Squircles and Sphubes

I came across a reference to a shape called a squircle recently, a portmanteau word formed from square and circle. I thought I'd investigate its mathematical properties. The squircle is a special case of the superellipse defined as:$$|\frac{x-a}{r_a}{|}^n+|\frac{y-b}{r_b}{|}^n=1$$where $r_a$ and $r_b$ are the semi-major and semi-minor axes and $a$ and $b$ are the $x$ and $y$ coordinates of the centre of the ellipse. The squircle is then defined as the superellipse with $r_a=r_b=r$ and $n=4$. So we have:$$(x-a{)}^4+(y-b{)}^4=r^2$$where $r$ is the minor radius of the squircle. In its simplest form, when $a=b=0$ and $r=1$, we have:$$x^4+y^4=1$$Figure 1 shows this:

Figure 1: created using GeoGebra

The area inside the circle can be expressed in terms of the gamma function $\mathrm{\Gamma }(x)$ as follows:$$\text{Area }=4r^2\frac{(\mathrm{\Gamma }(1+\frac{1}{4}){)}^2}{\mathrm{\Gamma }(1+\frac{2}{4})}=8r^2\frac{(\mathrm{\Gamma }(\frac{5}{4}){)}^2}{\sqrt{\pi }}=S\sqrt{2}{r}^2\approx 3.708r^2$$where $r$ is the minor radius of the squircle and $S$ is the lemniscate constant. I've no idea as to how that equation comes about and just summarising what's in Wikipedia.

Another squircle is of the form:$$x^2+y^2-\frac{s^2}{r^2}{x}^2{y}^2=r^2$$ where $r$ is the minor radius of the circle and $s$ is the squareness parameter. If $s=0$, we have a circle and if $s=1$, we have a square. Figure 2 shows the case where $r=1$ and $s=0.9$. This form of the squircle can be referred to as the FG-squircle after Fernández–Guasti, after one of its authors.

Figure 2: GeoGebra link

Squircles have also been used to construct dinner plates. A squircular plate has a larger area (and can thus hold more food) than a circular one with the same radius, but still occupies the same amount of space in a rectangular or square cupboard. See Figure 3:

Figure 3: source

There is a site that let's you create your own squircles. See Figure 4.

Figure 4: https://squircley.app/

The three dimensional equivalent of a squircle could be called a sphube as this article suggests: Three Dimensional Counterpart of the FG-squircle. The equation of such a 3-D shape is given by:$$x^2+y^2+z^2-\frac{s^2}{r^2}{x}^2{y}^2--\frac{s^2}{r^2}{y}^2{z}^2-\frac{s^2}{r^2}{x}^2{z}^2+\frac{s^4}{r^4}{x}^2{y}^2{z}^2=r^2$$Just like the FG-squircle, this shape has two parameters: squareness $s$ and radius $r$. When $s$ = 0, the shape is a sphere with radius $r$. When $s=1$, the shape is a cube with side length $2r$. In between, it is a three dimensional shape that resembles the sphere and the cube. The shape is shown in Figure 5 at varying values of squareness.

Figure 5: source

The article mentioned previously,  Three Dimensional Counterpart of the FG-squircle, treats other 3-D shapes among them being the: 

• squircular ellipsoid (sqellipsoid)
• squircular cylinder (sqylinder)
• squircular cone (sqone)
• sphylinder