How can you conformally transform a circle into a square? (conformal transformations are ones where angles are conserved) It has been shown that the disk can be conformally transformed into any other closed shape. There is even a theorem called Schwarz-Christoffel that gives the formula in terms of complex elliptic integrals. Unfortunately those formulas have no explicit formulation, so their use has been pretty much limited so far, even though the possibilities are large.
Here is one of the simplest but non trivial mappings between a disk and a square. It uses elliptic integrals (try looking for "Elliptic integral" on your calculator). This transformation is used in some cartographical projections: for example in Peirce's Quincuncial Map, named after Charles Peirce, an American polymath. Notice how the right angles of the square grid are still 90° in the the transformed image. Notice also how the center is almost unchanged and the only points that show great distortion are the ones in the corners.
Stay tuned for exciting conformal transformations of the sphere!