The Heilbronn problem for convex regions

A = $1$ = 1

Trivial.

3-fold dihedral symmetry · 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 = $\frac{5-\sqrt{5}}{10}$ = 0.2763932022500210…

Found by David Cantrell, June 2007.

5-fold dihedral symmetry · 5 minimal triangles.

details & viewer · coordinates record

A = $\frac{1}{6}$ = 0.1666666666666666…

Found by David Cantrell, June 2007.

6-fold dihedral symmetry · 6 minimal triangles.

details & viewer · coordinates record

A = $\frac{1}{9}$ = 0.1111111111111111…

Proved optimal by Zhenbing Zeng and Lu Yang, 1995.

120° rotationally symmetric · 6 minimal triangles.

details & viewer · coordinates proven reconstructed coords

A = 0.0800001393294662…

Found by David Cantrell, June 2007.

Not symmetric · 10 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0640647523079415…

Found by David Cantrell, June 2007.

Not symmetric · 11 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0519930715229428…

Found by David Cantrell, June 2007.

180° rotationally symmetric · 16 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = $\frac{2}{47}$ = 0.0425531914893617…

Found by David Cantrell, June 2007.

Not symmetric · 28 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = $\frac{2}{51}$ = 0.0392156862745098…

Found by David Cantrell, June 2007.

3-fold dihedral symmetry · 36 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0309368890348956…

Found by David Cantrell, June 2007.

Not symmetric · 15 minimal triangles.

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A = 0.0278355714584821…

Found by AlphaEvolve, June 2025.

Not symmetric · 16 minimal triangles.

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A = 0.0245640556132301…

Found by Tej Stead, July 2026.

Not symmetric · 25 minimal triangles.

details & viewer · coordinates record

A = 0.0222728771289807…

Found by David Cantrell, June 2007.

4-fold dihedral symmetry · 28 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0187314546401800…

Found by Tej Stead, July 2026.

Not symmetric · 29 minimal triangles.

details & viewer · coordinates record

A = $\frac{2-\sqrt[3]{\frac{3}{2}-\frac{\sqrt{\frac{239}{3}}}{6}}-\sqrt[3]{\frac{3}{2}+\frac{\sqrt{\frac{239}{3}}}{6}}}{18}$ = 0.0182388954634910…

Found by Tej Stead, July 2026.

6-fold dihedral symmetry · 36 minimal triangles.

details & viewer · coordinates record

A = 0.0150027043689313…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 32 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0144642050306690…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 40 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0127251600361903…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 37 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0122549627897763…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 38 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0110824852805632…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 38 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0107126406526936…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 48 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0088774237842822…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 44 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0084032681236144…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 45 minimal triangles.

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A = 0.0078037602737216…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 45 minimal triangles.

details & viewer · coordinates record

A = 0.0076277190031123…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 49 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0060788197206917…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 52 minimal triangles.

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A = 0.0059424727772124…

Found by Nathan Sudermann-Merx, August 2026.

Approximately 180° rotationally symmetric · 54 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0052147326735831…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 61 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0048867217592234…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 59 minimal triangles.

Record coordinates unpublished; best public configuration shown.

details & viewer · coordinates record

A = 0.0046143412220719…

Found by Tej Stead, August 2026.

Not symmetric · 60 minimal triangles.

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A = 0.0044135175036173…

Found by Tej Stead, August 2026.

Not symmetric · 61 minimal triangles.

details & viewer · coordinates record

A = 0.0040566972109141…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 65 minimal triangles.

Record coordinates unpublished; best public configuration shown.

details & viewer · coordinates record