In case you weren’t aware, Chalkdust make some really incredible-looking T-shirts. You can buy one here! However, the design tends to puzzle people. I haven’t yet met anyone that is good enough at curve sketching in their head to work out what the equation on it means.

The $\sqrt{-1}$ does give you a hint. Unless you’re an electrical engineer, you could read it as $\mathrm{i}$ and you might guess where the rest of it is going.

But understanding what the equation means is a nice curve sketching problem.

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

Using Wolfram Alpha is cheating! Curve sketching is a skill vastly underrated by almost everyone doing maths exams – but I think it’s hugely satisfying.

Since $x$ only appears as a function of $x^2$, the $y$ values will be the same for $x$ and $-x$ and thus the curve will be symmetric in the $y$ axis. There’s a $y^2$ term as well, so we should expect two $y$ values for each $x$ value.

It also looks a lot like the equation for a circle – but what is that $\sqrt[3]{x^2}$ doing? If we rewrite the equation as a function of $y$ we get more of an idea:

\[ y = \sqrt[3]{x^2} \pm \sqrt{1-x^2} .\]

The second term is just the equation of the circle, with the positive giving the upper half and the negative the lower half. So the first term is modifying both the top and bottom half of the circle. Since $\sqrt[3]{x^2}$ is increasing as you move away from the origin, it “pulls up” the curve by more as you move away from the centre.

Therefore, the top half on the circle now has two maxima, and the slope of the bottom half is much steeper. What have we drawn? A lovely heart!

However, our circle modifying expression $\sqrt[3]{x^2}$ isn’t unique. We need it to be even with a minimum at $(0,0)$, and to get the cusp in the middle, we need it to be not smooth at the origin. So quite a few functions of $\mid x \mid$ could work.

Paul Ma has looked at lots of different versions of equations for hearts. (However, I can’t be held responsible for anything else you might learn to plot from this article.)

Computers are now enormously helpful for understanding these kinds of problems – in three dimensions as well. But in the late 19th century, this was much more difficult – and serious geometers spent a lot of time thinking about strange looking surfaces. Without computers, they built intricate paper models, like this one by Olaus Henrici, which shows the surface given by \[ xyz = k^3(x+y+z-1)^3 .\]

Now that you also understand the true meaning of the Chalkdust T-shirt, you can join mathematical fashion pioneers Ian Stewart, David Colquhoun and Bearnoulli’s Principaw in getting your own!