Frank Farris begins his book “Creating Symmetry” with symmetric curves of N-fold rotational symmetry. An example:
He uses that we can interpret points (x,y) of the plane as complex numbers z=x+i*y. Thus a complex function f(t) of a real parameter t defines a curve in the complex plane. It becomes a closed curve if f(t+2*π)=f(t).
To get N-fold rotational symmetry we can rewrite Farris recipe as
f(t)=exp(i*m*t) * g(t),
where m is an integer number and the complex function g has a “faster” periodicity
Then, we can write the function g(t) as a complex Fourier series
where the coefficients a,b,c,d, … are complex numbers. If all numbers are real, then you get a mirror symmetric curve. For more details, you should look at Farris book.
I would be happy to see your favorite result. You might publish your code for “function updateZ(t)” as a comment to this post. Then, everyone could recreate it.