The Heilbronn Problem for Circles

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

3.


A = 3√3 / 4 = 1.299+

Trivial.

Completely symmetric.


4.


A = 1

Trivial.

Completely symmetric.


5.


A = √(50-10√5) / 8 = .657+

Trivial.

Completely symmetric.


6.


A = √3 / 4 = .433

Trivial.

Completely symmetric.


7.


A = .294+

Trivial.

Completely symmetric.


8.


A = .216+

Found by David Cantrell, July 2006.

Completely symmetric.


9.



A = .173+

Found by David Cantrell, July 2006.

Horizontally symmetric.


10.


A = .150+

Found by David Cantrell, July 2006.

Completely symmetric.


11.




A = .113+

Found by David Cantrell, August 2006.

Horizontally symmetric.


12.


A = .104+

Found by David Cantrell, July 2006.

Symmetry of an equilateral triangle.


13.



A = .0856+

Found by David Cantrell, June 2007.

Horizontally symmetric.


14.




A = .0758+

Found by David Cantrell, June 2007.

Horizontally symmetric.


15.




A = .0700+

Found by David Cantrell, August 2006.

Horizontally symmetric.


16.


A = .0661+

Found by David Cantrell, August 2006.

Symmetry of a square.


More information is available at Mathworld.

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