Squircles and Sparkles

You know what's easy to find in app icons these days? Squircles and "AI sparkle" stars.

How do you get the squircle? Well, we know the formula for the unit circle is

$$x^2 + y^2 = 1.$$

But we can generalize to any positive real power of $d$, with a radius of $r$:

$$|x|^d + |y|^d = r^d$$

Here's what you'll get when you let $d=7$:

As $d$ goes higher, the shape becomes a closer approximation of a rectangle. (Interestingly, if you let $d=1$ you get an actual rectangle, but tilted 45 degrees like a diamond.)

It's when $d$ is fractional that you get the sparkle. Here's what you'll get when you let $d=0.4$:

With the help of Claude Code, a made an Icon Builder Web App. It can export SVGs, a favicon.ico, and an iOS-ready PNG for the homescreen.

You can find the code on GitHub.