Circles in Squares

The following pictures show n unit circles packed inside the smallest known square (of side length s). Most of these have been proved optimal.


1.
           2.
           3.
s = 2
Trivial.
s = 2 + √2 = 3.414+
Trivial.
s = 2 + 1/√2 + √6/2 = 3.931+
Trivial.


4.
           5.
           6.
s = 4
Trivial.
s = 2 + 2√2 = 4.828+
Trivial.
s = 2 + 12/√13 = 5.328+
Proved by Graham in 1963.


7.
           8.
           9.
s = 4 + √3 = 5.732+
Proved by Schaer in 1964.
s = 2 + √2 + √6 = 5.863+
Proved by Schaer/Meir in 1964.
s = 6
Proved by Schaer in 1964.


10.
           11.
           12.
s = 6.747+
Proved by De Groot in 1990.
s = 7.022+
Proved by Peikert in 1991.
s = 2 + 15√(2/17) = 7.144+
Proved by Peikert in 1991.


13.
           14.
           15.
s = 7.463+
Proved by Peikert in 1991.
s = 6 + √3 = 7.732+
Proved by Wengerodt in 1987.
s = 4 + √2 + √6 = 7.863+
Proved by Peikert in 1991.


16.
           17.
           18.
s = 8
Proved by Wengerodt in 1983.
s = 8.532+
Proved by Peikert in 1991.
s = 2 + 24/√13 = 8.656+
Proved by Peikert in 1991.


19.
           20.
           21.
s = 8.907+
Proved by Peikert in 1991.
s = 130/17 + 16/17√2 = 8.978+
Proved by Peikert in 1991.
s = 9.358+


22.
           23.
           24.
s = 9.463+ s = 2 + 2√2 + 2√6 = 9.727+ s = 6 + √2 + √6 = 9.863+


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