Circles in Squares

The following pictures show n unit circles packed inside the smallest known square (of side length s). Most of these have been proved optimal.

 1. 2. 3. s = 2 Trivial. s = 2 + √2 = 3.414+ Trivial. s = 2 + 1/√2 + √6/2 = 3.931+ Trivial.

 4. 5. 6. s = 4 Trivial. s = 2 + 2√2 = 4.828+ Trivial. s = 2 + 12/√13 = 5.328+ Proved by Graham in 1963.

 7. 8. 9. s = 4 + √3 = 5.732+ Proved by Schaer in 1964. s = 2 + √2 + √6 = 5.863+ Proved by Schaer/Meir in 1964. s = 6 Proved by Schaer in 1964.

 10. 11. 12. s = 6.747+ Proved by De Groot in 1990. s = 7.022+ Proved by Peikert in 1991. s = 2 + 15√(2/17) = 7.144+ Proved by Peikert in 1991.

 13. 14. 15. s = 7.463+ Proved by Peikert in 1991. s = 6 + √3 = 7.732+ Proved by Wengerodt in 1987. s = 4 + √2 + √6 = 7.863+ Proved by Peikert in 1991.

 16. 17. 18. s = 8 Proved by Wengerodt in 1983. s = 8.532+ Proved by Peikert in 1991. s = 2 + 24/√13 = 8.656+ Proved by Peikert in 1991.

 19. 20. 21. s = 8.907+ Proved by Peikert in 1991. s = 130/17 + 16/17√2 = 8.978+ Proved by Peikert in 1991. s = 9.358+

 22. 23. 24. s = 9.463+ s = 2 + 2√2 + 2√6 = 9.727+ s = 6 + √2 + √6 = 9.863+

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