Hexagons in Hexagons

The following pictures show n regular hexagons with side 1 packed inside the smallest known regular hexagon (of side length s ).


1.

s = 1
Trivial.
2.

s = 2
Trivial.
3.

s = 2
Trivial.


4.

s = 7/3 = 2.333+
Trivial.
5.

s = 8/3 = 2.666+
Found by Erich Friedman
in March 2015.
6-7.

s = 5/√3 = 2.886+
Found by Maurizio Morandi
in April 2015.


8.

s = (51+5√3)/18 = 3.314+
Found by Maurizio Morandi
in April 2015.
9.

s = (24+5√3)/9 = 3.628+
Found by Maurizio Morandi
in April 2015.
10.

s = 2 + √3 = 3.732+
Found by Maurizio Morandi
in April 2015.


11.

s = 5(3 + √3)/6 = 3.943+
Found by Maurizio Morandi
in April 2015.
12-13.

s = 4
Trivial.
14.

s = 37√3/15 = 4.272+
Found by Maurizio Morandi
in April 2015.


15.

s = (73+35√3)/30 = 4.454+
Found by Maurizio Morandi
in April 2015.
16.

s = 55√3/21 = 4.536+
Found by Maurizio Morandi
in April 2015.
17-19.

s = 8/√3 = 4.618+
Found by Maurizio Morandi
in April 2015.


20-21.

s = 5
Trivial.
22.

s = (39+8√3)/10 = 5.285+
Found by Maurizio Morandi
in April 2015.
23.

s = 38/7 = 5.428+
Found by Maurizio Morandi
in April 2015.


24.

s = 19√3/6 = 5.484+
Found by Maurizio Morandi
in April 2015.


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