Maximum Radius of Sphere in the Gaps in a Face Centred Cubic Lattice

A cubic lattice consisting of layers of spheres of radius r with layers of spheres of radius r above and below, offset so that each ball of each layer sits between four spheres above and below,br>

between the spheres there are gaps. One of the layers is shown below.

The distance from A to B is  
\[\sqrt{(2r)^2 + (2r)^2 } = 2r \sqrt{2}\]

The distance from C to D is then  
\[\sqrt{(2r)^2 + (2r)^2 } -2r= 2r \sqrt{2} -2r\]

This face centred cubic structure has rotational symmety about the midpoint of AB, using the line AB as an axis of rotation, as well as vertical and horizontal rotational symmetry. This means that we can fit a sphere of radius  
\[\frac{ 2r \sqrt{2} -2r}{2} = r (\sqrt{2} -1)\]
  in this gap.

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