Convex, n = 30 record
A = 0.007160184530281…
full precision
Exact value of the published coordinate literals:
0.00716018453028160273929173468721
Classes
Symmetry
Symmetry
5-fold dihedral symmetry (group D5, order 10).
30 points in 4 orbits (10 + 10 + 5 + 5) — hover a point to see its orbit.
Minimal triangles
55 triangles tied (within relative 10−9) at the minimal area, in 6 congruence classes:
| count | side lengths | triangles (point indices) | |
|---|---|---|---|
| 10 | 0.1622 · 0.2001 · 0.3590 |
(10,21,28) (10,22,28) (11,22,29) (11,23,29) (12,23,25) (12,24,25) (13,20,26) (13,24,26) (14,20,27) (14,21,27) | |
| 10 | 0.1358 · 0.2994 · 0.4331 |
(5,6,13) (5,6,29) (5,9,12) (5,9,28) (6,7,14) (6,7,25) (7,8,10) (7,8,26) (8,9,11) (8,9,27) | |
| 10 | 0.2001 · 0.3918 · 0.5911 |
(10,11,21) (10,14,20) (11,12,22) (12,13,23) (13,14,24) (20,25,26) (21,26,27) (22,27,28) (23,28,29) (24,25,29) | |
| 10 | 0.1093 · 0.6340 · 0.7426 |
(0,10,12) (1,11,13) (2,12,14) (3,10,13) (4,11,14) (15,25,28) (16,26,29) (17,25,27) (18,26,28) (19,27,29) | |
| 10 | 0.1093 · 0.6650 · 0.7737 |
(0,10,25) (1,11,26) (2,12,27) (3,13,28) (4,14,29) (10,15,25) (11,16,26) (12,17,27) (13,18,28) (14,19,29) | |
| 5 | 0.3513 · 0.3513 · 0.7022 |
(0,9,19) (1,5,15) (2,6,16) (3,7,17) (4,8,18) |
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— 5-fold rotationally symmetric; 55 tied minimal triangles. - Verified in exact arithmetic: all 4060 triples enumerated, 55 tied at the minimum.
Record history
- 2026-09-15 value identified as the smallest positive root of 3939947491247037013671875A¹⁴ − 1809027443005920365234375A¹³ − 561316112395490153671875A¹² + 168266147723086346328125A¹¹ + 39107809958965241518750A¹⁰ − 5786579420962864331250A⁹ − 979935551940483859375A⁸ + 146070141928380282500A⁷ + 4969296164174601375A⁶ − 1750459881031176750A⁵ + 111249174865605350A⁴ − 2255035524340525A³ − 47180338100670A² + 2315489308395A − 12710533081 = 0.006561739536739… (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
