Convex, n = 32 record
A = 0.005850714333353…
full precision
Exact value of the published coordinate literals:
0.00585071433335319572689054899931
Symmetry
Symmetry
2-fold dihedral symmetry (group D2, order 4).
32 points in 8 orbits (4 + 4 + 4 + 4 + 4 + 4 + 4 + 4) — hover a point to see its orbit.
Minimal triangles
56 triangles tied (within relative 10−9) at the minimal area, in 14 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 4 | 0.0965 · 0.1015 · 0.1786 |
(0,20,30) (1,21,31) (2,22,28) (3,23,29) |
| 4 | 0.1015 · 0.3055 · 0.4046 |
(0,16,30) (1,17,31) (2,18,28) (3,19,29) |
| 4 | 0.1389 · 0.3029 · 0.4402 |
(8,17,26) (9,16,27) (10,19,24) (11,18,25) |
| 4 | 0.2228 · 0.2309 · 0.4525 |
(24,27,28) (24,27,31) (25,26,29) (25,26,30) |
| 4 | 0.1389 · 0.3377 · 0.4753 |
(4,19,24) (5,18,25) (6,17,26) (7,16,27) |
| 4 | 0.1792 · 0.3416 · 0.5198 |
(8,25,27) (9,24,26) (10,25,27) (11,24,26) |
| 4 | 0.2593 · 0.3217 · 0.5804 |
(16,22,25) (17,23,24) (18,20,27) (19,21,26) |
| 4 | 0.1672 · 0.4957 · 0.6623 |
(0,11,23) (1,10,22) (2,9,21) (3,8,20) |
| 4 | 0.1015 · 0.5853 · 0.6860 |
(0,22,30) (1,23,31) (2,20,28) (3,21,29) |
| 4 | 0.3029 · 0.4007 · 0.7033 |
(8,15,26) (9,14,27) (10,13,24) (11,12,25) |
| 4 | 0.2879 · 0.4580 · 0.7457 |
(8,18,22) (9,19,23) (10,16,20) (11,17,21) |
| 4 | 0.3377 · 0.4398 · 0.7772 |
(4,24,29) (5,25,28) (6,26,31) (7,27,30) |
| 4 | 0.3217 · 0.5358 · 0.8573 |
(4,18,20) (5,19,21) (6,16,22) (7,17,23) |
| 4 | 0.2681 · 0.6137 · 0.8816 |
(0,15,17) (1,14,16) (2,13,19) (3,12,18) |
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, greedy ruin-and-recreate, soft-min warm-up, trust-region SLP) and insertion ladders from neighbouring records; tie system refined to 80 digits by damped Gauss-Newton; verified in exact rational arithmetic— 2-fold dihedral symmetric; 56 tied minimal triangles. - Verified in exact arithmetic: all 4960 triples enumerated, 56 tied at the minimum.
Record history
- 2026-09-15 value identified as the smallest positive root of 82063347095239464055743104063373312A³⁷ + 235837838036448675737781290385014784A³⁶ + 182360774105834996680674968361500672A³⁵ + 64076922411287885475713814817144832A³⁴ − 24968750782137734522513207536386048A³³ − 8356915510808961074925021622697984A³² + 1947078531642221673921633956397056A³¹ + 1596413723665579001580299397103616A³⁰ − 2144427645379757065516684482183168A²⁹ + 124085821713214238909469948903424A²⁸ + 322932297783725889669775052767232A²⁷ + 37879206034496826572985388236800A²⁶ − 124457345212633923566077705453568A²⁵ + 37235602410838448045286454460416A²⁴ + 782652404584950039284446920704A²³ − 2308863458031299676765295738880A²² + 343170393179649394520042766336A²¹ + 20588482031166919293296574464A²⁰ − 10298983100777911998712053760A¹⁹ + 860435906085534422166978560A¹⁸ + 88458142362648429128679424A¹⁷ − 18316381320409802474299392A¹⁶ + 10534479751315674228736A¹⁵ + 125128929549093922367488A¹⁴ − 1265297500371437177856A¹³ − 511240441893047875840A¹² + 6957206434058964096A¹¹ + 910744287507848384A¹⁰ − 5383463503476416A⁹ − 524408693629264A⁸ − 13137916906040A⁷ − 524470185804A⁶ + 28221402710A⁵ + 817561204A⁴ − 9640587A³ − 167651A² + 2657A − 9 = 0.005846001142957… (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
