Comment by ogogmad

Have you considered finding a conformal transformation† that maps a square to any other possible shape, as long as the shape doesn't have any holes? Such a transformation always exists by the Riemann Mapping Theorem, and is unique as long as you specify in addition (1.) which point the square's centre maps to, and (2.) the angle of rotation around that point. Not sure if anyone's ever tried that.

If you actually want more aesthetic freedom, you can compose with an arbitrary diffeomorphism of the square to itself. But I think that might usually look worse.

† - That is, preserving all angles, including right angles. The terminology stems from the output angles conforming (???) to the input angles.