Triangle, n = 34 record
A = 0.004430910598389…
full precision
Exact value of the published coordinate literals:
0.00443091059838985521075750190747
Symmetry
Symmetry
3-fold dihedral symmetry (group D3, order 6).
34 points in 7 orbits (6 + 6 + 6 + 6 + 6 + 3 + 1) — hover a point to see its orbit.
Minimal triangles
60 triangles tied (within relative 10−9) at the minimal area, in 10 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 6 | 0.2022 · 0.5671 · 0.7689 |
(0,4,20) (0,4,21) (1,3,18) (1,3,23) (2,5,19) (2,5,22) |
| 6 | 0.2729 · 0.5080 · 0.7805 |
(0,25,31) (1,26,32) (2,24,30) (3,28,32) (4,29,31) (5,27,30) |
| 6 | 0.2754 · 0.5065 · 0.7815 |
(6,7,32) (6,8,31) (7,8,30) (9,10,31) (9,11,32) (10,11,30) |
| 6 | 0.2100 · 0.6230 · 0.8326 |
(6,16,29) (7,15,28) (8,17,27) (9,13,26) (10,12,25) (11,14,24) |
| 6 | 0.1246 · 0.7189 · 0.8430 |
(6,17,25) (7,16,26) (8,15,24) (9,14,28) (10,13,29) (11,12,27) |
| 6 | 0.3787 · 0.5081 · 0.8866 |
(0,24,26) (1,24,25) (2,25,26) (3,27,29) (4,27,28) (5,28,29) |
| 6 | 0.2754 · 0.7060 · 0.9812 |
(6,15,32) (7,17,30) (8,16,31) (9,12,31) (10,14,30) (11,13,32) |
| 6 | 0.3083 · 0.6998 · 1.0078 |
(0,12,28) (1,13,27) (2,14,29) (3,15,25) (4,16,24) (5,17,26) |
| 6 | 0.1717 · 0.8401 · 1.0116 |
(12,17,19) (12,17,21) (13,16,20) (13,16,23) (14,15,18) (14,15,22) |
| 6 | 0.5010 · 0.7141 · 1.2150 |
(0,16,18) (1,15,19) (2,17,20) (3,13,21) (4,12,22) (5,14,23) |
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; 60 tied minimal triangles. - Verified in exact arithmetic: all 5984 triples enumerated, 60 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
