Mandelbrot set, Fourier analysis, zero line in the Riemann zeta function and a cup of tea. There is a funny geometric pattern I have seen several different places and I wonder if someone know if they are somehow related.

*

*The Mandelbrot set:




*In Fourier analysis, when "the winding frequency matches up with the frequency of the signal":



*The image of the zero line of the Riemann zeta function:



*I my tea cup when the light hits it slightly side ways:


The main island in the Mandelbrot set
For a complex number $c$, consider the map $f_c(z) := z^2+c$.  We want to find the set of $c$ so that $f_c$ has an attracting fixed point.  That is, we want a point $z_0$ so that $f_c(z_0) = z_0$ and $|f'(z_0)|<1$.
Solve $f_c(z) = z$ to get
$$
z_0 = \frac{1+\sqrt{1-4c}}{2}
$$
Now $f_c'(z)=2z$, so we want $|1+\sqrt{1-4c}| < 1$.  The boundary of this region is parameterized as
$$
1+\sqrt{1-4c} = e^{it},\qquad 0 \le t < 2\pi .
$$
or
$$
c = \frac{2e^{it}-e^{i2t}}{4},\qquad 0 \le t < 2\pi
$$
This is a cardioid,

