Triangle, n = 24 record

A = the second-smallest positive root of

2936121559734991016957493040573897702439289 x^{40} + 125393009490931532227816796658494093211056055 x^{39} + 382320407668754647803857062671313474201961688 x^{38} - 1637734500061776995062565603397080655418705137 x^{37} + 4545975609176699923030827143991369765375564279 x^{36} - 7264314888720626474633452528315586180578794832 x^{35} + 8941035514700678599794539529814668864877481152 x^{34} - 8478458826920149415387441497236502455870146057 x^{33} + 6599035966495027782270087848058214125770960154 x^{32} - 4365773345390197826495386127499949172258727598 x^{31} + 2500637341873743536359885045862329326664277127 x^{30} - 1320705511459422773467253113693595079531756526 x^{29} + 640208241982735225043194268842042376674037964 x^{28} - 279044690997970337395141753868001954296771986 x^{27} + 106967510891508492934379783162927630663544074 x^{26} - 34795271517717023519198727300303880910117368 x^{25} + 11371491104881211411957649354420331479626532 x^{24} - 3980250706934512793851265038386609061634729 x^{23} + 1078424114249526591970776108095569744670434 x^{22} - 219058604055209752264093955642745693805620 x^{21} + 57756344618294813783152932153014813845683 x^{20} - 18550942308922421526405175335161388680773 x^{19} + 4501434812001843250815378867073945351690 x^{18} - 586797817129813952523540853966866090437 x^{17} + 113969891063464390244067935657419409847 x^{16} - 43377624149345483233200878407578869352 x^{15} + 7617135606786778806435720117422886109 x^{14} - 444212128481061575265450139483570393 x^{13} - 14034897497184492876624494245719237 x^{12} + 2216868864964708024439529336866510 x^{11} - 33678752175906840503808002707014 x^{10} - 1715938904136640542547034276951 x^{9} + 18290218267993965734726220881 x^{8} + 327500256179602258639548027 x^{7} - 2296357458534472549285755 x^{6} - 282094281744430415951 x^{5} - 285253487512807665839 x^{4} - 843074184701483745 x^{3} + 60945004716704774 x^{2} - 403903286251836 x + 786102807672

= 0.00898515407277903439564126295039590408338328341…

solved here as the exact critical point of the 7-parameter symmetry-reduced tie system (48 tied minimal triangles); minimal polynomial found by lattice reduction on 4400 digits and verified against an 8000-digit solution

48 triangles tie for the minimal area.

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	`0.3151 · 0.3569 · 0.6699`	(2,7,9) (3,6,8) (4,10,18) (5,11,19) (12,14,23) (13,15,22)
6	`0.2590 · 0.4411 · 0.6980`	(0,4,8) (1,5,9) (4,8,16) (5,9,17) (20,22,23) (21,22,23)
6	`0.2825 · 0.4545 · 0.7353`	(0,2,10) (1,3,11) (6,12,16) (7,13,17) (14,19,20) (15,18,21)
6	`0.2825 · 0.5687 · 0.8500`	(0,2,12) (1,3,13) (2,12,16) (3,13,17) (18,19,20) (18,19,21)
6	`0.2161 · 0.6393 · 0.8540`	(2,14,22) (3,15,23) (4,6,13) (5,7,12) (8,10,19) (9,11,18)
6	`0.1715 · 0.7284 · 0.8985`	(2,6,23) (3,7,22) (4,15,19) (5,14,18) (8,11,13) (9,10,12)
6	`0.4538 · 0.6084 · 1.0616`	(0,7,19) (1,6,18) (2,11,20) (3,10,21) (12,15,17) (13,14,16)
6	`0.3197 · 0.8500 · 1.1692`	(0,3,17) (0,12,21) (1,2,16) (1,13,20) (16,18,20) (17,19,21)

Provenance

Found by Tej Stead, August 2026.
Coordinates reconstructed by local optimization: heilbronn-site exact tie-system solve — exact solution of the symmetry-reduced tie system (7 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 second-smallest positive root of 2936121559734991016957493040573897702439289A⁴⁰ + 125393009490931532227816796658494093211056055A³⁹ + 382320407668754647803857062671313474201961688A³⁸ − 1637734500061776995062565603397080655418705137A³⁷ + 4545975609176699923030827143991369765375564279A³⁶ − 7264314888720626474633452528315586180578794832A³⁵ + 8941035514700678599794539529814668864877481152A³⁴ − 8478458826920149415387441497236502455870146057A³³ + 6599035966495027782270087848058214125770960154A³² − 4365773345390197826495386127499949172258727598A³¹ + 2500637341873743536359885045862329326664277127A³⁰ − 1320705511459422773467253113693595079531756526A²⁹ + 640208241982735225043194268842042376674037964A²⁸ − 279044690997970337395141753868001954296771986A²⁷ + 106967510891508492934379783162927630663544074A²⁶ − 34795271517717023519198727300303880910117368A²⁵ + 11371491104881211411957649354420331479626532A²⁴ − 3980250706934512793851265038386609061634729A²³ + 1078424114249526591970776108095569744670434A²² − 219058604055209752264093955642745693805620A²¹ + 57756344618294813783152932153014813845683A²⁰ − 18550942308922421526405175335161388680773A¹⁹ + 4501434812001843250815378867073945351690A¹⁸ − 586797817129813952523540853966866090437A¹⁷ + 113969891063464390244067935657419409847A¹⁶ − 43377624149345483233200878407578869352A¹⁵ + 7617135606786778806435720117422886109A¹⁴ − 444212128481061575265450139483570393A¹³ − 14034897497184492876624494245719237A¹² + 2216868864964708024439529336866510A¹¹ − 33678752175906840503808002707014A¹⁰ − 1715938904136640542547034276951A⁹ + 18290218267993965734726220881A⁸ + 327500256179602258639548027A⁷ − 2296357458534472549285755A⁶ − 282094281744430415951A⁵ − 285253487512807665839A⁴ − 843074184701483745A³ + 60945004716704774A² − 403903286251836A + 786102807672 = 0.008985154072779… (this site, from the symmetry-reduced tie system)

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