Circles in Circles
The following pictures show n unit circles packed inside the smallest known circle (of radius r).
1. |
2. |
3. |
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| r = 1 Trivial. |
r = 2 Trivial. |
r = 1 + 2 / √3 = 2.154+ Trivial. |
4. |
5. |
6. |
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| r = 1 + √2 = 2.414+ Trivial. |
r = 2.701+ Proved by Graham in 1968. |
r = 3 Proved by Graham in 1968. |
7. |
8. |
9. |
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| 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. |
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| 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. |
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| 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. |
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| 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. |
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| r = 1 + √2 + √6 = 4.863+ Proved by F. Fodor in 1999. |
r = 5.122+ Found by Goldberg in 1971. |
