Circles in Circles

The following pictures show n unit circles packed inside the smallest known circle (of radius r).


1.
           2.
           3.
r = 1
Trivial.
r = 2
Trivial.
r = 1 + 2 / √3 = 2.154+
Trivial.


4.
           5.
           6.
r = 1 + √2 = 2.414+
Trivial.
r = 2.701+
Proved by Graham in 1968.
r = 3
Proved by Graham in 1968.


7.
           8.
           9.
r = 3
Proved by Graham in 1968.
r = 3.304+
Proved by Braaksma in 1963.
r = 3.613+
Proved by Pirl in 1969.


10.
           11.
           12.
r = 3.813+
Proved by Pirl in 1969.
r = 3.923+
Proved by Melissen in 1994.
r = 4.029+
Found by Kravitz in 1967.


13.
           14.
           15.
r = 2 + √5 = 4.236+
Proved by F. Fodor in 2003.
r = 4.328+
Found by Goldberg in 1971.
r = 4.521+
Found by Kravitz in 1967.


16.
           17.
           18.
r = 4.615+
Found by Goldberg in 1971.
r = 4.792+
Found by Reis in 1975.
r = 1 + √2 + √6 = 4.863+
Found by Kravitz in 1967.


19.
           20.
r = 1 + √2 + √6 = 4.863+
Proved by F. Fodor in 1999.
r = 5.122+
Found by Goldberg in 1971.


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