Circles in Tans

The following pictures show n unit circles packed inside the smallest known tan (of leg length s).


1.            2.            3.
s = 2 + √2 = 3.414+
Trivial.
s = 2 + 2√2 = 4.828+
Proved by Yinfeng Xu
in 1996.
s = 4 + √2 = 5.414+
Proved by Yinfeng Xu
in 1996.


4.            5.            6.
s = 2 + 3√2 = 6.242+
Proved by Yinfeng Xu
in 1996.
s = 4 + √2 + √3 = 7.146+
Proved by Hans Melissen
in 1997.
s = 6 + √2 = 7.414+
Proved by Hans Melissen
in 1997.


7.            8.            9.
s = 8.181+
Proved by Yinfeng Xu
in 1996.
s = 2 + 3√2 + √6 = 8.692+
Found by Hans Melissen
in 1997.
s = 2 + 5√2 = 9.071+
Trivial.


10.            11.            12.
s = 8 + √2 = 9.414+
Trivial.
s = 5 + √(2/3) + 3√2 = 10.059+
Found by David W. Cantrell
in June 2005.
s = 10.422+
Found by David W. Cantrell
in June 2005.


13.            14.            15.
s = 10.798+
Found by Hans Melissen
in 1997.
s = 2 + 3√2 + 2√6 = 11.141+
Found by Erich Friedman
in April 2005.
s = 10 + √2 = 11.414+
Trivial.


16.            17.            18.
s = 11.892+
Found by David W. Cantrell
in January 2006.
s = 12.155+
Found by David W. Cantrell
in January 2006.
s = 12.542+
Found by Eckard Specht
in April 2009.


19.            20.            21.
s = 12.798+
Found by David W. Cantrell
in January 2006.
s = 13.092+
Found by Maurizio Morandi
in December 2008.
s = 12 + √2 = 13.414+
Trivial.


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