Circles in Circles

The following pictures show n circles with the largest possible sum of radii packed inside a unit circle.


1.
           2.
           3.
Σ r = 1
Trivial.
Σ r = 1
Trivial.
A continuum of solutions.
Σ r = 6√3 – 9 = 1.392+
Trivial.


4.
           5.
           6.
Σ r = 5 / 3 = 1.666+
Found by Veit Elser
in November 2009.
Σ r = 1.866+
Found by Veit Elser
in November 2009.
Σ r = 2.110+
Found by Veit Elser
in November 2009.


7.
           8.
           9.
Σ r = 7 / 3 = 2.333+
Found by Veit Elser
in November 2009.
Σ r = 2.512+
Found by Veit Elser
in November 2009.
Σ r = 2.660+
Found by Veit Elser
in November 2009.


10.
           11.
           12.
Σ r = 2.783+
Found by David W. Cantrell
in April 2011.
Σ r = 2.947+
Found by Veit Elser
in November 2009.
Σ r = 3.103+
Found by Veit Elser
in November 2009.


13.
           14.
           15.
Σ r = 3.227+
Found by David W. Cantrell
in April 2011.
Σ r = 3.359+
Found by Veit Elser
in November 2009.
Σ r = 3.480+
Found by David W. Cantrell
in April 2011.


16.
           17.
           18.
Σ r = 3.611+
Found by David W. Cantrell
in April 2011.
Σ r = 3.730+
Found by David W. Cantrell
in April 2011.
Σ r = 3.850+
Found by David W. Cantrell
in April 2011.


19.
           20.
           21.
Σ r = 3.981+
Found by David W. Cantrell
in May 2011.
Σ r = 4.076+
Found by David W. Cantrell
in April 2011.
Σ r = 4.181+
Found by David W. Cantrell
in June 2011.


22.
           23.
           24.
Σ r = 4.287+
Found by David W. Cantrell
in June 2011.
Σ r = 4.372+
Found by David W. Cantrell
in June 2011.
Σ r = 4.466+
Found by David W. Cantrell
in June 2011.


25.
           26.
           27.
Σ r = 4.564+
Found by David W. Cantrell
in June 2011.
Σ r = 4.664+
Found by David W. Cantrell
in June 2011.
Σ r = 4.762+
Found by David W. Cantrell
in June 2011.


28.
           29.
           30.
Σ r = 4.849+
Found by David W. Cantrell
in June 2011.
Σ r = 4.934+
Found by David W. Cantrell
in June 2011.
Σ r = 5.019+
Found by David W. Cantrell
in June 2011.


31.
           32.
Σ r = 5.104+
Found by David W. Cantrell
in June 2011.
Σ r = 5.191+
Found by David W. Cantrell
in June 2011.


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