The Heilbronn Problem for Squares

A = $\frac{1}{2}$ = 0.5

Trivial.

Mirror symmetric · 1 minimal triangle.

details & viewer · coordinates proven

A = $\frac{1}{2}$ = 0.5

Trivial.

4-fold dihedral symmetry · 4 minimal triangles.

details & viewer · coordinates proven

A = $\frac{\sqrt{3}}{9}$ = 0.1924500897298752…

Proved optimal by Yang Lu, Zhang Jingzhong, and Zeng Zhenbing, 1991.

Mirror symmetric · 4 minimal triangles.

details & viewer · coordinates proven

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

Proved optimal by A. Dress, L. Yang, and Z. B. Zeng, 1995.

2-fold dihedral symmetry · 6 minimal triangles.

details & viewer · coordinates proven

A = 0.0838590090075134…

Root of $152x^3+12x^2-14x+1$.

Found by F. Comellas and J. Yebra, December 2001.

Proved optimal by Zhenbing Chen and Liangyu Chen, 2011.

Not symmetric · 8 minimal triangles.

details & viewer · coordinates proven

A = $\frac{\sqrt{13}-1}{36}$ = 0.0723764243184441…

Found by F. Comellas and J. Yebra, December 2001.

Proved optimal by L. Dehbi and Z. Zeng, 2022.

180° rotationally symmetric · 12 minimal triangles.

details & viewer · coordinates proven

A = $\frac{9\sqrt{65}-55}{320}$ = 0.0548759991708967…

Found by F. Comellas and J. Yebra, December 2001.

Proved optimal by Nathan Sudermann-Merx, March 2026.

Mirror symmetric · 11 minimal triangles.

details & viewer · coordinates proven

A = 0.0465374195825417…

Root of $3456x^3-1764x^2+268x-9$.

Found by F. Comellas and J. Yebra, December 2001.

2-fold dihedral symmetry · 16 minimal triangles.

details & viewer · coordinates record

A = $\frac{1}{27}$ = 0.0370370370370370…

Found by Michael Goldberg, 1972.

Mirror symmetric · 28 minimal triangles.

details & viewer · coordinates record

A = 0.0325988586918196…

Root of $64x^3+80x^2+28x-1$.

Found by F. Comellas and J. Yebra, December 2001.

4-fold dihedral symmetry · 20 minimal triangles.

details & viewer · coordinates record

A = 0.0270199166691111…

Found by Peter Karpov, August 2011.

Not symmetric · 20 minimal triangles.

details & viewer · coordinates record

A = 0.0243039796209924…

Root of $320x^3+768x^2-60x+1$.

Found by Mark Beyleveld, August 2006.

2-fold dihedral symmetry · 26 minimal triangles.

details & viewer · coordinates record

A = 0.0212105410054906…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 25 minimal triangles.

details & viewer · coordinates record

A = $\frac{7}{341}$ = 0.0205278592375366…

Found by Mark Beyleveld, August 2006.

180° rotationally symmetric · 64 minimal triangles.

details & viewer · coordinates record

A = 0.0172616770725236…

Found by Chouaieb Nemri (AlphaEvolve, Google Cloud), August 2026.

Not symmetric · 27 minimal triangles.

details & viewer · coordinates record

A = 0.0155260599407024…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

180° rotationally symmetric · 30 minimal triangles.

details & viewer · coordinates record

A = 0.0139478620814845…

Found by Tej Stead, August 2026.

Not symmetric · 31 minimal triangles.

details & viewer · coordinates record

A = 0.0129381967147270…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

2-fold dihedral symmetry · 42 minimal triangles.

details & viewer · coordinates record

A = 0.0114737607857012…

Found by Rob Gardiner, September 2026.

Not symmetric · 37 minimal triangles.

details & viewer · coordinates record

A = 0.0106728538531263…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

180° rotationally symmetric · 42 minimal triangles.

details & viewer · coordinates record

A = 0.0097572897133147…

Found by Rob Gardiner, September 2026.

Not symmetric · 40 minimal triangles.

details & viewer · coordinates record

A = 0.0090890080772487…

Found by Ilkyoo Choi, May 2026.

2-fold dihedral symmetry · 48 minimal triangles.

details & viewer · coordinates record

A = 0.0082300694265250…

Found by Rob Gardiner, September 2026.

Not symmetric · 44 minimal triangles.

details & viewer · coordinates record

A = 0.0075775910098290…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

180° rotationally symmetric · 50 minimal triangles.

details & viewer · coordinates record

A = 0.0071024049798860…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

Mirror symmetric · 47 minimal triangles.

details & viewer · coordinates record

A = 0.0069005249837478…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

90° rotationally symmetric · 52 minimal triangles.

details & viewer · coordinates record

A = 0.0062759392131717…

Found by Rhys Chappell, September 2026.

Not symmetric · 51 minimal triangles.

details & viewer · coordinates record

A = 0.0060385274949059…

Found by Tej Stead, September 2026.

180° rotationally symmetric · 54 minimal triangles.

details & viewer · coordinates record

A = 0.0058370688458137…

Found by Tej Stead, September 2026.

Not symmetric · 56 minimal triangles.

details & viewer · coordinates record

A = 0.0058109022076319…

Root of $1063488763600070974359809499856896x^{37}-8708947682935878940847296682655744x^{36}+27577714282136100418854598127648768x^{35}-55035030690536916167751488543653888x^{34}+79687572059409491003836999405666304x^{33}-86826735461229463308968387772678144x^{32}+72067826641458876537382577805197312x^{31}-46335759900261621424264305652531200x^{30}+23430186694467081518946821175508992x^{29}-9454582878840204434198615004545024x^{28}+3084403549801143870217348781703168x^{27}-829469968483488745392496858628096x^{26}+188837683301790012459719927529472x^{25}-37096470646574585140327426818048x^{24}+6205977861803315266514396905472x^{23}-838905237099045124810043555840x^{22}+87050961805397770983574601728x^{21}-5788956513771157868054052864x^{20}-53645277330314592912408576x^{19}+85824647714248783021735936x^{18}-14822557651557114148028416x^{17}+1626705843100370505564160x^{16}-126880757188644236951552x^{15}+5193565622842539835392x^{14}+460396789394005049344x^{13}-131578910398117171200x^{12}+16532249616344451072x^{11}-1415430989144391168x^{10}+90352776750379392x^9-4443900578565568x^8+169656869516896x^7-4977902015600x^6+109709554136x^5-1755470784x^4+19451614x^3-139226x^2+571x-1$.

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

4-fold dihedral symmetry · 64 minimal triangles.

details & viewer · coordinates record

A = 0.0050713441584671…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

Not symmetric · 56 minimal triangles.

details & viewer · coordinates record

A = 0.0048348915769357…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

Not symmetric · 62 minimal triangles.

details & viewer · coordinates record

A = 0.0044322874443073…

Found by Rob Gardiner, September 2026.

Not symmetric · 64 minimal triangles.

details & viewer · coordinates record

A = 0.0041848886660263…

Found by Fable 5.1 with Shengtong Zhang, September 2026.

Not symmetric · 65 minimal triangles.

details & viewer · coordinates record