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


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


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


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


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


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


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


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


Back to