I recently was doing some complex number work and found this guy:
Is there a formula for a graph like this in two - dimensions? I know that the values are the same on every curve, but they are rotated (in polar coordinates) by $ 2 \pi / 3 $. A similar graph with the case $ n = 2 $ is a perfect hyperbola and I have a formula, but this would be the case where $ n = 3 $.
Note that the center of the formulas are the roots of the cyclotomic polynomials (This is a graph in the complex plane).
The natural way to get this using complex numbers is to look at $z^3$ where $z = x + i y.$ Then take a level set of, say, the real part, so set $$ x^3 - 3 x y^2 = 1 $$ This is not quite the same as the comments, this is $$ r^3 = \sec 3 \theta $$