Triangle, n = 27 record
A = 0.007029328169141…
full precision
Exact value of the published coordinate literals:
0.00702932816914157184094758788248
Symmetry
Not symmetric (group C1, order 1).
Minimal triangles
49 triangles tied (within relative 10−9) at the minimal area, in 49 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 1 | 0.1860 · 0.2018 · 0.3807 |
(3,17,20) |
| 1 | 0.1889 · 0.2018 · 0.3837 |
(3,17,25) |
| 1 | 0.1846 · 0.2821 · 0.4625 |
(6,14,16) |
| 1 | 0.1861 · 0.2885 · 0.4707 |
(2,10,24) |
| 1 | 0.1879 · 0.3018 · 0.4861 |
(13,15,26) |
| 1 | 0.1922 · 0.3031 · 0.4919 |
(1,5,22) |
| 1 | 0.1865 · 0.3122 · 0.4952 |
(0,8,17) |
| 1 | 0.1873 · 0.3146 · 0.4985 |
(0,3,21) |
| 1 | 0.2594 · 0.3828 · 0.6406 |
(4,21,25) |
| 1 | 0.2629 · 0.3856 · 0.6470 |
(7,8,20) |
| 1 | 0.2255 · 0.5181 · 0.7425 |
(4,11,17) |
| 1 | 0.2301 · 0.5260 · 0.7550 |
(3,7,23) |
| 1 | 0.1879 · 0.6631 · 0.8501 |
(7,15,26) |
| 1 | 0.4161 · 0.4387 · 0.8542 |
(6,11,20) |
| 1 | 0.4155 · 0.4424 · 0.8573 |
(13,18,19) |
| 1 | 0.1846 · 0.6747 · 0.8583 |
(7,14,16) |
| 1 | 0.4239 · 0.4387 · 0.8620 |
(11,12,20) |
| 1 | 0.1922 · 0.6734 · 0.8648 |
(1,4,5) |
| 1 | 0.4223 · 0.4449 · 0.8666 |
(23,24,25) |
| 1 | 0.1861 · 0.6828 · 0.8680 |
(2,4,10) |
| 1 | 0.4316 · 0.4449 · 0.8759 |
(9,23,25) |
| 1 | 0.1798 · 0.7196 · 0.8986 |
(5,17,18) |
| 1 | 0.2009 · 0.7168 · 0.9170 |
(3,19,26) |
| 1 | 0.2116 · 0.7135 · 0.9244 |
(10,15,23) |
| 1 | 0.2144 · 0.7224 · 0.9361 |
(1,11,14) |
| 1 | 0.3726 · 0.6301 · 1.0023 |
(3,4,6) |
| 1 | 0.3618 · 0.6486 · 1.0099 |
(4,12,16) |
| 1 | 0.3781 · 0.6413 · 1.0191 |
(7,17,24) |
| 1 | 0.4854 · 0.5341 · 1.0191 |
(9,21,26) |
| 1 | 0.3660 · 0.6575 · 1.0230 |
(2,7,9) |
| 1 | 0.4882 · 0.5377 · 1.0256 |
(5,8,12) |
| 1 | 0.3151 · 0.7252 · 1.0398 |
(1,6,15) |
| 1 | 0.3155 · 0.7252 · 1.0403 |
(1,15,24) |
| 1 | 0.3733 · 0.7535 · 1.1265 |
(0,6,7) |
| 1 | 0.2657 · 0.8625 · 1.1278 |
(6,17,21) |
| 1 | 0.3733 · 0.7662 · 1.1392 |
(0,7,13) |
| 1 | 0.3781 · 0.7654 · 1.1432 |
(0,4,24) |
| 1 | 0.2781 · 0.8699 · 1.1477 |
(2,5,13) |
| 1 | 0.3781 · 0.7781 · 1.1559 |
(0,4,22) |
| 1 | 0.2808 · 0.8758 · 1.1563 |
(16,22,26) |
| 1 | 0.4749 · 0.7155 · 1.1902 |
(8,9,18) |
| 1 | 0.4728 · 0.7330 · 1.2056 |
(12,19,21) |
| 1 | 0.4625 · 0.7461 · 1.2083 |
(6,14,25) |
| 1 | 0.4707 · 0.7584 · 1.2289 |
(10,20,24) |
| 1 | 0.4424 · 0.8601 · 1.3023 |
(13,19,22) |
| 1 | 0.6301 · 0.6816 · 1.3115 |
(4,6,9) |
| 1 | 0.6486 · 0.6786 · 1.3270 |
(4,12,13) |
| 1 | 0.6413 · 0.6907 · 1.3319 |
(7,12,24) |
| 1 | 0.6575 · 0.6884 · 1.3457 |
(7,9,22) |
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— not symmetric; 49 tied minimal triangles. - Verified in exact arithmetic: all 2925 triples enumerated, 49 tied at the minimum.
Record history
- 2026-09-15 value identified as the second-smallest positive root of 483033378918252657197180798068873245566038984492377336276366798398926960297151402767398024402A⁸⁵ + 4191023138645823137100831678333426811180348004894124282602306157377669032496708394192067759878A⁸⁴ + 8858611232602429342354962404888304395998659825645006876170683599719500943389453840208460605529A⁸³ + 32078659003215688111611720169536416781268542789449873034554985932498256917251458022364471026764A⁸² + 94535231028104484123166237399706697073748995217122795979746732035199573557283782605398260793371A⁸¹ − 19244257086350233877096191553739982688018382125233635205428839345820309737517914805036540136296A⁸⁰ − 392331670204065946319530218277534089110568801125448325789926201521201433462262244951962368265179A⁷⁹ + 641280112197960730096224852048569388272480005990028027048498843114813071197973776784431515300157A⁷⁸ − 751612900738408728405787177200598953547219664352037424322618423181857219761291189263563359856027A⁷⁷ + 591453748520177937988090200421064842920827279715889014076868930186676111233633803676036603492012A⁷⁶ − 276230950534897142928598655380684312167708726633570504193720249502872650942810748759623301712207A⁷⁵ + 75659842030621900231834937208496755615820001985138078309522956779040567466994779506188472414213A⁷⁴ + 14535873412489157778511905719101278946750176819072821430708407929560723777066055076075513386915A⁷³ − 17523838521909626523435165236814252196323506436867698435844641459202290743173919448860846192675A⁷² − 6769178350296683375980738588956009957263289931645976875886843343508382518614285939178691058031A⁷¹ + 11580787450699492035626150213465289667071836594450105561328111056106058152473269125307569658437A⁷⁰ − 7189131772707977054224363708915207064748416574462595285818679850748921665294062855034991350847A⁶⁹ + 3335950950299603553655180786365263806673126142853948405677396357884735296120432895005444170849A⁶⁸ − 1303315478419791576291847152749221520533261249119209934812244388810161215987887238513084186480A⁶⁷ + 430835792147845183796247548649519625457817511174170513037204610083861057545435743693113499184A⁶⁶ − 119970841363175128211869996893418132529599000834597149024569223273692925351656959655172939343A⁶⁵ + 28172801346068437051717893614466926596989133978099847435318290271673795618940267261033149818A⁶⁴ − 5571381854102491978644654189782625705729040271225558873719464346082947111211666970671090746A⁶³ + 919935246543057323524288132626194846328965142881489637361653041845617692055260006812738928A⁶² − 124221447101063732053006339899229357440879302881737760220402591142330157159312377681280105A⁶¹ + 12996683442260508198348044105555287342511132837097330010583785011599124156795100181278575A⁶⁰ − 876425700333017902444893061298677544921760184070388642416705058172103192091841718435282A⁵⁹ + 585947601328070701976733757655756256574331507863718407310252896848753093124266455858A⁵⁸ + 6788948783964122865016825144015273674933379967489049771045059367240498596041029667664A⁵⁷ + 207946570267029054681110102228565601114278501140610683737100956906643734162848010436A⁵⁶ − 420352136821733422576361557252636928906978702338997665951690069365172885698675508206A⁵⁵ + 116167573824248363512201500786860228705288679347183668714650413497832186790328969415A⁵⁴ − 20480153451774620049441933324300713541107209540841745485499017418332347139053314607A⁵³ + 2713003435935458298653308292791704537201166993375462594683300906935647052358692422A⁵² − 283647389395647221318324904930871869923886814737312698546565989821931227268198502A⁵¹ + 23669205746961906035993137129912146064430375041081617935027325128342279622565792A⁵⁰ − 1572827210149786149702700564267279484551691457152277801156549909026435845317881A⁴⁹ + 85307630047108435361764428270022362571154901703436487658342176979315853688767A⁴⁸ − 4416721098771580061681991230447725165379055178432297791525385335375623471348A⁴⁷ + 281608779825970274083947043202726158619165815229100060273152290697890083132A⁴⁶ − 18836921550408549457144968428857275517630670801441094005613258459917881408A⁴⁵ + 814056383766002411632547005495895895806244027104746386688336318775617435A⁴⁴ − 4805380933293901191103558469576468066046697181728633502974635299707192A⁴³ − 821753013162996253165133469072103516108744971760271163194903551703993A⁴² − 123513264746230417070722142235638649788661883326603047383679939508285A⁴¹ + 32135455812826513884666922710825454146232059633950282431563207424175A⁴⁰ − 4434162297880726720509947380289773897880793013505081092018463830091A³⁹ + 478443870334868556243916262970994358102667724622014948330959428913A³⁸ − 38322983951670290088260604549374553284500798838453059652597024066A³⁷ + 1999956701167991610267302501461123049742388831076980871752870882A³⁶ − 44980821867450708751620648931635591588602578742547347772953891A³⁵ − 1939003431710395191982693557786816737621187380683463833416545A³⁴ + 217502964781839012260524364554240341830313022114021895449153A³³ − 11943485090715452638795522445687588934683737202759330290143A³² + 775632466050702220948035533702312410855103639354886735050A³¹ − 50548005958683746608052351807706292188595716344429467258A³⁰ + 1311518928012931520583587556156377560324430746402614862A²⁹ + 125701554469935440526711822010594220844554917228966945A²⁸ − 14847847806918856519062698036805995931969120829160093A²⁷ + 590360265563144568979957447861260980634748870275778A²⁶ + 4562489168872673220784875152364971155057900815177A²⁵ − 1514008174735003964032974412314217602370483958833A²⁴ + 58702716092634127044631407408490932872029721201A²³ + 120254286906876608495704610891782123108271468A²² − 85190034864509513132980731794425670692592981A²¹ + 2390656922575513021015185091593351685088138A²⁰ + 18737988254128912389650160957925023809197A¹⁹ − 1694591188593847500835377906266308531018A¹⁸ − 6354726236224840159318140389369986766A¹⁷ + 797625338338668320269872971465203682A¹⁶ + 21819126706622890754556662759463056A¹⁵ − 754428584318941664912547339524512A¹⁴ − 13334376725942899504135783588754A¹³ + 392600307899550218611842388085A¹² + 7599954677080018824023968882A¹¹ − 105915966903331596264255142A¹⁰ − 4300639663947662980319900A⁹ − 33390915228809104488879A⁸ + 1441504057609716121898A⁷ + 26961419501216172395A⁶ + 252878589601627544A⁵ − 2680960008605724A⁴ − 22509340687304A³ + 204695651520A² − 760993248A + 1403136 = 0.006877860347970… (this site, from the symmetry-reduced tie system)
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