# My life with mathematical tattoos

… and how Chalkdust played a role in one of them

A few years ago, I started dating a ridiculously attractive goth. Dating a goth was a brand new thing to a nerdy mathematician like me, as I’d previously imagined they were all scary, death-obsessed depressives. Turns out I’d been stereotyping, and they’re actually just into a certain style: wearing black, listening to industrial music and body modification. Including tattoos. Lots and lots of tattoos.

This was a new one on me. I’ve always been a nerd, long before being a nerd was in. I was the nerdiest person in my high school, publicly teased for my love of maths and Doctor Who. I was so nerdy at university that I founded the maths club. So nerdy I became professor and combined my love of maths with my love of pop culture and made a mathematical model of zombies. Admittedly, that had gone viral in the media and won me a Guinness World Record, so being nerdy perhaps wasn’t as socially unacceptable as it once had been. But still, the point stands. I’m a big old nerd and proud of it!

/adjusts glasses and straightens pocket protector

In the past, I’d dated the odd person with a butterfly or mermaid tattoo, but Goth Girl had a bunch, all black, and was eager for more. I even went along to a tattoo session with her to see what the deal was and hold her hand when it hurt. And yeah, it hurt. (My hand, that is, because her grip was really tight.)

I was in no danger of becoming a goth myself (my irrepressible love of dance-able eighties music put paid to that idea before it began), but I found myself wearing a lot more black than I used to. She asked me early on if I would ever get a tattoo, and I said no way. Fine for her, and yeah they actually looked really good on her, but it just didn’t seem like the thing for me. Besides, I had a nerd rep to protect!

But, after about a year or so, she asked me a different question: “If you were going to get a tattoo, what would you get?” I thought about it for a moment and then had a sudden flash of inspiration, so I replied “Oh, I’d get the Mandelbrot Set on my back!”

## Fractal designs

I wasn’t being serious. But man was that a bold idea. Hmm.

I had to explain to her what the Mandelbrot Set was: “It’s the set of $c$ values such that the iterative map $z_{n+1}=z_n+c$ converges, except that the border is very complicated, infinitely so in fact, yet still entirely deterministic, which means… I’m losing you, aren’t I? Here, let me show you this.”

This gets the point across a lot better than explaining the convergence, I discovered.

And when I showed her a picture, she was really wowed by it. I did some googling and found a picture of some guy with exactly that tattoo. It was a reddish colour and filled about a third of his back.

So it got me thinking. And thinking. And thinking some more. (Hey, what can I say? I’m a nerd; I like that sort of thing.)

After about six months of thinking about it, I decided I was going to go for it. The Mandelbrot Set on my back! I figured if I was going to get a tattoo, I’d just get the one but make it a big one. And nobody need know except any fellow sunbathers. (I live in Canada, so not too much of a problem there.)

During this time, I went to a conference and told a bunch of my professor friends about this idea. Most were quite taken aback and I think somewhat appalled but too polite to outright say so. One said “Why don’t you just get the equation instead? We’ll know what it is.” Another said “You know, I’ve never wanted to see the back of another mathematician before.”

Two days before the appointment, I was at a departmental lunch, sitting between a colleague and the chair of my department. I quietly told the colleague about my crazy plan and showed him the photo of the random guy online who had the tattoo. My chair leaned over and said “What are we looking at?” I replied “Me, in two days.” She shrieked, grab my phone out of my hands and proceeded to show the entire table. It so happened that I was about to start teaching my third-year course on dynamical systems, which included a unit on chaos theory. So she said “As chair, I give you permission to take your shirt off in class.”
Hmm. Probably not, but maybe I could at least show them a photo…

My appointment was with the same tattoo artist that Goth Girl used. I was quite nervous, because I didn’t want her to mess it up. I started to explain the concept of self-similarity, but she interrupted me mid-flow and said “I just draw.” So she sketched an outline. That’s where a huge mistake was made. And I do mean huge.

I’d sent her the picture, but I’d miscommunicated the scale. So when she produced the outline, it was enormous. Rather than taking up a discreet(ish) third of my back, it was so large it basically filled my entire back. Whoa. This wasn’t what I’d been imagining… but it was pretty bold. And looked good. Okay, hell yes, let’s go for it…

The way it works is that first they print a sketch onto you in purple ink. This can be adjusted if you don’t like it. In my case, only the vertical tail was slightly off centre. So that was wiped off and reapplied. And… we’re off.

Everyone always asks me if it hurt. Yeah, it hurt. But it hurt like a sunburn hurts. A bit more when she went over bone, but I was lying face-down on a massage table the entire time. And, unlike when you get a massage, here I could actually read a book while doing so! (Did I mention I was a bit nerdy? Oh right, I see that I did…) I found the pain was a lot less if I was reading, so this was win-win.

About four hours later, I had myself a Mandelbrot Set on my back. Holy guacamole. My tattoo artist said “You know, I normally do dragons and skulls and roses. This was… different.”

The back of another mathematician.

I didn’t take my shirt off in class. That just didn’t seem right. But I did show them a photo. The overriding emotion in my students? Sheer bewilderment. “Our prof has a tattoo?” they appeared to be collectively thinking. “Nothing makes sense any more…”

The next conference I went to, one of the colleagues I’d talked to previously was there. Someone relatively famous, in fact. There was a problem with the computer in one talk, so there was a delay, and he sidled up to me and asked “Did you do it?” I nodded and said “Oh yes!” He asked if he could see it, but we were in a room full of mathematicians, professors and students alike, so I didn’t want to take my shirt off. Instead, he crouched down behind me, lifted up my T-shirt and stuck his head inside, loudly exclaiming “Oh my god!” and “I can’t believe what I’m seeing!”

We got the room’s attention. I think it might have been less scandalous if I’d simply taken my shirt off.

At a later conference, we were at the conference banquet, which was spread over two floors: the main dining hall and a more elite balcony, where the senior professors had naturally gathered. A friend came and found me and told me the senior professors had requested my presence upstairs. I went up and found five of them standing there. One said “We understand you have the Mandelbrot Set on your back.” I nodded, and they said “Well then. We need to see this.” Goth Girl later summed it up like this: “So the senior professors summoned you upstairs and ordered you to disrobe? Uh-huh.” My response: “That sounds way more sordid than these conferences ever are.”

So now I had a tattoo, and I loved it. Except… I couldn’t actually see it. I kept angling in the mirror or twisting my head to catch a glimpse. About a year later, I decided I really wanted another one. But what to get?

## I heart tattoos

Here’s where Chalkdust came in. As thanks for writing an earlier article (on the Jacobian, which I love but have no plans to have etched into my skin), they sent me a T-shirt saying
$$\sqrt{-1}\qquad\quad\\ x^2+\left(y-\sqrt[3]{x^2}\right)^2=1\\ \text{Chalkdust}\qquad$$
I figured out that the first word was “$i$” and then realised that the equation must be the equation for a heart. This was super cute, so I mentioned to Goth Girl that I was thinking of getting an equation for a heart over my heart. She said she loved that idea. I was also hoping that the mathematicians who requested the equation instead of the picture of the Mandelbrot Set would be satisfied. Or at least amused.

$\sqrt{-1}\ \heartsuit\ x^2+\left(y-\sqrt[3]{x^2}\right)^2=1$

This tattoo was a lot smaller and consequently a lot easier. It barely even hurt. The only funny thing was that the original tattoo artist wasn’t available, so I went to a new one. He treated it like I was getting Chinese characters or something, saying “I will draw exactly that on you. So you need to make sure it’s right.” When I pointed out that not only could I read the equation but that I was the one who’d generated it, he looked a lot less nervous.

Shortly after getting this, I went to a party with a lot of nerds (hey, they’re my people). So I showed them the equation without telling them what it was. There was a flurry of activity with people pulling out phones and trying to solve it using the Wolfram Alpha app they had pre-installed (yep, definitely my people). My one friend was quickest, except she forgot that there were two roots, so she only drew the right-hand part. “Some kind of J?” she posited. “A Christmas candy cane? Oh wait, I need to input the negative root… Ah!”

## Doubling tattoos

Okay, so now I’d been well and truly bitten by the tattoo bug. They’re actually quite addictive. But you don’t want to rush into any of them, as they are on your body for life. That said, both my maths tattoos were hidden under my shirt, and I’m not really the kind of person who hides his geekery. So I wanted a more visible one.

I decided a cool thing to get would be a bifurcation diagram showing the period doubling route to chaos. This comes from the discrete-time dynamical system
$$x_{n+1}=rx_n(1-x_n),$$
where $r$ is a parameter you can vary. For small values of $r$, there’s a single, stable equilibrium at the origin. If $r$ gets a bit larger, the origin becomes unstable and a second, stable, equilibrium appears. If you increase $r$ a bit more, both equilibria are unstable, but a periodic orbit arises. Increase $r$ yet again and that periodic orbit becomes unstable just as a period-4 orbit appears. Then period-8, period-16 and so forth. In fact, Sharkovskii’s theorem says that if you have period-3 (which you do for this map), then you have every period, along with chaos. That’s a really incredible result… and it makes for a really stunning visual image if you plot the periods versus $r$.

If you look closely, you’ll see the word “Vegan” on my hand. I also got a number of non-maths tattoos, because these things are really addictive. (Not shown: Doctor Who tattoo, because I might be a bit nerdy.)

Goth Girl loved this one, saying that it was one of the most unusual and visually striking tattoos she’d ever seen. Except… when I’d had the idea, I’d googled it and discovered a single picture of someone with it on their forearm. I really wanted a unique tattoo, but Goth Girl told me not to worry about that, because it was almost impossible to find a unique tattoo. I should just enjoy them for what they were and not worry about that.

However… this was a problem I thought I could solve. The answer, it seemed to me, was to simply go higher. When searching online, I found lots of people who had the quadratic formula, Euler’s identity and the like. I met a woman at a conference who had a Fourier transform. I did some informal asking around to see if any other mathematicians had tattoos. People knew of one or two people here or there, but they were mostly millennials. And almost none had mathematical tattoos.

So, with existence sorted, I realised my solution to the problem of uniqueness. I simply needed something from advanced mathematics, because anyone who merely liked maths wouldn’t have anything super advanced, while those who actually knew advanced maths probably didn’t have a tattoo.

I decided on a continued fraction, one of the coolest things I’d learned that was way outside my own field. You can represent any number as a series of nested fractions. The more perfect versions have a 1 as the ultimate numerator, but they aren’t the only versions. And I remembered from my final semester of undergraduate that $\pi$ had a really beautiful continued fraction. A quick check online and I found it.

More accurate than 22/7.

Originally I was going to go bigger, but my tattoo artist pointed out that to fit that on my upper arm would mean shrinking the font. So I figured seven iterations was enough. What I love about this is that it takes an irrational number $\pi$ but, by viewing it in just the right way, shows a beautiful pattern. Order within chaos. Huh. I guess my tattoos have a bit of a theme.

Most people who get tattoos start small. But because I was in my forties (and tenured!) before I started, I went big and bold from the beginning. I love having spectacular tattoos. Of course, you run out of real estate pretty quickly, but that’s the price of doing business.

I showed a colleague in my field both the period doubling and the continued fraction tattoos. She was totally wowed by the continued fraction. I pointed out that the period doubling was in our area. She nodded and said that was fine, but the continued fraction was just amazing. It’s a narrow audience who can appreciate it, I guess, but it sure is an enthusiastic one.

## Blowing them away

It was four hours of needles, but what really hurt was holding my arm above my head while lying on my side for that long. Still, no pain, no gain… as my high school sports teacher once said, never imagining the circumstances under which I’d eventually embrace it.

However, as much as I love the equations, it’s the visual representations that are the most striking and get the most interest from non-mathematicians. So I decided I wanted one more. It needed to be a) visual, b) unique and c) impressive. No pressure then.

So I decided on a soap bubble from geometric measure theory. I saw this presented by Frank Morgan at my first-ever conference, and it blew me away, both for how striking it was and also for the fact that you could actually apply mathematics to real-world problems (in this case soap bubbles). That was amazing and, in many ways, set me on the path I’m on today, where I apply differential equations to infectious disease problems. So I got that on my left side, over my hip, waist and ribcage.

I went to a university about five hours’ drive away to give a talk and walked into the maths lounge in a T-shirt. Immediately, one professor (whom I’d never met) exclaimed “Hey! That’s a bifurcation diagram on your arm! Everybody, look at this.” So I had the room’s attention. I then lifted up my other sleeve, and he exclaimed “That’s a continued fraction!” He got the heart equation and the Mandelbrot Set. This guy was good. And when I showed him the soap bubble, he said “That looks like the work of Frank Morgan.” I said “That’s because it is the work of Frank Morgan. He came to Australia when I was a student, and it really inspired me.” Dumbfounded, he said “But Frank Morgan is a friend of mine. He lives an hour from here…” (I had no idea, having not had the courage to reach out to him.)

So there we go. I’m now a fairly heavily tattooed maths professor. My mother was a bit dumbfounded, saying that in her day the only people with tattoos were prisoners and sailors, but I just love the nerd tattoo. Goth Girl taught me about body modification, but I embraced it in my own nerdy way. And that is kind of cool, actually.

Robert Smith? (rsmith43@uottawa.ca) is a professor of biomathematics at the University of Ottawa. He once combined his day job with pop culture and in doing so accidentally invented the academic sub-discipline of mathematical modelling of zombies. He uses mathematical models to predict the spread of diseases, from HIV to malaria to Ebola. He’s also the foremost authority on the spread of Bieber Fever, but let’s not worry about that one. He has eleven books on academia and/or pop culture to his name, most recently Mathematical Modelling of Zombies and The Doctors Are In: The Essential and Unofficial Guide to TV’s Greatest Time Lord.
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