Convex, n = 19 record

A = the second-smallest positive root of

16654996327987663164080307837682632049874839205015076556456543273852809727444121317152538714517183919 x^{51} - 50741002518352730600264528339104376773434838790202632540805147086034247252763865792753920464267635573 x^{50} + 59523505893012158559437235867853244947179692023685017476500188651227296076109329215656086064011267489 x^{49} - 36987041733778503631684570054578915904064764481393835129403048466929628726892550253194595633252726850 x^{48} + 13802020942359781359532835670331400292457190606793717904934340951894516144229404778573568979157901556 x^{47} - 3296658808903291667381651953620346122872866425015747469513998802768001079781870454051426962094273277 x^{46} + 522699078360979182096354950715984808762864042609018058610467299133481973843165505430892133091818631 x^{45} - 56082102368193268784691303908377332071558230085164345974612381847171956162447511292919077809772193 x^{44} + 3779588158414745759040253926156411201169394223850594499546797341163372099615339644694043803476810 x^{43} + 38804158227810064872881847143965708389640299045318202795044060326916513806603210230514813480670 x^{42} - 76245308599054938450279905392109547756658057869894850434217737857123867652222317431472472454500 x^{41} + 17213914510096799610910794433407976213793308524909473858407512374332360333733441195768093958666 x^{40} - 2581698546922737206754552938808007819834442770590777748460326465749984379596151951704067940505 x^{39} + 312072360548528036615085784780657382510240660922647503094597198643874680225097890033187808289 x^{38} - 34094820277978157885246445053088308041049796551224894512623542089173869593754653498973936641 x^{37} + 3410402811145246131813770717309440263360780712183061972564114609654437120334322647184416544 x^{36} - 271282760586160591069969069140476945510429502380734626569881260277916827821253614589450353 x^{35} + 8964213447929428305402665467314421858263171553663157732197428154274832533890558641276837 x^{34} + 1923203007732281067391499795416781297576812695277433611936479858013445273721466583889417 x^{33} - 501518281647971586746624890890317734768769703924228123088295643341747807703660169964670 x^{32} + 74901741414760772803431635718001969326647968436687168820562144741951877663763333348762 x^{31} - 8793339854703183362216942014089270298583083848430875674639726423156497665143583856599 x^{30} + 884440462041187107283475346039472588977291523754286257033259570465574394383577964060 x^{29} - 78974785991390863296101086854453347114231599106556602496429768566048683284604195488 x^{28} + 6351440938225235091307229476747584664822770409807711680636450788763662338660730736 x^{27} - 461577386020991827608535129901123426274312148150149667208924550289794040921123158 x^{26} + 30241062227720993053683464944230198235675014316896137072924910087824613616621533 x^{25} - 1777889704731817957933477778233549802357792023928514745543045283724426249599528 x^{24} + 93267061234360602967296674081132845199112348187406824265727268012016566200583 x^{23} - 4338371522575923233034210097677980674235870009879661484100060767182893535897 x^{22} + 177640264174182468779901368262860951792277130596752158820640835566528124760 x^{21} - 6346615055201503020261343318716637981874577322541383923515904824683154528 x^{20} + 195719278657162866559477668957744832524686044349121546932247027308617521 x^{19} - 5146798111408492011733625064073866364135976197359254613756039953565431 x^{18} + 114342597713325086360466620824031491178078220012926733476841878296701 x^{17} - 2166532041106191143395488189771983665844918905009609494849319862555 x^{16} + 37534499535017718712938026546492630945885931493825986952019257529 x^{15} - 686311606973090058567849302474565048555704337440088851152041666 x^{14} + 13463640572723940217486049731712260212765015277098533721414827 x^{13} - 208766904360611863076421478201045069313767677355624072803244 x^{12} + 523821011921006477340257754559207277970409454884057162450 x^{11} + 85093250513592250373365314115539360225677251028427399015 x^{10} - 2714743802786573644617033845589881585012310918640093409 x^{9} + 45990737626797492919030822727366787337965820963078068 x^{8} - 487271946585283985377454739508037532093635633664116 x^{7} + 2432093267720660392912770667937619993847163656888 x^{6} + 5979744051687564902194282372076326165161580304 x^{5} + 4273919079255637923529042315667407153215968 x^{4} - 892839406092143487318163123564942857358272 x^{3} - 5635659728305912723837243683640292006016 x^{2} - 1029008599106478248990748958653643776 x + 87191354287695850324598344835925504

= 0.0159310923501429601958965531934385297732250146…

solved here as the exact critical point of the 12-parameter symmetry-reduced tie system (33 tied minimal triangles); minimal polynomial found by lattice reduction on 11000 digits and verified against a 20000-digit solution

33 triangles tie for the minimal area.

Symmetry

Not symmetric (group C1, order 1).

Minimal triangles

33 triangles tied (within relative 10⁻⁹) at the minimal area, in 33 congruence classes:

count	side lengths	triangles (point indices)
1	`0.1961 · 0.4213 · 0.6068`	(6,11,12)
1	`0.1965 · 0.4217 · 0.6076`	(7,9,13)
1	`0.1964 · 0.4221 · 0.6080`	(8,10,14)
1	`0.2738 · 0.3833 · 0.6493`	(9,14,18)
1	`0.2740 · 0.3840 · 0.6503`	(10,12,18)
1	`0.2743 · 0.3839 · 0.6505`	(11,13,18)
1	`0.2721 · 0.4742 · 0.7408`	(7,12,14)
1	`0.2725 · 0.4746 · 0.7417`	(6,13,14)
1	`0.2725 · 0.4751 · 0.7422`	(8,12,13)
1	`0.1961 · 0.6167 · 0.8075`	(6,11,17)
1	`0.1964 · 0.6171 · 0.8081`	(8,10,16)
1	`0.1965 · 0.6178 · 0.8090`	(7,9,15)
1	`0.3539 · 0.4967 · 0.8471`	(0,8,15)
1	`0.3542 · 0.4969 · 0.8477`	(1,6,16)
1	`0.3546 · 0.4976 · 0.8487`	(2,7,17)
1	`0.2721 · 0.6068 · 0.8753`	(6,7,12)
1	`0.2725 · 0.6076 · 0.8765`	(7,8,13)
1	`0.2725 · 0.6080 · 0.8769`	(6,8,14)
1	`0.3413 · 0.6598 · 0.9988`	(0,3,10)
1	`0.3418 · 0.6604 · 0.9999`	(1,4,11)
1	`0.3419 · 0.6611 · 1.0007`	(2,5,9)
1	`0.4404 · 0.5847 · 1.0230`	(0,2,13)
1	`0.4411 · 0.5855 · 1.0247`	(1,2,12)
1	`0.4410 · 0.5857 · 1.0248`	(0,1,14)
1	`0.3833 · 0.6525 · 1.0337`	(9,16,18)
1	`0.3840 · 0.6534 · 1.0354`	(10,17,18)
1	`0.3839 · 0.6536 · 1.0355`	(11,15,18)
1	`0.5190 · 0.5502 · 1.0675`	(4,10,15)
1	`0.5190 · 0.5505 · 1.0679`	(5,11,16)
1	`0.5199 · 0.5512 · 1.0694`	(3,9,17)
1	`0.4404 · 0.7669 · 1.2059`	(2,4,13)
1	`0.4410 · 0.7678 · 1.2076`	(0,5,14)
1	`0.4411 · 0.7683 · 1.2081`	(1,3,12)

Provenance

Found by Nathan Sudermann-Merx, August 2026.
Coordinates by an external contributor, re-verified here: spiralulam geometry-problems record_hunt pipeline, 2026-08-27 — cold start restricted to cyclic and dihedral symmetry classes, then symmetry-breaking polish; the remaining flat direction closed by trust-region SLP.
Verified in exact arithmetic: all 969 triples enumerated, 33 tied at the minimum.

Record history

2026-09-15 value identified as the second-smallest positive root of 16654996327987663164080307837682632049874839205015076556456543273852809727444121317152538714517183919A⁵¹ − 50741002518352730600264528339104376773434838790202632540805147086034247252763865792753920464267635573A⁵⁰ + 59523505893012158559437235867853244947179692023685017476500188651227296076109329215656086064011267489A⁴⁹ − 36987041733778503631684570054578915904064764481393835129403048466929628726892550253194595633252726850A⁴⁸ + 13802020942359781359532835670331400292457190606793717904934340951894516144229404778573568979157901556A⁴⁷ − 3296658808903291667381651953620346122872866425015747469513998802768001079781870454051426962094273277A⁴⁶ + 522699078360979182096354950715984808762864042609018058610467299133481973843165505430892133091818631A⁴⁵ − 56082102368193268784691303908377332071558230085164345974612381847171956162447511292919077809772193A⁴⁴ + 3779588158414745759040253926156411201169394223850594499546797341163372099615339644694043803476810A⁴³ + 38804158227810064872881847143965708389640299045318202795044060326916513806603210230514813480670A⁴² − 76245308599054938450279905392109547756658057869894850434217737857123867652222317431472472454500A⁴¹ + 17213914510096799610910794433407976213793308524909473858407512374332360333733441195768093958666A⁴⁰ − 2581698546922737206754552938808007819834442770590777748460326465749984379596151951704067940505A³⁹ + 312072360548528036615085784780657382510240660922647503094597198643874680225097890033187808289A³⁸ − 34094820277978157885246445053088308041049796551224894512623542089173869593754653498973936641A³⁷ + 3410402811145246131813770717309440263360780712183061972564114609654437120334322647184416544A³⁶ − 271282760586160591069969069140476945510429502380734626569881260277916827821253614589450353A³⁵ + 8964213447929428305402665467314421858263171553663157732197428154274832533890558641276837A³⁴ + 1923203007732281067391499795416781297576812695277433611936479858013445273721466583889417A³³ − 501518281647971586746624890890317734768769703924228123088295643341747807703660169964670A³² + 74901741414760772803431635718001969326647968436687168820562144741951877663763333348762A³¹ − 8793339854703183362216942014089270298583083848430875674639726423156497665143583856599A³⁰ + 884440462041187107283475346039472588977291523754286257033259570465574394383577964060A²⁹ − 78974785991390863296101086854453347114231599106556602496429768566048683284604195488A²⁸ + 6351440938225235091307229476747584664822770409807711680636450788763662338660730736A²⁷ − 461577386020991827608535129901123426274312148150149667208924550289794040921123158A²⁶ + 30241062227720993053683464944230198235675014316896137072924910087824613616621533A²⁵ − 1777889704731817957933477778233549802357792023928514745543045283724426249599528A²⁴ + 93267061234360602967296674081132845199112348187406824265727268012016566200583A²³ − 4338371522575923233034210097677980674235870009879661484100060767182893535897A²² + 177640264174182468779901368262860951792277130596752158820640835566528124760A²¹ − 6346615055201503020261343318716637981874577322541383923515904824683154528A²⁰ + 195719278657162866559477668957744832524686044349121546932247027308617521A¹⁹ − 5146798111408492011733625064073866364135976197359254613756039953565431A¹⁸ + 114342597713325086360466620824031491178078220012926733476841878296701A¹⁷ − 2166532041106191143395488189771983665844918905009609494849319862555A¹⁶ + 37534499535017718712938026546492630945885931493825986952019257529A¹⁵ − 686311606973090058567849302474565048555704337440088851152041666A¹⁴ + 13463640572723940217486049731712260212765015277098533721414827A¹³ − 208766904360611863076421478201045069313767677355624072803244A¹² + 523821011921006477340257754559207277970409454884057162450A¹¹ + 85093250513592250373365314115539360225677251028427399015A¹⁰ − 2714743802786573644617033845589881585012310918640093409A⁹ + 45990737626797492919030822727366787337965820963078068A⁸ − 487271946585283985377454739508037532093635633664116A⁷ + 2432093267720660392912770667937619993847163656888A⁶ + 5979744051687564902194282372076326165161580304A⁵ + 4273919079255637923529042315667407153215968A⁴ − 892839406092143487318163123564942857358272A³ − 5635659728305912723837243683640292006016A² − 1029008599106478248990748958653643776A + 87191354287695850324598344835925504 = 0.015931092350142… (this site, from the symmetry-reduced tie system)

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