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Because while the circle (an object given by a smooth differentiable function in the frist quadrant, if the origin is at the coordinate origin) is the limit in pointwise convergence of this process, the limit lenghts of the rigid figures (which are always 4) isn't that of the circle.
Of course I've just rephrased what the picture says, so my answer is - why would it be? You're dealing with real analysis here, limits over infinite sequences, and barely anyone understand analysis.
