To achieve this: http://en.wikipedia.org/wiki/Square_wheel, what should $L$ be?
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The height of the catenary of a square of side length 1 is $(1/\sqrt2)-(1/2)$ since the diagonal of a square is $\sqrt2$ times as long as the side length and the centre of the square must remain at a constant height. The length of the catenary is 1 because the square rolls without slipping.