Convex, n = 24 record
A =
the smallest positive root of
= 0.0107126406526939440129974540914427932771893935…
solved here as the exact critical point of the 8-parameter symmetry-reduced tie system (48 tied minimal triangles); minimal polynomial found by lattice reduction on 11000 digits and verified against a 20000-digit solution
Symmetry
Symmetry
3-fold dihedral symmetry (group D3, order 6).
24 points in 4 orbits (6 + 6 + 6 + 6) — hover a point to see its orbit.
Minimal triangles
48 triangles tied (within relative 10−9) at the minimal area, in 8 congruence classes:
| count | side lengths | triangles (point indices) |
|---|---|---|
| 6 | 0.1763 · 0.3594 · 0.5338 |
(1,11,17) (1,17,21) (4,8,9) (4,9,10) (5,6,12) (6,12,23) |
| 6 | 0.2219 · 0.3468 · 0.5672 |
(0,2,21) (3,8,20) (5,13,16) (7,15,23) (10,18,22) (11,14,19) |
| 6 | 0.2721 · 0.3637 · 0.6348 |
(1,4,19) (1,6,15) (2,12,17) (4,6,18) (9,12,16) (9,17,20) |
| 6 | 0.2219 · 0.4424 · 0.6633 |
(0,11,14) (0,14,21) (3,8,22) (3,10,22) (5,7,13) (7,13,23) |
| 6 | 0.1847 · 0.4966 · 0.6802 |
(0,12,15) (1,2,7) (3,17,19) (4,14,20) (6,16,22) (9,13,18) |
| 6 | 0.1417 · 0.5697 · 0.7102 |
(0,6,19) (1,3,18) (2,9,14) (4,13,15) (7,16,17) (12,20,22) |
| 6 | 0.3648 · 0.4668 · 0.8312 |
(0,18,23) (2,5,22) (3,11,15) (7,20,21) (8,14,16) (10,13,19) |
| 6 | 0.2526 · 0.6633 · 0.9155 |
(0,10,11) (3,10,23) (5,7,8) (5,14,21) (8,21,22) (11,13,23) |
Provenance
- Found by Nathan Sudermann-Merx, August 2026.
- Coordinates reconstructed by local optimization:
heilbronn-site exact tie-system solve— exact solution of the symmetry-reduced tie system (8 parameters, 48 tied minimal triangles); literals are the exact values rounded to 30 decimals. - Verified in exact arithmetic: all 2024 triples enumerated, 48 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 397777245318657641320538922492702173952372763389389043694980360764148997791A⁷² − 2716620312646442145010329127161073018234118866271499021149062332236475422355A⁷¹ + 9419238705687373592093473464202689405045883184265793655568319891909814648529A⁷⁰ − 21363559330289112405526772870010341073826778349323518609930264029437609525985A⁶⁹ + 34922553732538168330320775202965368380111520750791423287634157820473364142040A⁶⁸ − 43463500814547433326200680827812277790675600149828727395331496847005029976757A⁶⁷ + 42762092455077781469974130900789698235184710865471422742196060252943413940169A⁶⁶ − 34207814063061502743709291775216222413487113175613793080911469655305001203810A⁶⁵ + 22743450828039780442490178278158233769963102225819546113850930779893488303890A⁶⁴ − 12790556131402495427934990318865807806703013256359443957966601572892199273318A⁶³ + 6172973744289483501261716515449892230471495200712868270308211486663117920378A⁶² − 2588159085704047919861241280848496560196123937655520803064849798046593516953A⁶¹ + 953122342638692898472219741681118482514637334387072445062321604437686997437A⁶⁰ − 311640843008317992853866796378943034535416516567188374933402930213847826426A⁵⁹ + 91563675508150054810109770002265510679615570148831251773152499418751024579A⁵⁸ − 24539623530518244827109247789030550326840819114793650796911975034041300553A⁵⁷ + 6117978518660378680353492588801081426562890503925033093450081125711626416A⁵⁶ − 1453411092461016726785484718781861670725755990616793729372143282954833092A⁵⁵ + 336954068454701221549376197824873369950100800810524811451891546660174900A⁵⁴ − 77347285076717518615452896134159764589941968436555078692531420262615623A⁵³ + 17542746271945369131999007436694149164156578142755584425773009143156560A⁵² − 3867043687263816233277545255489000307628677245183327326380483776948859A⁵¹ + 810725491303092456773931368104030179634457538756204507999184328417302A⁵⁰ − 158534713109751283348588878443963141268135631299282603395101145566665A⁴⁹ + 28480935863285876677160383060670035664932176478938473084928768136777A⁴⁸ − 4643389698098243908306276087735052373330246326802532587487487337615A⁴⁷ + 678578602876838205416281380674070976528808855551598058880303290202A⁴⁶ − 87421520179259343760603480686435194198480366042730984338377618707A⁴⁵ + 9646738564008262320947758588019710572132701065455957839577488209A⁴⁴ − 855769033429430904297725287948530983410948666509364557667600152A⁴³ + 49518462643018888988448982529079717435903138996755573454076447A⁴² + 748451296285745102069006654462173402536295838731052706137118A⁴¹ − 699057640410458581591382207070495473376941247673729336498724A⁴⁰ + 115610992047326666723331646523292772859499526824076804113993A³⁹ − 12638733371508100778612358130644295756168175094406087297932A³⁸ + 985220526536407929967062475625769458393437052077012873604A³⁷ − 45773343410408087064826165186217409661264701566385736361A³⁶ − 846167375175331975510253352138336194754240809216816562A³⁵ + 446718775658654635474382890518409228731059955894493178A³⁴ − 56556078128055014118455338011176259201411800931595426A³³ + 5012860743832344043730931429909056193476716505066893A³² − 356623280635903031119857155364721862006630424747125A³¹ + 20629598232991049235282616056194233821785787435973A³⁰ − 832968883643296327906673132401712278727829534470A²⁹ + 222552108952131024700483377620778479285074025A²⁸ + 3792287476409091730072438757309046957135741603A²⁷ − 384471930777311650091077793274173906797649010A²⁶ + 21574140985758085719380148285322981089512634A²⁵ − 700166703842012814142173085063281742703552A²⁴ + 15108554647318290324980212311406663279723A²³ − 1092052821316786026782625249313066823786A²² + 92053083060887357716436740707541412945A²¹ − 1279257840451294718129076541434977205A²⁰ − 333127863609644227505836414831572123A¹⁹ + 25101609723681746182669234366017910A¹⁸ − 684261038229017093272789787365479A¹⁷ + 12973536723168598964128641312171A¹⁶ − 1803428991463021218267596633604A¹⁵ + 138407362454210616137358761696A¹⁴ − 3510237864164149092691696852A¹³ − 69266602678385283568630063A¹² + 6035926628328881219403562A¹¹ − 106473004460484424702413A¹⁰ + 953350610725696299608A⁹ − 263807465683231568900A⁸ + 23980367108587283440A⁷ − 1107535648797423520A⁶ + 32958528333527712A⁵ − 672227047557184A⁴ + 9569009605632A³ − 97173328640A² + 668183040A − 2236416 = 0.010712640652693… (this site, from the symmetry-reduced tie system)
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