# Mathematical Tattoos: The Ink Equation

# BLOG: Heidelberg Laureate Forum

So often, mathematicians speak of the beauty of equations, not only in the way they can convey complex relationships and ideas succinctly, but also from a purely visual point of view. Take, for example, Euler’s identity: \( e^{i\pi} + 1 = 0\). Aesthetically simple and elegant, it also highlights the deep links between foundational mathematical operations and numbers. Three operations – addition, multiplication and exponentiation – glue together five of the most important and fundamental mathematical constants – 0, 1, \(e\), \(i\), \(\pi\) – that on face value have no business intermingling in the same equation.

Of course, even with these convincing arguments, whether the Euler equation is beautiful or not is purely subjective. As the saying goes, beauty is in the eye of the beholder. So, when a person decides to get body art, it is often a highly personal expression of their idea of what constitutes beauty.

For young researcher Leo Liang, an applied mathematician and computer scientist from the US who is also an extremely accomplished musician, his biggest tattoo is an abstract representation of the essence of his love for music. “I like music that just exists purely by itself; there’s no underlying words or, I guess, meaning to it in a way,” he told us when we spoke at the 11th Heidelberg Laureate Forum, where he was attending as a young researcher. “And that really inspired my tattoo: there’s no meaning to it necessarily, but I just like it because of the way it is.”

### Tattoos That Count

Tom Crawford – a mathematician at the University of Oxford and the University of Cambridge, mathematics advocate and communicator, and moderator at the 11th HLF – is no stranger to the tattoo artist’s needle either. Of the ~120 and counting tattoos dotted across him (mostly by @Nat_Von_B at Tattoo Crazy Cambridge), many are mathematics-themed. Unsurprisingly, Euler’s identity features, as do other equations well-recognised for their importance, including Maxwell’s equations of electromagnetism, Heisenberg’s uncertainty principle from quantum mechanics and the Navier– Stokes equation that represents fluid flow.

But there is another, perhaps less familiar, equation that was Crawford’s first maths-themed tattoo, and is proudly emblazoned on his inner forearm:

\[ \frac{H}{H_{0}} = \frac{1 – \alpha_{T}}{1 – \alpha_{i}} \]

“This was referred to by my PhD examiner as genius in my viva, but I hadn’t even realised that it was the main result of my thesis,” recalls Crawford. His research was studying river outflows into the ocean, ultimately to improve models of how pollution is spread from river systems. During his studies, he conducted experiments of how water would behave in a large lab-based tank, and observed the equivalent of a rotating vortex near the river mouth, with a boundary current propagating along the coast.

Most importantly, because he could run his experiments for much longer timescales than experiments in real rivers, he observed that the depth of the current was not a constant, as was previously thought. This meant rewriting the standard potential vorticity equation, which connects the vorticity (whirlpool-like rotation labelled \(\alpha\)) of a river outflow with its depth (labelled \(H\)), to account for time dependence. “I wasn’t planning to get this particular tattoo until my viva happened, but it commemorates or represents my research.”

### Artistic Representation of the Basel Problem

Some of his other maths-themed tattoos move away from simple equations. Perhaps one of the most interesting, and Crawford’s personal favourite, maths-themed tattoo is a series of bands around his other forearm that are a visualisation of the Basel (or Basler) problem.

First proposed in 1644 by Italian mathematician and priest Pietro Mengoli, the Basel problem can be stated as follows:

Find the numerical value of

\[ 1 +\frac{1}{4} + \frac{1}{9}+ \frac{1}{16} + \frac{1}{25} + … = \sum_{x=1}^{\infty} \frac{1}{x^2} \]

Because it converges extremely slowly, finding an exact solution stumped the greatest minds of the time, including the Bernoulli brothers, Christian Goldbach and Gottfried Wilhelm Leibniz. Leonard Euler finally solved the problem in 1735. “The answer is \(\frac{\pi^2}{6}\); I can prove it in 12 different ways, but it doesn’t make sense,” Crawford says. “And where does \(\pi\) come from? It’s literally the most unexpected \(\pi\) ever – I love the result so much.”

To visualise this through body art, a thick initial band around Crawford’s wrist represents zero, and another about 10 centimetres up his arm represents 1. Then both the distance and the thickness of the bands shrink according to \(\frac{1}{x^2}\), until they are too thin to tattoo. A little way above that point is a dotted line representing the limit of the sum that you can never get to, i.e. \(\frac{\pi^2}{6}\).

### Judge the Researcher, Not the Tattoos

Crawford receives a lot of attention because of his ink, and has channelled this attention for good, always happy to explain what a certain tattoo represents to anyone who asks and presenting talks in bars and comedy clubs explaining the maths of his tattoos. But he also fields questions from many mathematicians, via email or social media, who are thinking of getting a tattoo but are wary of it hurting their career prospects.

“Many people say they want to get such and such a tattoo, but are not sure if it’s going to impact their career, and they’ve asked me for advice on how I’ve navigated that potential judgment,” says Crawford. “I always say, unless there are cultural or religious factors, you should just be judged on your ability to do maths or science or whatever academic subject you’re doing – who cares what you choose to wear or what you do with your body?”

“Have I ever faced situations where I feel like it has held me back in my career? There have been one or two instances where I have felt negativity, but if they see me and don’t judge me on my ability to do maths or to communicate maths, and instead judge me on the fact that I have tattoos, they’re not really people I particularly want to collaborate with.”

Benjamin Skuse wrote (09. Oct 2024):

>

[…] Tom Crawford – a mathematician at the University of Oxford and the University of Cambridge, mathematics advocate and communicator, and moderator at the 11th HLF – is no stranger to the tattoo artist’s needle either. […]>

[… The] equation that was Crawford’s first maths-themed tattoo, and is proudly emblazoned on his inner forearm:\[ \frac{H}{H_{0}} = \frac{1 – \alpha_{T}}{1 – \alpha_{i}} \]

>

“This was referred to by my PhD examiner as genius in my viva, but I hadn’t even realised that it was the main result of my thesis,” recalls Crawford. […]Had Crawford written the relevant equation (expressing the accounting for time dependence; eq. (7.4) in his thesis) instead as

\[ \frac{H[ \, t \, ]}{H[ \, t_0 \, ]} = \frac{1 – \alpha[ \, t \, ]}{1 – \alpha[ \, t_0 \, ]} \]

it could perhaps even have been referred to as “a piece of art”. …