Square, n = 28 record
A = 0.006900524983747…
full precision
Exact value of the published coordinate literals:
0.00690052498374786327735081807886
Symmetry
Symmetry
90° Rotationally symmetric (group C4, order 4).
28 points in 7 orbits (4 + 4 + 4 + 4 + 4 + 4 + 4) — hover a point to see its orbit.
Minimal triangles
52 triangles tied (within relative 10−9) at the minimal area, in 13 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 4 | 0.1738 · 0.3079 · 0.4779 |
(0,19,22) (1,18,23) (2,16,21) (3,17,20) |
| 4 | 0.1974 · 0.3186 · 0.5130 |
(12,21,26) (13,20,27) (14,23,25) (15,22,24) |
| 4 | 0.1871 · 0.3380 · 0.5222 |
(0,15,27) (1,14,26) (2,12,24) (3,13,25) |
| 4 | 0.1927 · 0.4779 · 0.6690 |
(0,7,9) (1,6,8) (2,4,11) (3,5,10) |
| 4 | 0.2703 · 0.4712 · 0.7405 |
(8,10,16) (8,11,19) (9,10,18) (9,11,17) |
| 4 | 0.1927 · 0.5753 · 0.7668 |
(4,11,21) (5,10,20) (6,8,23) (7,9,22) |
| 4 | 0.3166 · 0.5241 · 0.8400 |
(0,8,12) (1,9,13) (2,10,14) (3,11,15) |
| 4 | 0.2209 · 0.6211 · 0.8412 |
(0,4,25) (1,5,24) (2,6,27) (3,7,26) |
| 4 | 0.1905 · 0.6686 · 0.8582 |
(4,13,16) (5,12,17) (6,15,18) (7,14,19) |
| 4 | 0.3079 · 0.5546 · 0.8618 |
(0,16,19) (1,17,18) (2,16,18) (3,17,19) |
| 4 | 0.1905 · 0.7265 · 0.9163 |
(4,13,24) (5,12,25) (6,15,26) (7,14,27) |
| 4 | 0.3166 · 0.6892 · 1.0053 |
(8,12,27) (9,13,26) (10,14,24) (11,15,25) |
| 4 | 0.2885 · 0.7609 · 1.0490 |
(4,12,19) (5,13,18) (6,14,16) (7,15,17) |
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, ruin-and-recreate, soft-min L-BFGS warm-up, trust-region SLP polish); tie system refined to 80 digits by damped Gauss-Newton; verified in exact rational arithmetic— C4 (90-degree rotational) symmetric, no mirrors; 52 tied minimal triangles. Found by C4-restricted basin hopping.. - Verified in exact arithmetic: all 3276 triples enumerated, 52 tied at the minimum.
Record history
- 2026-09-15 coordinates replaced by the exact tie-system solution (30 decimals); value identified as the third-smallest positive root of 129855381559548191126323200A²² + 1449710972320347984683335680A²¹ − 2478659442167013686701457408A²⁰ + 799463337263994585732874240A¹⁹ + 582844340495777607776731136A¹⁸ − 300004839192639580226191360A¹⁷ − 26536855816606898469797888A¹⁶ + 15880766042285724417654784A¹⁵ + 2595306939146842889994240A¹⁴ + 277927529954923803981824A¹³ + 30828271753934824910336A¹² + 358185307838960007168A¹¹ − 169592977963250861696A¹⁰ − 4134782508965742336A⁹ + 348746682161613952A⁸ + 3609289394806864A⁷ − 384576262753936A⁶ + 797849507416A⁵ + 42062264670A⁴ − 137277605A³ − 1196071A² + 3250A + 9 = 0.006775810239337… (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
