Square, n = 32 record
A =
the smallest positive root of
= 0.00581090220763199441877943342965668494369584136…
solved here as the exact critical point of the 7-parameter symmetry-reduced tie system (64 tied minimal triangles); minimal polynomial found by lattice reduction on 4400 digits and verified against an 8000-digit solution
Symmetry
Symmetry
4-fold dihedral symmetry (group D4, order 8).
32 points in 5 orbits (8 + 8 + 8 + 4 + 4) — hover a point to see its orbit.
Minimal triangles
64 triangles tied (within relative 10−9) at the minimal area, in 8 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 8 | 0.2044 · 0.3687 · 0.5715 |
(0,10,22) (1,11,20) (2,8,21) (3,9,23) (4,14,18) (5,15,17) (6,12,19) (7,13,16) |
| 8 | 0.2082 · 0.3971 · 0.6039 |
(0,14,31) (1,12,30) (2,13,29) (3,15,28) (4,10,31) (5,9,28) (6,11,30) (7,8,29) |
| 8 | 0.3546 · 0.3971 · 0.7510 |
(8,22,29) (9,20,28) (10,21,31) (11,23,30) (12,18,30) (13,17,29) (14,19,31) (15,16,28) |
| 8 | 0.3687 · 0.3872 · 0.7552 |
(0,6,12) (0,7,10) (1,4,14) (1,5,11) (2,4,8) (2,5,15) (3,6,9) (3,7,13) |
| 8 | 0.2044 · 0.7429 · 0.9468 |
(0,16,22) (1,17,20) (2,18,21) (3,19,23) (4,18,20) (5,17,21) (6,19,22) (7,16,23) |
| 8 | 0.3932 · 0.6039 · 0.9968 |
(0,14,20) (1,12,22) (2,13,23) (3,15,21) (4,10,16) (5,9,19) (6,11,17) (7,8,18) |
| 8 | 0.4614 · 0.5639 · 1.0251 |
(8,17,26) (9,16,26) (10,19,27) (11,18,27) (12,23,24) (13,22,25) (14,21,25) (15,20,24) |
| 8 | 0.5639 · 0.6192 · 1.1830 |
(0,17,26) (1,16,26) (2,19,27) (3,18,27) (4,23,24) (5,22,25) (6,21,25) (7,20,24) |
Provenance
- Found by Marc-Emmanuel Coupvent des Graviers, September 2026.
- Coordinates by an external contributor, re-verified here:
Heilbronn toolkit, D4 multistart over a C-kernel orbit-reduced triangle evaluator (decomposition libre^3 + axe_h + diag, dimension 8); refined to 600 digits by Gauss-Newton on the equal-critical-area system— D4 symmetric, eight boundary points (two per side); 64 critical triangles in 8 congruence classes for 56 degrees of freedom, so the configuration is rigid. - Verified in exact arithmetic: all 4960 triples enumerated, 64 tied at the minimum.
Record history
- 2026-09-15 value identified as the smallest positive root of 1063488763600070974359809499856896A³⁷ − 8708947682935878940847296682655744A³⁶ + 27577714282136100418854598127648768A³⁵ − 55035030690536916167751488543653888A³⁴ + 79687572059409491003836999405666304A³³ − 86826735461229463308968387772678144A³² + 72067826641458876537382577805197312A³¹ − 46335759900261621424264305652531200A³⁰ + 23430186694467081518946821175508992A²⁹ − 9454582878840204434198615004545024A²⁸ + 3084403549801143870217348781703168A²⁷ − 829469968483488745392496858628096A²⁶ + 188837683301790012459719927529472A²⁵ − 37096470646574585140327426818048A²⁴ + 6205977861803315266514396905472A²³ − 838905237099045124810043555840A²² + 87050961805397770983574601728A²¹ − 5788956513771157868054052864A²⁰ − 53645277330314592912408576A¹⁹ + 85824647714248783021735936A¹⁸ − 14822557651557114148028416A¹⁷ + 1626705843100370505564160A¹⁶ − 126880757188644236951552A¹⁵ + 5193565622842539835392A¹⁴ + 460396789394005049344A¹³ − 131578910398117171200A¹² + 16532249616344451072A¹¹ − 1415430989144391168A¹⁰ + 90352776750379392A⁹ − 4443900578565568A⁸ + 169656869516896A⁷ − 4977902015600A⁶ + 109709554136A⁵ − 1755470784A⁴ + 19451614A³ − 139226A² + 571A − 1 = 0.005810902207631… (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
