The following pictures show n circles with the largest possible sum of radii packed inside an ellipse with perimeter 2π.
1.
2.
3.
Σ r = 1
Circular.
Σ r = 1.185+
Found by David W. Cantrell
in September 2011.
Σ r = 1.407+
Found by David W. Cantrell
in September 2011.
4.
5.
6.
Σ r = 1.684+
Found by David W. Cantrell
in September 2011.
Σ r = 1.887+
Found by David W. Cantrell
in September 2011.
Σ r = 2.110+
Circular.
Found by Veit Elser
in November 2009.
7.
8.
9.
Σ r = 7 / 3 = 2.333+
Circular.
Found by Veit Elser
in November 2009.
Σ r = 2.512+
Circular.
Found by Veit Elser
in November 2009.
Σ r = 2.660+
Circular.
Found by Veit Elser
in November 2009.
10.
11.
12.
Σ r = 2.809+
Found by David W. Cantrell
in September 2011.
Σ r = 2.950+
Found by David W. Cantrell
in September 2011.
Σ r = 3.103+
Circular.
Found by Veit Elser
in November 2009.
13.
14.
15.
Σ r = 3.231+
Found by David W. Cantrell
in September 2011.
Σ r = 3.370+
Found by David W. Cantrell
in September 2011.
Σ r = 3.485+
Found by David W. Cantrell
in September 2011.
16.
17.
18.
Σ r = 3.231+
Circular.
Found by David W. Cantrell
in April 2011.
Σ r = 3.732+
Found by David W. Cantrell
in September 2011.
Σ r = 3.851+
Found by David W. Cantrell
in September 2011.
19.
20.
21.
Σ r = 3.981+
Circular.
Found by David W. Cantrell
in May 2011.
Σ r = 4.077+
Found by David W. Cantrell
in September 2011.
Σ r = 4.182+
Found by David W. Cantrell
in September 2011.
22.
23.
24.
Σ r = 4.287+
Circular.
Found by David W. Cantrell
in June 2011.
Σ r = 4.378+
Found by David W. Cantrell
in September 2011.
Σ r = 4.478+
Found by David W. Cantrell
in September 2011.
25.
26.
27.
Σ r = 4.570+
Found by David W. Cantrell
in September 2011.
Σ r = 4.667+
Found by David W. Cantrell
in September 2011.
Σ r = 4.762+
Circular.
Found by David W. Cantrell
in June 2011.
28.
29.
30.
Σ r = 4.849+
Found by David W. Cantrell
in September 2011.
Σ r = 4.938+
Found by David W. Cantrell
in September 2011.
Σ r = 5.030+
Found by David W. Cantrell
in September 2011.
31.
32.
Σ r = 5.105+
Found by David W. Cantrell
in September 2011.
Σ r = 5.192+
Found by David W. Cantrell
in September 2011.