Abstract: It is almost impossible for humans to imagine higher dimensional spaces, yet mathematicians can prove things about them nevertheless, so how do they do it? In my research project, I looked at the Hopf fibration, which is a phenomenon that occurs with specific high dimensional spheres. I discuss how I grappled with these abstract mathematical ideas during my project, and what makes me interested in studying these things.
Ever since I had heard about higher dimensions for the first time, I have been fascinated by them. There’s something intriguing about the idea that there are objects that we cannot hope to ever visualise in their entirety, which makes it incredibly difficult to gain a solid intuition for how they work. Of course, it is entirely possible to describe them precisely using our mathematical tools and prove facts about them that are very much nontrivial.
In my research, I had the chance to properly work with some high dimensional objects, with a focus on spheres. In particular, there is this thing called a Hopf fibration which exists for certain spheres in dimensions related to powers of two. It turns out that there is this pattern in the Hopf fibrations which, strangely, breaks down suddenly when you go beyond 16 dimensions. It is not at all obvious why this would be the case, and it is basically impossible to visualise these kinds of objects. Nevertheless, after my six-week research project, I can now say that it “makes sense” to me why this happens.
Of course, I did not magically gain the ability to see these higher dimensional spaces in these six weeks. What happened instead, was that I had learned how the Hopf fibration connects to various other objects in mathematics, which were much easier to understand in terms of its algebraic properties. This led to a deep dive into all sorts of references about projective spaces, Cayley-Dickson algebras, spin groups, and more, each of which provided a different perspective to the Hopf map and allowed me to slowly refine my intuition bit by bit.
This is, in essence, what mathematics is really about. Many people think of mathematics as a bunch of dry, tedious computations, which is understandable as this is what most people experience during maths classes in school. This is because school mathematics covers topics that have already been thoroughly studied and people have already put in the effort to present the material as cleanly as possible. However, the part of mathematics that people don’t see is the messy intuitions and trial and error necessary to come up with new mathematical ideas in the first place. This is the creative process required to do mathematics, and it is the part which I find most exciting.
Many people ask me why we care about these bizarre, abstract objects which live in spaces that are so alien compared to the 3-dimensional Euclidean space we are familiar with. One reason is that we often only discover things to apply mathematical theories to in retrospect, so it is worth developing theories which may become useful one day. While this is true, as someone interested in pure mathematics, I honestly don’t really care much about the applications myself. Instead, I enjoy mathematics because it gives me a constant supply of new perspectives to view things from and ways to see connections between seemingly unrelated ideas.
Anders Yu
Adelaide University
