Triangle, n = 28 record
A = 0.006694341604594…
full precision
Exact value of the published coordinate literals:
0.00669434160459487243059851886647
Symmetry
Symmetry
3-fold dihedral symmetry (group D3, order 6).
28 points in 6 orbits (6 + 6 + 6 + 6 + 3 + 1) — hover a point to see its orbit.
Minimal triangles
54 triangles tied (within relative 10−9) at the minimal area, in 9 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 6 | 0.1821 · 0.5931 · 0.7741 |
(0,22,23) (1,21,22) (2,21,23) (3,19,20) (4,18,19) (5,18,20) |
| 6 | 0.1867 · 0.6302 · 0.8160 |
(0,11,24) (1,10,25) (2,9,26) (3,8,24) (4,7,26) (5,6,25) |
| 6 | 0.1821 · 0.6878 · 0.8691 |
(0,10,23) (1,9,22) (2,11,21) (3,7,20) (4,6,19) (5,8,18) |
| 6 | 0.3481 · 0.6806 · 1.0283 |
(6,13,24) (7,14,25) (8,12,26) (9,16,24) (10,17,26) (11,15,25) |
| 6 | 0.3479 · 0.6878 · 1.0354 |
(0,10,12) (1,9,13) (2,11,14) (3,7,15) (4,6,16) (5,8,17) |
| 6 | 0.3481 · 0.7174 · 1.0651 |
(6,15,24) (7,17,25) (8,16,26) (9,12,24) (10,14,26) (11,13,25) |
| 6 | 0.2748 · 0.8908 · 1.1652 |
(0,8,15) (1,6,17) (2,7,16) (3,11,12) (4,9,14) (5,10,13) |
| 6 | 0.5317 · 0.6848 · 1.2162 |
(0,14,21) (1,12,23) (2,13,22) (3,17,18) (4,15,20) (5,16,19) |
| 6 | 0.4836 · 0.9103 · 1.3937 |
(12,13,21) (12,14,22) (13,14,23) (15,16,18) (15,17,19) (16,17,20) |
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— fully (D3) symmetric; 54 tied minimal triangles. - Verified in exact arithmetic: all 3276 triples enumerated, 54 tied at the minimum.
Record history
- 2026-09-15 value identified as the smallest positive root of 846946762630674412914894488973679014320417836574432516725308816A⁶² + 1040463958094198170293594446198860050537785395989723565364660796A⁶¹ + 4693098577077516293191912107821382037764937476077409228845061081A⁶⁰ + 5315785815746919436170845254278880568215666941744170905882341135A⁵⁹ + 10131302854853364336104004948107740930730446386395081353512748386A⁵⁸ + 10203032322964115627748396370353722009608026173731513908536638554A⁵⁷ + 9466758384051369579645179085286528572241009594585057151589371133A⁵⁶ + 9307473806332753174189565681869391357338998917414974738947760401A⁵⁵ + 2481272557792435767091344490924205826904024347759777531908895306A⁵⁴ + 460571144668986810464785600923429183338667242496457741195505406A⁵³ + 962750087418730885145714231780644885907310537787048833602931450A⁵² + 43716755458109619417010943524209550310552595974855404877430882A⁵¹ − 23924631387360919502819257694126686048961422579221822345663087A⁵⁰ − 8426995484126094447429559004347571551463644442805347932319106A⁴⁹ − 4096670214901701956598381820129918900530576500558992043491416A⁴⁸ − 652459554194789933935938318154199236265623581731044393148709A⁴⁷ + 347188921844507220604528367239507458249347849013857729014878A⁴⁶ − 1478319522255936653331953141880314624827591797130318526381229A⁴⁵ + 1026784092066548802390134047914103466641313811567709601530230A⁴⁴ − 37136721635893545287184261564676217884642987887827416008993A⁴³ − 161058901494832056356904600201637394866609979900998734513849A⁴² + 55013448962323150466252655183106144610734037927316150145047A⁴¹ − 1725006846787181320345319785017759436646778509095217197131A⁴⁰ − 3428725102413237990358544302973436379891989581206693823376A³⁹ + 1051845967088273865409539481924679272972027721505929428095A³⁸ − 124302030988654697659912880437801174236948924958784785749A³⁷ − 7834285174402048117167823326216348186694731750141596958A³⁶ + 6017056697249438164804958444397097970979204321156492084A³⁵ − 1292873424646543476474891082984209186466594484998224738A³⁴ + 174719945530250969905024327318497732144980622048876579A³³ − 14980849692559344551464724718567451387267730814148839A³² + 212051388305661140340588536912849651140049576372142A³¹ + 220848283149160872990603836086675885290132674484777A³⁰ − 55574793025026898163877813925895347514357003506653A²⁹ + 9282261449002141914556472251879090878258716230713A²⁸ − 1256451538732390443283776799260983210532470305185A²⁷ + 145724633705229664822089961015778560663267325682A²⁶ − 14785426990586465566907455085702371632704433735A²⁵ + 1322038933143406319616580646403450257501952156A²⁴ − 104272099232701396222114945001473061596271994A²³ + 7237030147600943110077075674980215219489543A²² − 440120687628031593164742101241955459584411A²¹ + 23321247001459781964033066432778908450135A²⁰ − 1068770606494687122340645567167224954223A¹⁹ + 41918019201725812367327053799781195672A¹⁸ − 1383580756547360070653602062610629822A¹⁷ + 37251742944809687195160058099356693A¹⁶ − 760647935348493312411301098430668A¹⁵ + 8962736799691699876669626349893A¹⁴ + 89826937188253724411427218270A¹³ − 8697339042659371158869531441A¹² + 292942109995317195208331627A¹¹ − 7038805733217372382933794A¹⁰ + 137233621568375334458932A⁹ − 2306406439806791821446A⁸ + 34586582248843400631A⁷ − 465100156849482046A⁶ + 5451186930367743A⁵ − 52831547855201A⁴ + 396184991848A³ − 2112250652A² + 7003296A − 10640 = 0.006165171202500… (this site, from the symmetry-reduced tie system)
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