Triangle, n = 19 record
A = 0.013481891382806…
full precision
Exact value of the published coordinate literals:
0.0134818913828067437130718338319
Symmetry
Not symmetric (group C1, order 1).
Minimal triangles
30 triangles tied (within relative 10−9) at the minimal area, in 30 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 1 | 0.1377 · 0.2417 · 0.3412 |
(0,4,15) |
| 1 | 0.1309 · 0.2557 · 0.3496 |
(11,14,16) |
| 1 | 0.1309 · 0.2875 · 0.3903 |
(11,13,16) |
| 1 | 0.1547 · 0.2784 · 0.4107 |
(1,2,8) |
| 1 | 0.1547 · 0.2864 · 0.4199 |
(1,2,10) |
| 1 | 0.1377 · 0.3049 · 0.4198 |
(4,15,18) |
| 1 | 0.2438 · 0.2762 · 0.5090 |
(10,12,17) |
| 1 | 0.2403 · 0.2932 · 0.5233 |
(3,6,18) |
| 1 | 0.2581 · 0.2965 · 0.5458 |
(6,12,13) |
| 1 | 0.2476 · 0.3286 · 0.5682 |
(0,5,17) |
| 1 | 0.2417 · 0.4288 · 0.6651 |
(0,4,10) |
| 1 | 0.2557 · 0.4169 · 0.6674 |
(8,14,16) |
| 1 | 0.2784 · 0.4169 · 0.6907 |
(2,8,14) |
| 1 | 0.2834 · 0.4661 · 0.7457 |
(3,8,13) |
| 1 | 0.2875 · 0.4699 · 0.7539 |
(11,13,18) |
| 1 | 0.3049 · 0.4699 · 0.7715 |
(13,15,18) |
| 1 | 0.2712 · 0.5090 · 0.7768 |
(10,12,18) |
| 1 | 0.3402 · 0.4723 · 0.8096 |
(1,7,17) |
| 1 | 0.3159 · 0.5233 · 0.8366 |
(3,14,18) |
| 1 | 0.4536 · 0.4723 · 0.9240 |
(1,4,17) |
| 1 | 0.2834 · 0.6663 · 0.9477 |
(2,3,13) |
| 1 | 0.4288 · 0.5987 · 1.0262 |
(0,7,10) |
| 1 | 0.5256 · 0.5268 · 1.0511 |
(6,11,15) |
| 1 | 0.3159 · 0.7736 · 1.0882 |
(3,14,15) |
| 1 | 0.3655 · 0.7453 · 1.1095 |
(5,10,16) |
| 1 | 0.3446 · 0.7850 · 1.1285 |
(0,11,12) |
| 1 | 0.5268 · 0.6173 · 1.1431 |
(4,6,11) |
| 1 | 0.3975 · 0.7473 · 1.1437 |
(4,12,14) |
| 1 | 0.5089 · 0.6623 · 1.1703 |
(5,6,7) |
| 1 | 0.5644 · 0.6674 · 1.2310 |
(7,8,16) |
Provenance
- Found by Marc-Emmanuel Coupvent des Graviers, September 2026.
- Coordinates by an external contributor, re-verified here:
One-point deletion from the n=20 record, refined to 120 digits by Gauss-Newton (refine_tri.py)— No symmetry; obtained by deleting one point of the n=20 record (Fable 5.1 with Shengtong Zhang): two points on each side, 30 minimal triangles, not rigid (rank 30/33).. - Verified in exact arithmetic: all 969 triples enumerated, 30 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 121700424194596706563914852976470923088332989288383432429600338116608A⁴⁰ − 1624241589124961109259204575872237896185904791843192582294453624637440A³⁹ + 1485600632751214499546000486610622732784091962556279572623155744079776A³⁸ − 651695021414292482179378312596166211074208096511735267711479528699097A³⁷ + 219085371916340910322795817799139047096725837854446111919052972646381A³⁶ − 41531361943274856337287866832322699953713604349062302785228787464851A³⁵ + 4799525575408198441132146858904269349824944250316514904140911368993A³⁴ + 143432086031696841116271530027217078851512039130150725081416086103A³³ − 179289384224524583348064157284113542250479429664968162119593535642A³² − 41843339707982849748852965058784735011005975324833715468116761363A³¹ + 16313055637294098933235610527501317048652284879873292588356272454A³⁰ − 2799308376791849551462067350568632156732171498818122203458591405A²⁹ + 234792285619574463616553793419739598184978953587012308567478993A²⁸ − 44351206689213226968375725135994569317842670477342822002970593A²⁷ + 4804213284643109261872828123586836957938751554361546629109905A²⁶ − 630862533839864548520963038320431494409427342294224406232803A²⁵ + 69709475616889176214603191992455944315601694273278520227884A²⁴ − 15866321323869662771294838589623695342858281940949112283061A²³ + 2176811396375652335888660658914962580584258687309543760304A²² − 197649184763002872067257257313736893877657186431664376502A²¹ + 9919695767337541034167406869959322247687818260077499864A²⁰ − 2027444449816312765124565806351050433887336679555261076A¹⁹ + 470137433878226610790501993624206678885085095789926880A¹⁸ − 35267500054821856270925594677771613478947180431457304A¹⁷ + 2550700931423984295525297867511659845664194795901360A¹⁶ − 532744756084283521072958876284649994924612037034064A¹⁵ + 60191257530067372085596978546915458686983310910912A¹⁴ − 3170810707482397648776609704565004275815344828224A¹³ + 99507899729798315181667272601534713583763522304A¹² − 5889767910179697661120975645902243307530230784A¹¹ + 524728597444972252545401047192743704799516672A¹⁰ − 29202533240741699718776871354362858223818752A⁹ + 1005020107307790696646780062439504442345472A⁸ − 23715158183134211834821542612944747175936A⁷ + 430662953147421009389535647789523566592A⁶ − 6563572757550263333016818596624121856A⁵ + 83966917297191372140341779141918720A⁴ − 880228654544429809906505703358464A³ + 7687794935721245140360595767296A² − 50933541244180629691939946496A + 171593111953606186648469504 = 0.013453543425964… (this site, from the symmetry-reduced tie system)
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