Circles in Hexagons

The following pictures show n unit circles packed inside the smallest hexagon (of side length s).


1.
           2.
           3.
s = 2/√3 = 1.154+
Trivial.
s = 1 + 2/√3 = 2.154+
Trivial.
s = 4/√3 = 2.309+
Trivial.


4.
           5.
           6.
s = 2/√3 + 4/√7 = 2.666+
Found by Shahriar Manzoor
in April 2002.
s = 2.999+
Found by Erich Friedman
in 1999.
s = 2 + 2/√3 = 3.154+
Trivial.
 


7.
           8.
           9.
s = 2 + 2/√3 = 3.154+
Trivial.
 
s = 3.709+
Found by David W. Cantrell
in July 2002.
s = 20/7 + 2/√3 = 4.011+
Found by David W. Cantrell
in July 2002.


10.
           11.
           12.
s = 4.119+
Found by David W. Cantrell
in July 2002.
s = 2 + 4/√3 = 4.309+
Found by Erich Friedman
in 1999.
s = 2 + 4/√3 = 4.309+
Found by Erich Friedman
in 1999.


13.
           14.
           15.
s = 8/√3 = 4.618+
Found by Erich Friedman
in 1999.
s = 2 + 2/√3 + 4/√7 = 4.666+
Found by David W. Cantrell
in July 2002.
s = 4.961+
Found by David W. Cantrell
in January 2006.


16.
           17.
           18.
s = 4.999+
Found by David W. Cantrell
in July 2002.
s = 5.133+
Found by David W. Cantrell
in January 2006.
s = 4 + 2/3 = 5.154+
Trivial.
 


19.
           20.
           21.
s = 4 + 2/√3 = 5.154+
Trivial.
 
s = 5.676+
Found by David W. Cantrell
in July 2012.
s = 5.709+
Found by David W. Cantrell
in July 2012.


22.
           23.
           24.
s = 5.952+
Found by David W. Cantrell
in July 2012.
s = 34/7 + 2/√3 = 6.011+
Found by David W. Cantrell
in July 2012.
s = 6.119+
Found by David W. Cantrell
in July 2012.


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