If you want to go deeper, this is a subject with an immense number of connections. But to find them you'll probably have to know what people call them.

You can read more about the curves of Lamé plotted in this article at https://en.wikipedia.org/wiki/Superellipse. If you're in Sweden, the layout of https://en.wikipedia.org/wiki/Sergels_torg is a superellipse design by Piet Hein. Martin Gardner wrote a delightful column about this in the September 01965 Scientific American: https://www.scientificamerican.com/article/mathematical-game... "The superellipse: a curve that lies between the ellipse and the rectangle" which I don't have a copy of, except the slightly corrupted copy at https://piethein.com/superellipse/. It begins lyrically:

> Civilized man is surrounded on all sides, indoors and out, by a subtle, seldom-noticed conflict between two ancient ways of shaping things: the orthogonal and the round. Cars on circular wheels, guided by hands on circular steering wheels, move along streets that intersect like the lines of a rectangular lattice. Buildings and houses are made up mostly of right angles, relieved occasionally by circular domes and windows. At rectangular or circular tables, with rectangular napkins on our laps, we eat from circular plates and drink from glasses with circular cross sections. We light cylindrical cigarettes with matches torn from rectangular packs, and we pay the rectangular bill with rectangular bank notes and circular coins.

This column is included in one of Martin Gardner's books, which is where I read it in my childhood.

Superquadrics are a generalization of the three-dimensional case (see https://en.wikipedia.org/wiki/Superquadrics); Ed Mackey's 01987 "Superquadrics" screensaver is included in xscreensaver, which you can easily install if you're running Debian or Android with F-Droid: https://f-droid.org/en/packages/org.jwz.xscreensaver/

Viewed as level sets of vector norms (https://en.wikipedia.org/wiki/Norm_(mathematics)) these curves are called "balls": https://en.wikipedia.org/wiki/Ball_(mathematics)#In_normed_v.... Vector norms are fundamental to approximation theory, and because people often do math on measurements from the real world [citation needed] which are always imprecise [citation needed], approximation theory is pretty widely applicable. It's often convenient to use one of the alternative norms mentioned in Michał's article for your proofs.