Circles in Rectangles

The following pictures show n circles with the largest possible sum of radii packed inside a rectangle of perimeter 4.


1.
           2.
           3.
Σ r = 1/2 = .500
Trivial.
Σ r = 2/3 = .666+
Trivial.
Σ r = 8(7 – 2√2) / 41 = .813+
Found by David W. Cantrell
in July 2011.


4.
           5.
           6.
Σ r = 1.006+
Found by David W. Cantrell
in July 2011.
Σ r = 1.112+
Found by David W. Cantrell
in July 2011.
Σ r = 1.525+
Found by David W. Cantrell
in July 2011.


7.
           8.
           9.
Σ r = 1.324+
Found by David W. Cantrell
in July 2011.
Σ r = 1.430+
Found by David W. Cantrell
in July 2011.
Σ r = 1.528+
Found by David W. Cantrell
in July 2011.


10.
           11.
           12.
Σ r = 1.610+
Found by David W. Cantrell
in July 2011.
Σ r = 1.685+
Found by David W. Cantrell
in July 2011.
Σ r = 1.765+
Found by David W. Cantrell
in July 2011.


13.
           14.
           15.
Σ r = 1.835+
Found by David W. Cantrell
in July 2011.
Σ r = 1.916+
Found by David W. Cantrell
in July 2011.
Σ r = 1.985+
Found by David W. Cantrell
in July 2011.


16.
           17.
           18.
Σ r = 2.059+
Found by David W. Cantrell
in July 2011.
Σ r = 2.117+
Found by David W. Cantrell
in July 2011.
Σ r = 2.180+
Found by David W. Cantrell
in July 2011.


19.
           20.
           21.
Σ r = 2.241+
Found by David W. Cantrell
in July 2011.
Σ r = 2.305+
Found by David W. Cantrell
in July 2011.
Σ r = 2.364+
Found by David W. Cantrell
in July 2011.


22.
           23.
           24.
Σ r = 2.425+
Found by David W. Cantrell
in July 2011.
Σ r = 2.484+
Found by David W. Cantrell
in July 2011.
Σ r = 2.535+
Found by David W. Cantrell
in July 2011.


25.
           26.
           27.
Σ r = 2.592+
Found by David W. Cantrell
in July 2011.
Σ r = 2.638+
Found by David W. Cantrell
in July 2011.
Σ r = 2.687+
Found by David W. Cantrell
in July 2011.


28.
           29.
           30.
Σ r = 2.740+
Found by David W. Cantrell
in July 2011.
Σ r = 2.792+
Found by David W. Cantrell
in July 2011.
Σ r = 2.844+
Found by David W. Cantrell
in July 2011.


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