At work the other day, we had a bit of a break to celebrated the month’s birthdays. During this time — my coworkers sitting around a large table, munching on pie and ice cream — a conversation broke out. How many holes are there in a paper towel roll: 1 or 2?
Of course, there is the joke that if you claim 1, then there is only one hole of consequence in the human body: that which leads from your mouth to the escape of your digestive system. At that point you’re just talking out of your ass.
But, being a company full of nerds and pedants, those of us who were sensible had to claim 1; after all, a paper towel has width to it and is thus homeomorphic to a torus (think, bagel).
Topology is a fun subject. Imagine taking geometry, and saying “I don’t really care about angles, or lengths, or these things that make geometry what it is. I mainly just care if I can stretch something to make it into something else, and how many holes there are in the shape.” That’s the base level of topology. An introductory topology course teaches you about homeomorphisms, which means you can continuously stretch, pull, push, or generally manipulate an object without cutting or tearing it.
Imagine you had a very flexible, putty-like bagel. If you mad the bagel very thin, skinny, and tall, you can imagine forming yourself a paper towel roll. This is what we mean when we say they are homeomorphic. We don’t put an extra hole anywhere, we just squish and shape things to our will.
Some people may ask why this is useful, particularly when you learn a coffee mug is also homeomorphic to a torus. The easiest explanation is math is about abstractions and patterns. Sure, a cube and a non-cubic rectangular prism have some differences, insofar as the rectangular prism doesn’t have three equal edge lengths. A cube and a triangular prism seem even further removed. Yet, any prism, or more generally polyhedron (which includes pyramids as well!), are all homeomorphic to a sphere. These three dimensional figures share certain properties, most notably that they are solid objects with no holes in them. And there’s something to be said for that!
While topology goes well beyond this idea, it gives a good grasp at what mathematicians are interested in. They noticed that we can generalize some of these properties, and map between objects using these homeomorphisms. So, instead of topology being about classes of objects which are the same in some small way, it is really about how we relate these similar objects. Topology is about showing the similarities and patterns between objects. It’s interesting that with a consistent set of rules, we can say “cube is to sphere as bagel is to paper towel roll.” We notice this pattern, and figure out how to make it rigorous and useful.
This is an under-appreciated aspect of math. It is not about rules that have no reason, or about abstraction for no reason. It is about finding a pattern, and seeing how other objects follow a similar pattern. This is what make mathematics a very fun subject, yet it is a mindset very few people are able to have. I hope this can change in the future.