Square, n = 31 record
A = 0.005837068845813…
full precision
Exact value of the published coordinate literals:
0.0058370688458137627921035586805
Symmetry
Not symmetric (group C1, order 1).
Minimal triangles
56 triangles tied (within relative 10−9) at the minimal area, in 56 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 1 | 0.2028 · 0.3667 · 0.5678 |
(5,15,17) |
| 1 | 0.2052 · 0.3664 · 0.5700 |
(3,9,22) |
| 1 | 0.2061 · 0.3659 · 0.5704 |
(6,12,18) |
| 1 | 0.2054 · 0.3707 · 0.5745 |
(7,13,16) |
| 1 | 0.2062 · 0.3699 · 0.5746 |
(1,11,19) |
| 1 | 0.2049 · 0.3752 · 0.5786 |
(0,10,21) |
| 1 | 0.2079 · 0.3732 · 0.5796 |
(2,8,20) |
| 1 | 0.2067 · 0.3941 · 0.5994 |
(3,15,27) |
| 1 | 0.2086 · 0.3937 · 0.6009 |
(5,9,27) |
| 1 | 0.2072 · 0.3959 · 0.6017 |
(0,14,30) |
| 1 | 0.2068 · 0.3999 · 0.6053 |
(4,10,30) |
| 1 | 0.2084 · 0.4002 · 0.6073 |
(2,13,28) |
| 1 | 0.2140 · 0.3986 · 0.6113 |
(6,11,29) |
| 1 | 0.2116 · 0.4030 · 0.6133 |
(7,8,28) |
| 1 | 0.2150 · 0.4064 · 0.6202 |
(7,23,25) |
| 1 | 0.2062 · 0.4816 · 0.6868 |
(1,3,19) |
| 1 | 0.2061 · 0.4835 · 0.6886 |
(4,6,18) |
| 1 | 0.2049 · 0.4941 · 0.6981 |
(0,2,21) |
| 1 | 0.2507 · 0.4944 · 0.7444 |
(10,24,29) |
| 1 | 0.2503 · 0.4951 · 0.7446 |
(15,25,29) |
| 1 | 0.3523 · 0.3959 · 0.7476 |
(14,18,30) |
| 1 | 0.3560 · 0.3937 · 0.7490 |
(9,19,27) |
| 1 | 0.3659 · 0.3839 · 0.7491 |
(0,6,12) |
| 1 | 0.3521 · 0.4002 · 0.7517 |
(13,17,28) |
| 1 | 0.3699 · 0.3825 · 0.7518 |
(1,5,11) |
| 1 | 0.3588 · 0.3941 · 0.7523 |
(15,16,27) |
| 1 | 0.3707 · 0.3840 · 0.7540 |
(3,7,13) |
| 1 | 0.3667 · 0.3918 · 0.7579 |
(2,5,15) |
| 1 | 0.3664 · 0.3930 · 0.7587 |
(3,6,9) |
| 1 | 0.2500 · 0.5100 · 0.7593 |
(11,23,30) |
| 1 | 0.3732 · 0.3867 · 0.7593 |
(2,4,8) |
| 1 | 0.3608 · 0.3999 · 0.7600 |
(10,20,30) |
| 1 | 0.3597 · 0.4030 · 0.7620 |
(8,21,28) |
| 1 | 0.2898 · 0.5043 · 0.7934 |
(22,24,27) |
| 1 | 0.2929 · 0.5016 · 0.7939 |
(16,26,28) |
| 1 | 0.2897 · 0.5056 · 0.7947 |
(17,26,27) |
| 1 | 0.1598 · 0.7023 · 0.8614 |
(3,11,12) |
| 1 | 0.2062 · 0.7401 · 0.9459 |
(1,17,19) |
| 1 | 0.2028 · 0.7445 · 0.9468 |
(5,17,20) |
| 1 | 0.2052 · 0.7428 · 0.9475 |
(3,18,22) |
| 1 | 0.3925 · 0.6017 · 0.9940 |
(0,14,19) |
| 1 | 0.3954 · 0.6009 · 0.9961 |
(5,9,18) |
| 1 | 0.3907 · 0.6073 · 0.9977 |
(2,13,22) |
| 1 | 0.3892 · 0.6113 · 1.0002 |
(6,11,17) |
| 1 | 0.4012 · 0.6053 · 1.0062 |
(4,10,16) |
| 1 | 0.4584 · 0.5614 · 1.0195 |
(10,18,26) |
| 1 | 0.4551 · 0.5686 · 1.0235 |
(9,16,25) |
| 1 | 0.4590 · 0.5692 · 1.0280 |
(14,20,24) |
| 1 | 0.4694 · 0.5618 · 1.0309 |
(8,17,25) |
| 1 | 0.4702 · 0.5637 · 1.0337 |
(12,22,23) |
| 1 | 0.4665 · 0.5721 · 1.0383 |
(13,21,24) |
| 1 | 0.5686 · 0.6127 · 1.1811 |
(1,16,25) |
| 1 | 0.5692 · 0.6132 · 1.1823 |
(6,20,24) |
| 1 | 0.5637 · 0.6289 · 1.1925 |
(4,22,23) |
| 1 | 0.2920 · 0.9006 · 1.1924 |
(7,18,29) |
| 1 | 0.5721 · 0.6237 · 1.1956 |
(5,21,24) |
Provenance
- Found by Tej Stead, September 2026.
- Coordinates by an external contributor, re-verified here:
heilbronn-site search/refine.py — square n=32 record minus one point, re-polished— Marc-Emmanuel Coupvent des Graviers' n=32 configuration with one point deleted and re-polished; removing a point cannot lower the minimum area, and the n=32 value already exceeded the n=31 record.. - Verified in exact arithmetic: all 4495 triples enumerated, 56 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
