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.

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 = .0266+

Found by Mark Beyleveld, August 2006.

Horizontally symmetric.

14.

A = .0243+

Found by Mark Beyleveld, August 2006.

Symmetries of a rectangle.

15.


A = 29 / 1395 = .0207+

Found by David Cantrell, August 2006.

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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