I’ve noticed there seems to be a general unspoken rule about the relationship between mathematics and science: any mathematics, no matter how abstract or seemingly disconnected from reality, eventually finds use or representation in the natural world. For example, most people are probably familiar with the idea of an “imaginary number,” namely the square root of -1. The term “imaginary” was coined by famed mathematician René Descartes, who really meant it as a derogatory term: imaginary numbers were thought to be useless artifacts of the imagination. Today, such numbers are a fundamental part of physics, used in every branch from quantum mechanics to optics to mechanics to describe the properties of physical systems. They are almost the opposite of useless.
Other examples abound. Number theory, which is the branch of mathematics devoted to the study of the integers, i.e. 1,2,3,4,…, and their relationships, would seem to be completely devoid of practical interest. However, it plays an important role in modern cryptography, helping to keep our data secure in the information age. Another example is the study of quaternions, objects which may be considered three-dimensional generalizations of imaginary numbers. These quantities were almost forgotten by the early 20th century but have become extremely useful in computer graphics and robotics, among other technologies.
Even with a knowledge that even the most abstract math can be of real-world relevance, I still can find myself caught of guard, even stunned, when such math peeks out from within a very physical problem. Today, I was working on an optics problem when I realized that the problem in question was a direct demonstration of the strangeness of infinite sets! The optics problem in question involves light beams with so-called optical vortices in them, something I’ve talked about on the blog before. And the property of infinite sets in question is the very strange booking practices of a hotel with infinite rooms!
Skulls in the Stars 
