Circles in Ellipses

The following pictures show n unit circles packed inside the ellipse with smallest known area A (with semi-axes a and b).

 1. 2. 3. A = π(1)a = 1     b = 1Trivial. A = π(3√3/2)a = 3/√2     b = √(3/2)James BuddenhagenJune 2004 A = π(16√6/9)a = 8/√6     b = 4/3James BuddenhagenJune 2004

 4. 5. 6. A = π(2+2√3)a = 1+√3     b = 2David W. CantrellJune 2007 A = π(7.279+)a = 3.822+     b = 1.904+David W. CantrellJune 2007 A = π(7.803+)a = 2+√3     b = 2.090+David W. CantrellJune 2007

 7. 8. 9. A = π(9)a = 3     b = 3David W. CantrellJune 2007 A = π(10.546+)a = 4.718+     b = 2.235+David W. CantrellJune 2007 A = π(12.403+)a = 4.283+     b = 2.895+David W. CantrellJune 2007

 10. 11. 12. A = π(12.826+)a = 4.117+     b = 3.115+David W. CantrellJune 2007 A = π(14.578+)a = 4.772+     b = 3.054+David W. CantrellJune 2007 A = π(15.921+)a = 4.905+     b = 3.245+David W. CantrellJune 2007

 13. 14. 15. A = π(17.008+)a = 5.168+     b = 3.291+David W. CantrellJune 2007 A = π(18.014+)a = 4.730+     b = 3.808+David W. CantrellJune 2007 A = π(19.824+)a = 6.361+     b = 3.116+David W. CantrellJune 2007

 16. 17. 18. A = π(20.761+)a = 7.289+     b = 2.847+David W. CantrellJune 2007 A = π(22.033+)a = 7.998+     b = 2.754+David W. CantrellJune 2007 A = π(22.819+)a = 5.730+     b = 3.982+David W. CantrellJune 2007

 19. 20. 21. A = π(23.655+)a = 1 + √2 + √6     b = 1 + √2 + √6David W. CantrellJune 2007 A = π(25.723+)a = 9.005+     b = 2.856+David W. CantrellJune 2007 A = π(27.084+)a = 6.497+     b = 4.168+David W. CantrellJune 2007

 22. 23. 24. A = π(27.998+)a = 6.751+     b = 4.146+David W. CantrellJune 2007 A = π(29.514+)a = 9.918+     b = 2.975+David W. CantrellJune 2007 A = π(30.224+)a = 5.996+     b = 5.040+David W. CantrellJune 2007

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