Circles in Squares

The following pictures show n unit circles packed inside the smallest square (of side length s). Most of these (n=1 through n=20, plus n=25 and n=36) have been proved optimal.

1.            2.            3.
s = 2
Trivial. 		s = 2 + 2 = 3.414+
Trivial. 		s = 2 + 1/2 + 6/2 = 3.931+
Trivial.

4.            5.            6.
s = 4
Trivial. 		s = 2 + 22 = 4.828+
Trivial. 		s = 2 + 12/13 = 5.328+
Proved by Graham in 1963.

7.            8.            9.
s = 4 + 3 = 5.732+
Proved by Schaer in 1964. 		s = 2 + 2 + 6 = 5.863+
Proved by Schaer/Meir in 1964. 		s = 6
Proved by Schaer in 1964.

10.            11.            12.
s = 6.747+
Proved by De Groot in 1990. 		s = 7.022+
Proved by Peikert in 1991. 		s = 2 + 15(2/17) = 7.144+
Proved by Peikert in 1991.

13.            14.            15.
s = 7.463+
Proved by Peikert in 1991. 		s = 6 + 3 = 7.732+
Proved by Wengerodt in 1987. 		s = 4 + 2 + 6 = 7.863+
Proved by Peikert in 1991.

16.            17.            18.
s = 8
Proved by Wengerodt in 1983. 		s = 8.532+
Proved by Peikert in 1991. 		s = 2 + 24/13 = 8.656+
Proved by Peikert in 1991.

19.            20.            21.
s = 8.907+
Proved by Peikert in 1991. 		s = 130/17 + 16/172 = 8.978+
Proved by Peikert in 1991. 		s = 9.358+

22.            23.            24.
s = 9.463+ s = 2 + 22 + 26 = 9.727+ s = 6 + 2 + 6 = 9.863+

Circles in Squares Links

| Peter Szabo | Ronald Peikert |

Back to Erich's Packing Center.