Circles in Triangles

The following pictures show n circles with the largest possible sum of radii packed inside a unit equilateral triangle.


1.
           2.
           3.
Σ r = √3/6 = .288+
Trivial.
Σ r = 2√3/9 = .384+
Trivial.
Σ r = (3√3 – 3) / 4 = .549+
Trivial.


4.
           5.
           6.
Σ r = .617+
Found by David W. Cantrell
in June 2011.
Σ r = .693+
Found by David W. Cantrell
in June 2011.
Σ r = 6 – 3√3 = .803+
Trivial.


7.
           8.
           9.
Σ r = .855+
Found by David W. Cantrell
in June 2011.
Σ r = .912+
Found by David W. Cantrell
in June 2011.
Σ r = .976+
Found by David W. Cantrell
in June 2011.


10.
           11.
           12.
Σ r = (15 – 5√3) / 6 = 1.056+
Trivial.
Σ r = 1.098+
Found by David W. Cantrell
in June 2011.
Σ r = 1.143+
Found by David W. Cantrell
in June 2011.


13.
           14.
           15.
Σ r = 1.191+
Found by David W. Cantrell
in June 2011.
Σ r = 1.243+
Found by David W. Cantrell
in June 2011.
Σ r = (60 – 15√3) / 26 = 1.308+
Trivial.


16.
           17.
           18.
Σ r = 1.343+
Found by David W. Cantrell
in June 2011.
Σ r = 1.381+
Found by David W. Cantrell
in June 2011.
Σ r = 1.420+
Found by David W. Cantrell
in June 2011.


19.
           20.
           21.
Σ r = 1.462+
Found by David W. Cantrell
in June 2011.
Σ r = 1.505+
Found by David W. Cantrell
in July 2011.
Σ r = (105 – 21√3) / 44 = 1.559+
Trivial.


22.
           23.
           24.
Σ r = 1.589+
Found by David W. Cantrell
in July 2011.
Σ r = 1.622+
Found by David W. Cantrell
in July 2011.
Σ r = 1.655+
Found by David W. Cantrell
in July 2011.


25.
           26.
           27.
Σ r = 1.690+
Found by David W. Cantrell
in July 2011.
Σ r = 1.727+
Found by David W. Cantrell
in July 2011.
Σ r = 1.763+
Found by David W. Cantrell
in July 2011.


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