A while ago, I showed how to visualize holomorphic self-maps of the sphere by drawing the pre-image of the standard polar coordinate system of the sphere (aka latitudes and longitudes). I mentioned that it should be possible to have these 3D printed, and here they are.
They are printed with a gypsum printer, which is the only one have access to that can do color. That means that they are definitely neither suitable for golf or table tennis, nor for the bath tub. But I could use these for an exam in a Complex Analysis class. Each student gets one of these balls, and has to find out what rational function it represents.
The red lines (being the preimages of the longitudes) come together in the preimages of the two poles. Hence we can locate the zeroes and poles of the function. The only problem is that these pictures don’t distinguish between zero and infinity, nor tell they anything about scaling.
They do tell about branching, i.e. the location of the zeroes of the derivative. For instance, in the blurry ball to left up above, we see a branched point where four of the red lines meet (instead of the expected 8 of the polar grid).