The Heilbronn Problem for Squares

The following pictures show n points inside a unit square so that the area A of the smallest triangle formed by these points is maximized. The smallest area triangles are shown.

3.



A = 1/2 = .500

Trivial.

One of an infinite family of solutions.

Symmetric about a diagonal.


4.



A = 1/2 = .500

Trivial.

Completely symmetric.


5.



A = √3 / 9 = .192+

Proved optimal by A. Dress, L. Yang, J. Z. Zhang, and Z. B. Zeng, 1992.

Symmetric about a diagonal.


6.



A = 1/8 = .125

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

One of an infinite family of solutions.

Vertically and horizontally symmetric.


7.



A = .0838+

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

Proved optimal by Zhenbing Chen and Liangyu Chen, 2008.

Not symmetric.


8.



A = (√13-1) / 36 = .0723+

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

180o Rotationally symmetric.


9.



A = (9√65-55) / 320 = .0548+

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

Symmetric about a diagonal.


10.



A = .0465+

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

Symmetric about both diagonals.


11.



A = 1/27 = .0370+

Found by Michael Goldberg, 1972.

Horizontally symmetric.


12.



A = .0325+

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

Completely symmetric.


13.



A = .0270+

Found by Peter Karpov, August 2011.

Not symmetric.


14.



A = .0243+

Found by Mark Beyleveld, August 2006.

Symmetries of a rectangle.


15.



A = .0211+

Found by Peter Karpov, August 2011.

Not symmetric.


16.



A = 7 / 341 = .0205+

Found by Mark Beyleveld, August 2006.

180o Rotationally symmetric.


More information is available at Mathworld, and the article by Comellas and Yebra.

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