The Heilbronn problem for triangles

Place n points in a triangle of unit area. The problem is affine-invariant, so the triangle's shape does not matter: coordinates are stored in the right triangle (0,0),(1,0),(0,1) and drawn equilateral. Papers that use the unit right triangle (area ½) quote values half as large as these. Optimality is proven for n ≤ 8.

A = $1$ = 1

Trivial.

3-fold dihedral symmetry · 1 minimal triangle.

details & viewer · coordinates trivial

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

Trivial.

3-fold dihedral symmetry · 3 minimal triangles.

details & viewer · coordinates trivial

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

Proved optimal by Royce Peng, 1989.

Mirror symmetric · 4 minimal triangles.

details & viewer · coordinates proven reconstructed coords

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.

3-fold dihedral symmetry · 6 minimal triangles.

details & viewer · coordinates proven reconstructed coords

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 reconstructed coords

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 reconstructed coords

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

Found by David Cantrell, July 2006.

120° rotationally symmetric · 13 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0433767406780494…

Found by David Cantrell, June 2007.

Not symmetric · 14 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0365298898800298…

Found by AlphaEvolve, June 2025.

Mirror symmetric · 17 minimal triangles.

details & viewer · coordinates record

A = 0.0310047817435253…

Found by David Cantrell, July 2006.

3-fold dihedral symmetry · 21 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0265565289197382…

Found by Tej Stead, July 2026.

Not symmetric · 20 minimal triangles.

details & viewer · coordinates record

A = 0.0237757731075718…

Found by David Cantrell, June 2007.

Not symmetric · 23 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0210907694602489…

Found by David Cantrell, June 2007.

Approximately 120° rotationally symmetric · 15 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0179762759872343…

Found by David Cantrell, June 2007.

120° rotationally symmetric · 27 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0155336750278416…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 29 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0148943429879589…

Found by Nathan Sudermann-Merx, August 2026.

3-fold dihedral symmetry · 33 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0117052759216522…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 30 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0111377526629232…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 35 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0106228655398858…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 37 minimal triangles.

details & viewer · coordinates record

A = 0.0093132281025523…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 38 minimal triangles.

details & viewer · coordinates record

A = 0.0090941626030574…

Found by Nathan Sudermann-Merx, August 2026.

Not symmetric · 41 minimal triangles.

details & viewer · coordinates record

A = 0.0089851540727786…

Found by Nathan Sudermann-Merx, August 2026.

3-fold dihedral symmetry · 48 minimal triangles.

details & viewer · coordinates record

A = 0.0069938719140268…

Found by Nathan Sudermann-Merx, August 2026.

3-fold dihedral symmetry · 45 minimal triangles.

details & viewer · coordinates record reconstructed coords

A = 0.0064916538398907…

Found by Tej Stead, August 2026.

Not symmetric · 47 minimal triangles.

details & viewer · coordinates record

A = 0.0063611755735585…

Found by Tej Stead, August 2026.

Not symmetric · 47 minimal triangles.

details & viewer · coordinates record

A = 0.0057026321597513…

Found by Tej Stead, August 2026.

Not symmetric · 51 minimal triangles.

details & viewer · coordinates record

A = 0.0054100992943801…

Found by Tej Stead, August 2026.

Not symmetric · 53 minimal triangles.

details & viewer · coordinates record

A = 0.0050980547585095…

Found by Tej Stead, August 2026.

Not symmetric · 55 minimal triangles.

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A = 0.0047559479619046…

Found by Tej Stead, August 2026.

Not symmetric · 57 minimal triangles.

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A = 0.0041779546673439…

Found by Tej Stead, August 2026.

Not symmetric · 59 minimal triangles.

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A = 0.0039674252663312…

Found by Tej Stead, August 2026.

Not symmetric · 60 minimal triangles.

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A = 0.0037581802156109…

Found by Tej Stead, August 2026.

Not symmetric · 62 minimal triangles.

details & viewer · coordinates record

A = 0.0036062262305574…

Found by Tej Stead, August 2026.

Not symmetric · 64 minimal triangles.

details & viewer · coordinates record