Circles in Circles

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.
posted @ 2015-01-03 02:34  Unikanade  阅读(309)  评论(0)    收藏  举报