Convex, n = 28 record
A = 0.007718840433941…
full precision
Exact value of the published coordinate literals:
0.00771884043394155152651360829623
Symmetry
Symmetry
2-fold dihedral symmetry (group D2, order 4).
28 points in 8 orbits (4 + 4 + 4 + 4 + 4 + 4 + 2 + 2) — hover a point to see its orbit.
Minimal triangles
48 triangles tied (within relative 10−9) at the minimal area, in 12 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 4 | 0.1105 · 0.1593 · 0.2560 |
(0,8,12) (1,9,13) (2,10,14) (3,11,15) |
| 4 | 0.1367 · 0.2699 · 0.4027 |
(4,6,8) (4,6,10) (5,7,9) (5,7,11) |
| 4 | 0.1593 · 0.2915 · 0.4481 |
(0,12,15) (1,13,14) (2,13,14) (3,12,15) |
| 4 | 0.1501 · 0.3389 · 0.4867 |
(4,14,16) (5,15,17) (6,12,18) (7,13,19) |
| 4 | 0.1590 · 0.3416 · 0.4984 |
(8,16,18) (9,17,19) (10,16,18) (11,17,19) |
| 4 | 0.1501 · 0.4038 · 0.5522 |
(4,16,26) (5,17,27) (6,18,27) (7,19,26) |
| 4 | 0.3016 · 0.3760 · 0.6769 |
(0,11,27) (1,10,26) (2,9,26) (3,8,27) |
| 4 | 0.2168 · 0.4668 · 0.6828 |
(12,21,23) (13,20,22) (14,21,23) (15,20,22) |
| 4 | 0.2253 · 0.5022 · 0.7268 |
(8,17,22) (9,16,23) (10,19,20) (11,18,21) |
| 4 | 0.3444 · 0.4443 · 0.7883 |
(4,15,21) (5,14,20) (6,13,23) (7,12,22) |
| 4 | 0.1590 · 0.6493 · 0.8075 |
(8,14,18) (9,15,19) (10,12,16) (11,13,17) |
| 4 | 0.2523 · 0.6139 · 0.8658 |
(0,19,21) (1,18,20) (2,17,23) (3,16,22) |
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; 48 tied minimal triangles. - Verified in exact arithmetic: all 3276 triples enumerated, 48 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 141505808528413609398104A²¹ − 347201852078890289815572A²⁰ + 209729327252935753033652A¹⁹ − 19209568557609663339407A¹⁸ − 18402477785685131421794A¹⁷ + 5037232699028666468780A¹⁶ + 49158865077336142912A¹⁵ − 156440592887326627509A¹⁴ + 17907236430244757890A¹³ − 703766695024870589A¹² + 76173578588367617A¹¹ − 7144891802298985A¹⁰ − 250457266947307A⁹ + 15528870692727A⁸ + 702882992336A⁷ + 73193246084A⁶ − 2711427691A⁵ − 174437795A⁴ + 6324626A³ + 60417A² − 4284A + 27 = 0.007627719003112… (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
