Diagonals in an n-cube

Geometry Level pending

Take a unit n-cube (an n dimensional analogue of a cube) in \(d\) dimensions. The length of the longest segment connecting two points in the cube, or longest diagonal, is \(D\). Let \(S\) be the sum of all the distinct diagonals of length \(D\). For this particular dimension, you are given that

\(\frac { { S }^{ 2 } }{ d } =64\)

What is \(d\)?

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