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: $$\bigg |\frac{x-a}{r_a}\bigg |^n+\bigg |\frac{y-b}{r_b}\bigg |^n=1$$ 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 $\Gamma(x)$ as follows: $$\text{Area }=4r^2 \frac{\big ( \Gamma(1+\frac{1}{4}) \big )^2}{\Gamma(1+\frac{2}{4})}=8r^2 \frac{ \big ( \Gamma(\frac{5}{4}) \big )^2}{\sqrt{ \pi}}=S \sqrt{2} r^2 \approx 3.708 r^2$$ 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: $$x^2+y^2-\frac{s^2}{r^2}x^2y^2=r^2$$ 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: $$x^2+y^2+z^2-\frac{s^2}{r^2}x^2y^2--\frac{s^2}{r^2}y^2z^2-\frac{s^2}{r^2}x^2z^2+\frac{s^4}{r^4}x^2y^2z^2=r^2$$ 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