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+
Found by Kravitz in 1967. 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+
Found by Kravitz in 1967. r = 5.122+
Found by Goldberg in 1971.

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+ Found by Kravitz in 1967.		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+ Found by Kravitz in 1967.		r = 5.122+ Found by Goldberg in 1971.