There is a conformal map between any two simply connected open proper subsets of the complex plane. This means, for example, there is a one-to-one analytic map from the interior of a square onto the interior of a a circle. Or from the interior of a triangle onto the interior of a pentagon. Or from the Mickey Mouse logo to the Batman logo (see here).
So we can map (the interior of) a rectangle conformally onto a very different shape. Can we map a rectangle onto a rectangle? Yes, clearly we can do this with a linear polynomial, f(z) = az + b. Are there any other possibilities? Surprisingly, the answer is no: if an analytic function takes any rectangle to another rectangle, that analytic function must be a linear polynomial.
Since a linear polynomial is the composition of a scaling, a rotation, and a translation, this says that if a conformal map takes a rectangle to a rectangle, it must take it to a similar rectangle.
These statements are proved in [1]. Furthermore, the authors prove that “An analytic function mapping some closed convex n-gon R onto another closed convex n-gon S is a linear polynomial.”
More posts on conformal mapping
[1] Joseph Bak and Pisheng Ding. Shape Distortion by Analytic Functions. The American Mathematical Monthly. Feb. 2009, Vol. 116, No. 2.
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