The Heilbronn problem for squares

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

Trivial.

Mirror symmetric · 1 minimal triangle.

details & viewer · coordinates trivial

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

Trivial.

4-fold dihedral symmetry · 4 minimal triangles.

details & viewer · coordinates trivial

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…

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…

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

4-fold dihedral symmetry · 20 minimal triangles.

details & viewer · coordinates record

A = 0.0270188309265

Found by Peter Karpov, August 2011.

Not symmetric · 1 minimal triangle.

details & viewer · coordinates record

A = 0.024303833806

Found by Mark Beyleveld, August 2006.

2-fold dihedral symmetry · 4 minimal triangles.

details & viewer · coordinates record

A = 0.021105359687

Found by Peter Karpov, August 2011.

Not symmetric · 1 minimal triangle.

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 = 0.0164812220057778…

Found by Tej Stead, July 2026.

Not symmetric · 29 minimal triangles.

details & viewer · coordinates record

A = 0.0144326564130846…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 29 minimal triangles.

Record coordinates unpublished; best public configuration shown.

details & viewer · coordinates record

A = 0.0133861776173687…

Found by William Shanley, July 2026.

Not symmetric · 32 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0129115556839418…

Found by William Shanley, July 2026.

C4 · 36 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0108149295425821…

Found by Tej Stead, August 2026.

Not symmetric · 36 minimal triangles.

details & viewer · coordinates record

A = 0.0095698170553216…

Found by Nathan Sudermann-Merx, August 2026.

2-fold dihedral symmetry · 40 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0088127537336762…

Found by Tej Stead, August 2026.

Not symmetric · 39 minimal triangles.

details & viewer · coordinates record

A = 0.0084950131265235…

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

4-fold dihedral symmetry · 56 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0072446758601343…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 41 minimal triangles.

Record coordinates unpublished; best public configuration shown.

details & viewer · coordinates record

A = 0.0068268500402073…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 39 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0067903492413679…

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

Not symmetric · 49 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0067758102393368…

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

4-fold dihedral symmetry · 56 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0056196694691326…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 51 minimal triangles.

details & viewer · coordinates record

A = 0.0054451230827491…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 54 minimal triangles.

details & viewer · coordinates record

A = 0.0053903722019738…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 56 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0046903315380751…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 58 minimal triangles.

details & viewer · coordinates record

A = 0.0041463803677157…

Found by Tej Stead, August 2026.

Not symmetric · 61 minimal triangles.

details & viewer · coordinates record

A = 0.0040047585523910…

Found by Tej Stead, August 2026.

Not symmetric · 62 minimal triangles.

details & viewer · coordinates record

A = 0.0037024937676992…

Found by Tej Stead, August 2026.

Not symmetric · 65 minimal triangles.

details & viewer · coordinates record