### Squircles

You can put a square peg in a round hole.

Shapes between circles and squares have proved invaluable to engineers and have also found their way onto our dinner tables. A plate in the shape of a `squircle’ is shown in this figure .

Squircular plate: holds more food and is easier to store.

The Equation of a Squircle

An ellipse with centre at the origin and major and minor semi-axes ${a}$ and ${b}$ is described by the equation

$\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \,.$

In particular, if ${a = b}$ we get a circle of radius ${a}$

$\displaystyle x^2 + y^2 = a^2 \,.$

The index 2 corresponds to the Euclidean or ${L_2}$ metric and ${a}$ is the distance of the point ${(x, y)}$ from the origin. We can control the shape by varying the value of the index (corresponding to the use of an ${L_p}$-norm). In the following figure we plot the graph of

$\displaystyle \left|\frac{x}{a}\right|^p + \left|\frac{y}{b}\right|^p = 1 \,. \ \ \ \ \ (1)$

Supercircles: p=0.5 (orange), p=1 (red), p=2 (thick black), p=4 (magenta), p=50 (blue).

for ${a=b=1}$ and ${p\in\{0.5,1,2,4,50\}}$. For ${p>2}$ the curve bulges outwards from the circle, as shown here, and we get a supercircle.

The particular case ${p=4}$,

$\displaystyle x^4 + y^4 = a^4 \ \ \ \ \ (2)$

is often called a squircle. It is also known as Lamé’s special quartic.

The equation for a squircle in polar coordinates is easily found. Substituting ${x=r\cos\theta}$ and ${y=r\sin\theta}$ into (2) we get

$\displaystyle r = \frac{a}{\sqrt[4]{1-\frac{1}{2}\sin^2 2\theta}} \approx \left(1+\frac{1}{8}\sin^2 2\theta \right) a \,.$

Clearly ${r\ge a}$ (${a}$ is called the minor radius) and ${r}$ takes its maximum value (or major radius) ${\sqrt[4]{2} a\approx 1.2a}$ for ${\theta\in\{\pm\frac{\pi}{4},\pm\frac{3\pi}{4}\}}$.

Taking a cue from the approximate form, we can construct curves with more oscillations by replacing the argument ${2\theta}$ of the sine function by ${2n\theta}$ for larger ${n}$. Curves with equation

$\displaystyle r = 1 + \frac{1}{8}\sin^2 2n\theta \ \ \ \ \ (3)$

for values ${n\in\{1,2,8\}}$ are shown (first quadrant only) in the figure below (left panel).

Left: graphs of (3) for n=1 (black) n=2 (blue) and n=8 (magenta). Right: Nest of squircles with varying radii.

It is a simple matter to scale the squircle. A nest of squircles is shown above (right panel).

Some Applications

The superellipse has some of the best aspects of both the ellipse and the rectangle. Just as the supercircle with index 4 is called the squircle, the superellipse with that index has been called the rectellipse.

Supercircles have been found to be useful in optics. The central spot in the diffusion pattern of a beam of light passing through a square aperture can be modelled by a squircle. Closer to home, squircular dinner plates hold more food for a given (minor) radius while they occupy the same space in a kitchen cupboard

One of the most notable applications of the superellipse was in the roundabout at Sergels torg in the heart of Stockholm. The challenge was to design the space to ensure efficient traffic flow within constraints imposed by the surrounding buildings. The architect was the renowned Danish innovator Piet Hein. An old image of the roundabout is shown in the figure below.

Sergels torg, Stockholm (Image © Piet Hein, 1959).