“Hairy balls” in optics?

The hairy ball theorem is a fascinating aspect of the subject of mathematics known as topology, which loosely speaking is concerned with the mathematical characterization of shapes. But hairy balls are of significance in other fields, as well: in fact, many optical scientists are unaware that a hairy ball is, in essence, right in their face every time they do experiments! It is hidden in the polarization of light, i.e. in light’s inherent transverse wave properties. In this post we will uncover this mysterious hairy ball of light.