a/b=r
Circles in Circles
The following pictures show n unit circles packed inside the smallest known circle (of radius r).
1.
2.
r = 1
Trivial. r = 2
Trivial.
3.
4.
r = 1 + 2 / 3 = 2.154+
Trivial. r = 1+2 = 2.414+
Trivial.
5.
6.
r = 2.701+
Proved by Graham in 1968. r = 3
Proved by Graham in 1968.
7.
8.
r = 3
Proved by Graham in 1968. r = 3.304+
Proved by Braaksma in 1963.
9.
10.
r = 3.613+
Proved by Pirl in 1969. r = 3.813+
Proved by Pirl in 1969.
11.
12.
r = 3.923+
Proved by Melissen in 1994. r = 4.029+
Found by Kravitz in 1967.
13.
14.
r = 2 + 5 = 4.236+
Found by Kravitz in 1967. r = 4.328+
Found by Goldberg in 1971.
15.
16.
r = 4.521+
Found by Kravitz in 1967. r = 4.615+
Found by Goldberg in 1971.
17.
18.
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+
Found by Kravitz in 1967. r = 5.122+
Found by Goldberg in 1971.
Back to Erich's Packing Center.
Circles in Circles
The following pictures show n unit circles packed inside the smallest known circle (of radius r).
1.
2.
r = 1
Trivial. r = 2
Trivial.
3.
4.
r = 1 + 2 / 3 = 2.154+
Trivial. r = 1+2 = 2.414+
Trivial.
5.
6.
r = 2.701+
Proved by Graham in 1968. r = 3
Proved by Graham in 1968.
7.
8.
r = 3
Proved by Graham in 1968. r = 3.304+
Proved by Braaksma in 1963.
9.
10.
r = 3.613+
Proved by Pirl in 1969. r = 3.813+
Proved by Pirl in 1969.
11.
12.
r = 3.923+
Proved by Melissen in 1994. r = 4.029+
Found by Kravitz in 1967.
13.
14.
r = 2 + 5 = 4.236+
Found by Kravitz in 1967. r = 4.328+
Found by Goldberg in 1971.
15.
16.
r = 4.521+
Found by Kravitz in 1967. r = 4.615+
Found by Goldberg in 1971.
17.
18.
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+
Found by Kravitz in 1967. r = 5.122+
Found by Goldberg in 1971.
Back to Erich's Packing Center.