The Heilbronn Problem for Triangles

Place n points in a triangle of unit area. Because the problem is invariant under affine transformations, the specific triangle shape does not affect the normalized area: coordinates are stored in the standard right triangle (0,0), (1,0), (0,1) and displayed as an equilateral figure. Note that literature using the unnormalized right triangle with vertices (0,0),(1,0),(0,1) has area 1/2, and therefore reports values exactly half of those shown here. Optimality is proven for n ≤ 8.

A = $1$ = 1

Trivial.

3-fold dihedral symmetry · 1 minimal triangle.

details & viewer · coordinates proven

A = $\frac{1}{3}$ = 0.3333333333333333…

Trivial.

3-fold dihedral symmetry · 3 minimal triangles.

details & viewer · coordinates proven

A = $3 - 2 \sqrt{2}$ = 0.1715728752538099…

Proved optimal by Royce Peng, 1989.

Mirror symmetric · 4 minimal triangles.

details & viewer · coordinates proven

A = $\frac{1}{8}$ = 0.125

Found by L. Yang, J. Z. Zhang, and Z. B. Zeng, 1991.

Proved optimal by L. Yang, J. Z. Zhang, and Z. B. Zeng, 1994.

Not symmetric · 6 minimal triangles.

details & viewer · coordinates proven

A = $\frac{7}{72}$ = 0.0972222222222222…

Found by David Cantrell, July 2006.

Proved optimal by Z. Zeng and L. Chen, 2019.

120° rotationally symmetric · 9 minimal triangles.

details & viewer · coordinates proven

A = 0.0677892132541972…

Root of $1216512 t^{7} + 1022976 t^{6} - 2745456 t^{5} + 2218730 t^{4} + 32206304 t^{3} - 19148615 t^{2} + 3156103 t - 86233$ .

Found by David Cantrell, July 2006.

Proved optimal by Nathan Sudermann-Merx, 2026.

Not symmetric · 11 minimal triangles.

details & viewer · coordinates proven

A = $\frac{43}{784}$ = 0.0548469387755102…

Found by David Cantrell, July 2006.

120° rotationally symmetric · 13 minimal triangles.

details & viewer · coordinates record

A = 0.0433767406780494…

Found by David Cantrell, June 2007.

Not symmetric · 14 minimal triangles.

details & viewer · coordinates record

A = 0.0365298898800302…

Root of $386726293821797392786129 x^{16} + 5390497410699904416698271 x^{15} + 9150784972225868037074965 x^{14} + 29882099349975917905402053 x^{13} + 96907959773628435631853994 x^{12} + 8630833231832902477977498 x^{11} + 163485900461635219362256358 x^{10} - 73728098063632444097933977 x^{9} - 62591326820175894704348590 x^{8} + 19409148088019531147406410 x^{7} + 8189217351544212941245001 x^{6} - 440725435064635936977209 x^{5} - 172988796750685677368738 x^{4} + 6949815954004975270119 x^{3} + 907432336598652742572 x^{2} - 61466944226230595232 x + 1011466570038909840$ .

Found by AlphaEvolve, June 2025.

Mirror symmetric · 17 minimal triangles.

details & viewer · coordinates record

A = 0.0310047817435254…

Root of $729 x^{6} - 291168 x^{5} + 390241 x^{4} - 43317 x^{3} - 4386 x^{2} + 5 x + 5$ .

Found by David Cantrell, July 2006.

3-fold dihedral symmetry · 21 minimal triangles.

details & viewer · coordinates record

A = 0.0265565289197382…

Found by Tej Stead, July 2026.

Not symmetric · 20 minimal triangles.

details & viewer · coordinates record

A = 0.0237757731075727…

Found by David Cantrell, June 2007.

Not symmetric · 23 minimal triangles.

details & viewer · coordinates record

A = 0.0210907694602666…

Root of $5250987 x^{5} + 1609650 x^{4} + 161469 x^{3} + 5438 x^{2} - 13 x - 4$ .

Found by David Cantrell, June 2007.

120° rotationally symmetric · 15 minimal triangles.

details & viewer · coordinates record

A = 0.0183555463554034…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

Not symmetric · 24 minimal triangles.

details & viewer · coordinates record

A = 0.0162423284071037…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

Not symmetric · 28 minimal triangles.

details & viewer · coordinates record

A = 0.0148943429879598…

Root of $43200 x^{12} - 310177539 x^{11} + 381486159 x^{10} + 10547986913 x^{9} - 5356562854 x^{8} + 3623073265 x^{7} + 565010257 x^{6} - 356137529 x^{5} - 40275162 x^{4} + 2253891 x^{3} + 6619 x^{2} - 717 x + 4$ .

Found by Nathan Sudermann-Merx, August 2026.

3-fold dihedral symmetry · 33 minimal triangles.

details & viewer · coordinates record

A = 0.0134818913828067…

Found by Marc-Emmanuel Coupvent des Graviers, September 2026.

Not symmetric · 30 minimal triangles.

details & viewer · coordinates record

A = 0.0126093915286413…

Found by Rob Gardiner, September 2026.

Not symmetric · 35 minimal triangles.

details & viewer · coordinates record

A = 0.0113959769796381…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

120° rotationally symmetric · 39 minimal triangles.

details & viewer · coordinates record

A = 0.0102661276072629…

Root of $4113894419667609830181603 x^{22} + 29807468238423455008126578 x^{21} - 39085790603220143290920177 x^{20} + 24488515905115680120305898 x^{19} - 10736525499024455231632668 x^{18} + 3768816750540027903994113 x^{17} - 1158442379073845833748328 x^{16} + 324799853969658361401171 x^{15} - 71937195458392144391355 x^{14} + 11920737923277897236508 x^{13} - 1491784289359000650099 x^{12} + 155491081480323732357 x^{11} - 18192256172188373745 x^{10} + 1431857137555076508 x^{9} - 104692732480960029 x^{8} + 11430606627963063 x^{7} - 705604556304666 x^{6} + 16028440098039 x^{5} + 15064749231 x^{4} - 2374399435 x^{3} - 25225143 x^{2} + 119664 x + 2704$ .

Found by Tej Stead, August 2026.

3-fold dihedral symmetry · 39 minimal triangles.

details & viewer · coordinates record

A = 0.0092697818729312…

Found by Tej Stead, August 2026.

Mirror symmetric · 42 minimal triangles.

details & viewer · coordinates record

A = 0.0089851540727790…

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

Found by Tej Stead, August 2026.

3-fold dihedral symmetry · 48 minimal triangles.

details & viewer · coordinates record

A = 0.0078026387410542…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

120° rotationally symmetric · 42 minimal triangles.

details & viewer · coordinates record

A = 0.0074715697268151…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

Mirror symmetric · 48 minimal triangles.

details & viewer · coordinates record

A = 0.0070293281691415…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

Not symmetric · 49 minimal triangles.

details & viewer · coordinates record

A = 0.0066943416045948…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

3-fold dihedral symmetry · 54 minimal triangles.

details & viewer · coordinates record

A = 0.0061236911848443…

Found by Marc-Emmanuel Coupvent des Graviers, September 2026.

Not symmetric · 53 minimal triangles.

details & viewer · coordinates record

A = 0.0059574010917487…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

120° rotationally symmetric · 57 minimal triangles.

details & viewer · coordinates record

A = 0.0051612843156243…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

120° rotationally symmetric · 57 minimal triangles.

details & viewer · coordinates record

A = 0.0048007554607870…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

Not symmetric · 59 minimal triangles.

details & viewer · coordinates record

A = 0.0047282133951204…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

3-fold dihedral symmetry · 66 minimal triangles.

details & viewer · coordinates record

A = 0.0044309105983898…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

3-fold dihedral symmetry · 60 minimal triangles.

details & viewer · coordinates record

A = 0.0040326843207058…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

Mirror symmetric · 65 minimal triangles.

details & viewer · coordinates record