Triangle, n = 33 record
A = 0.004728213395120…
full precision
Exact value of the published coordinate literals:
0.00472821339512040532324946879678
Symmetry
Symmetry
3-fold dihedral symmetry (group D3, order 6).
33 points in 6 orbits (6 + 6 + 6 + 6 + 6 + 3) — hover a point to see its orbit.
Minimal triangles
66 triangles tied (within relative 10−9) at the minimal area, in 11 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 6 | 0.1565 · 0.2553 · 0.4091 |
(0,7,21) (1,8,23) (2,6,22) (3,10,18) (4,11,20) (5,9,19) |
| 6 | 0.1067 · 0.3322 · 0.4360 |
(6,12,25) (7,13,26) (8,14,24) (9,15,28) (10,16,29) (11,17,27) |
| 6 | 0.2533 · 0.4163 · 0.6690 |
(0,20,29) (1,18,28) (2,19,27) (3,23,26) (4,21,25) (5,22,24) |
| 6 | 0.3690 · 0.4360 · 0.8047 |
(12,25,27) (13,26,29) (14,24,28) (15,24,28) (16,26,29) (17,25,27) |
| 6 | 0.3639 · 0.4558 · 0.8194 |
(0,3,8) (0,3,11) (1,5,6) (1,5,10) (2,4,7) (2,4,9) |
| 6 | 0.3528 · 0.4868 · 0.8393 |
(6,19,24) (7,20,25) (8,18,26) (9,22,27) (10,23,28) (11,21,29) |
| 6 | 0.3825 · 0.5884 · 0.9707 |
(12,19,23) (13,20,22) (14,18,21) (15,20,22) (16,19,23) (17,18,21) |
| 6 | 0.4091 · 0.5967 · 1.0056 |
(0,7,31) (1,8,32) (2,6,30) (3,10,32) (4,11,31) (5,9,30) |
| 6 | 0.1976 · 0.8666 · 1.0640 |
(0,6,26) (1,7,24) (2,8,25) (3,9,29) (4,10,27) (5,11,28) |
| 6 | 0.6262 · 0.6460 · 1.2722 |
(6,14,31) (7,12,32) (8,13,30) (9,17,32) (10,15,31) (11,16,30) |
| 6 | 0.4360 · 0.8639 · 1.2998 |
(12,13,25) (12,14,24) (13,14,26) (15,16,28) (15,17,27) (16,17,29) |
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— fully (D3) symmetric; 66 tied minimal triangles. - Verified in exact arithmetic: all 5456 triples enumerated, 66 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
