Following Jack’s suggestion, let’s try to visualize holomorphic maps from the Riemann sphere to itself. As a reference, here is the Riemann sphere with polar coordinates, and on the right hand side its cylindrical projection.
The preimage of this under a quadratic polynomial looks like this:
And here a rational function function of degree 4:
One can get quick images using uv-mapping, but there are some rendering artifacts I don’t know how to get rid of yet. Degree 5:
Finally, the Gamma function, which has an essential singularity at infinity.
Martha says their gypsum printer can print these in color. We’ll see, I hope.