 4 | 0.3026 · 0.6192 · 0.9199 |
(0,14,16) (1,15,16) (2,12,17) (3,13,17) |
| 4 | 0.2634 · 0.6748 · 0.9361 |
(0,4,8) (1,5,9) (2,6,10) (3,7,11) |
| 4 | 0.2856 · 0.6763 · 0.9601 |
(8,11,12) (8,11,15) (9,10,13) (9,10,14) |
| 4 | 0.2519 · 0.8436 · 1.0941 |
(0,6,18) (1,7,19) (2,4,18) (3,5,19) |
| 4 | 0.3797 · 0.7163 · 1.0950 |
(4,14,17) (5,15,17) (6,12,16) (7,13,16) |
| 4 | 0.5107 · 0.9040 · 1.4142 |
(4,7,12) (4,7,15) (5,6,13) (5,6,14) |
| 2 | 0.5007 · 0.5007 · 1.0000 |
(4,5,16) (6,7,17) |
Provenance
- Found by Fable 5.1 with Shengtong Zhang, September 2026.
- Coordinates by an external contributor, re-verified here:
Fable51 search campaign, September 2026: symmetry-reduced basin hopping (orbit parametrization, ruin-and-recreate, soft-min L-BFGS warm-up, trust-region SLP polish); tie system refined to 80 digits by damped Gauss-Newton; verified in exact rational arithmetic— D2 symmetric (both axis-parallel mirrors, one point pair on each axis), 42 tied minimal triangles; found by D2-restricted basin hopping. - Verified in exact arithmetic: all 1140 triples enumerated, 42 tied at the minimum.
Record history
- 2026-09-15 coordinates replaced by the exact tie-system solution (30 decimals); value identified as the smallest positive root of 5159652213303791781987483648A²⁰ − 14873731040215145274619133952A¹⁹ + 37276541653723146274006171648A¹⁸ − 52238240118507558517342552064A¹⁷ + 62502424423753181467918835712A¹⁶ − 54890303870678162355408859136A¹⁵ + 41296381321309451287346792448A¹⁴ − 23990692082786577852058909696A¹³ + 11697128400308059530041718784A¹² − 4290541018969399815006176256A¹¹ + 1307047908489004447138440448A¹⁰ − 295321305838074373049587520A⁹ + 62957223750583680729904416A⁸ − 8147410609690964182366144A⁷ + 1229653614127639776763512A⁶ − 81632120153696942187372A⁵ + 9337511571257634384074A⁴ − 510601603113382503914A³ + 9752576282457500787A² − 48198374124107420A − 139936077203380 = 0.012911555683942… (this site, from the symmetry-reduced tie system)
Downloads
- points.txt coordinates, tab-separated
- points.csv coordinates, CSV
- points.json full record: value, provenance, verification
- figure.svg this figure
