Mathematics as art, and as craft

I don’t think I’m too shy in the fact I have a somewhat non-standard approach to mathematics, but I had a recent realisation about my own mindset that I found interesting.

I grew up in a family with a strong emphasis on arts and crafts. Spinning, knitting, pottery, leadlighting, paper-making, printing, furniture restoration, garment construction, baking, drawing and so on. At the end of the day, there was something you could hold, touch, wear and so on. At one stage in high school I considered studying Design.

I think this idea that at the end of the day, one can actually make something, is one that pervades my mathematics. This ranges from my habit of physically printing stapled booklets of each paper I write, to wanting a concrete formulas or constructions for certain abstract objects. At one point I had found explicit transition functions for the nontrivial String bundle on S^5, and I collaborated with someone more expert at coding than me to generate an animation of part of this. Another time I really wanted to get my hands on (what an apt metaphor!) what amounts to a map of higher (non-concrete) differentiable stacks, so worked out the formula for my own satisfaction:

An explicit formula for a functor from the 5-sphere to the stack of SU(2)-bundles with connection

I love proving a good theorem, but if I can write it out in a really visually pleasing way, then it is much more satisfying. Such as the circle of ideas around the diagonal argument/Lawvere fixed point theorem.

The Yanofsky variation of Lawvere’s fixed point theorem, in a magmoidal category with diagonals. That’s the proof.

I very much like designing nice-looking diagrams, and at one point was trying to get working a string diagram calculus for working in the hom-bicategory of \mathbf{2Cat} (with objects the weak 2-functors), for the purpose of building amazing looking explicit calculations to verify a tricategorical universal property.

String diagram data for a transformation between weak 2-functors

Sadly, I never finished this, and now the reason—a second independent proof of a higher-categorical fact in the literature with many omitted details—is now moot, with another proof by other people.

I have a t-shirt with a picture of the data generated in my computational project on the Parker loop (joint with Ben Nagy), and I’m itching to make more clothes with my maths on them. I love the fun I’ve had with David Butler thinking about measures of convexity of deltahedra, built with Polydrons (it seems to be an open question how non-convex one can go!)

It just speaks to me when I can actually make something, or at least feel like I’ve made something when doing maths. Something I can point at and say “I made that”. I think there’s an opportunity in the market for really high-quality art prints of pieces of really visually beautiful category theory, for instance, or even just mathematics more broadly. I’ve experimented over the years with (admittedly naive, amateur, filter-heavy) photographs of mathematics, for the sake of striving for an aesthetic presentation.

Proof of an elementary result around t-th roots of integers, with the lions on Adelaide’s CML Building in the background

I want the mathematics I create to “feel real”. Sometimes that feeling comes when I can hold the whole conceptual picture in my head at once, but it’s ephemeral. Actually making the end product, making it tangible—no matter how painful it can feel in the process—is a real point of closure.

Even the process of making little summary notes of subjects I studied at school and uni has produced objects that I have kept, and have a fondness for. They are the distillation of that learning, the physical artifact that represents my knowledge

Notes from Year 12 Physics and Chemistry

Even the choice to work in physical notebooks, and slowly build that collection, rather than digital note-taking gives me something I can see slowly grow, and I can appreciate as being a reflection of my changing ideas and development in research. Having nice notebooks gives an aesthetic that motivates me to fill them up.

Stack of (mostly) Moleskine research notebooks

Given that mathematics is generally taught as a playground of the mind, though there is of course a push in places for more manipulables in maths education (physical or digital), more visualisation, I do wonder the extent to which students feel like they are missing an aspect like this. We don’t need a 3d printer in a classroom to have the students make something tangible, lasting and awesome. Somehow I’ve managed to avoid edutech, despite winning a graphics calculator at school back in the 90s—and never learning how to use it—and I’ve never been taught with manipulables past Polydrons at age 5 and MAB blocks at age 6 (both of which I still think are awesome). But I love actually making mathematical things.

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