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The complex squaring map \(f(z) = z^2\) "wraps the plane around itself twice" by doubling polar angles at \(0\). The domain plane is at the top, compare our Complex Square Roots products. The image, with a third coordinate added to clarify the mapping, is at bottom. The grid of hyperbolas at the top maps to a square Cartesian grid at bottom. Either surface may be viewed as the Riemann surface of the square root.
