I noticed that the Mandelbort Set is shaped like a cardioid, I saw this answer that gives a technical explanation of why this happens, but I thought of something that could lead to a different (and more intuitive) explanation.
Epicycloids are also formed by modular times tables, and when you raise a complex number $z=(r,\theta)$ to an exponent $n$ the result is a complex number $z'=(r^n,n\theta)$. If $r=1$, eponents would behave exactly as modular times tables. So maybe that's why a Multibrot of degree $d$ has similar properties to the modular times tables of $d$?
However, in the case of Mandelbrot, the cardioid is not placed tangent to the unit circle, (where $r=1$) as I would have expected. So, is this just a coincidence? Or is my intuition on the right track to something?
My main goal is having an intuitive answer of why Multibrots look like epicycloids, an answer that does not rely on a technical proof.
