Why does the mandelbrot set and its different variants follow similar patterns to epi/hypo trochodis and circular multiplication tables?

Question

So the $z^2 + c$ variant has a cardioid shape at the center. This shape is made by an epitrochoid with a ratio of the radi being one, or from the two times table when we display it in a circle (as seen in this video https://www.youtube.com/watch?v=qhbuKbxJsk8). The next variation $z^3 +c$ has a nephroid as its central bulb, his shape is made by an epitrochoid with a ratio of the radi being two (or from the 3 times table when we display it as above). $z^4 + c$ follows the pattern, its central bulb is produced by an epitrochoid with a ratio of the radi being three(or the 4 times table). So to generalise the central bulb has a shape made by an epitrochoid with ratio n-1, where n is the exponent in $z^n +c$.

This also hold true for the mandelbar sets (when we flip the sign on its "imaginary" component). The first fractal in the set $\bar{z}^ 2 +c$ has a central bulb of the shape made by a hypotrochoids with ratio 3. The next in the sequence $\bar{z}^ 3 +c$ has a central bulb of the shape made by a hypotrochoid with ratio 4. For the mandelbar set the cental bulb is produced by an hypotrochoid of ratio n+1, where n is the exponent in $\bar{z}^ n +c$.

What causes these links? This site has some diagrams of the different fractals mentioned http://www.relativitybook.com/CoolStuff/erkfractals_powers.html. On this site it also talks about the rotational symmetry of each fractal and how it follows the same pattern. (for the mandelbrot sets the rotaional symmetry is n-1 that of exponent, and the mandelbar set the rotational symmetry is n+1 of the exponent).

To me it seems odd that the mandelbrot set follows the same rule as epitrochoids and the mandelbar sets (the inverse of the mandelbrot sets to some extent) follows the same rule as the inverse of the epitrochoids, the hypotrochoids.

Answer 1

The components of the Mandelbrot set correspond to sets of $c$ values such that the corresponding function $f_c(z)=z^2+c$ has some specific behavior. For example, $c$ is in the main cardioid if and only if the function $f_c$ has a attractive fixed point. The disk of radius $1/4$ attached to the left of the main cardioid is the period two bulb; a point $c$ is in this bulb if and only if the function $f_c$ has an attractive orbit of period 2. The set of period 3 parameters is a bit more complicated, since it is not connected. It consists of two disk like components near the top and bottom of the main cardioid and a small copy of the cardioid on the negative real axis. Again, a parameter value $c$ is in this period 3 region if and only if the corresponding function $f_c$ has an attractive orbit of period 3. All of these regions are outlined in this image:

Note that those outlines are generated using actual parametrizations that explain exactly how those shapes arise. Again, the parameter value $c$ is in the main cardioid if and only if the corresponding function $f_c$ has an attractive fixed point. Algebraically: \begin{align} f_c(z) &= z \\ |f_c'(z)| &< 1. \end{align}

The equation on top states that $z$ is a fixed point. The inequality on the bottom states that the fixed point is attractive. Now, on the boundary we should have: \begin{align} f_c(z) &= z \\ |f_c'(z)| &= 1. \end{align} That is, the fixed point is no longer strictly attractive on the boundary but just neutral. Since any complex number of absolute value 1 can be expressed in the form $e^{it}$, this pair of equations can be written without the absolute value: \begin{align} f_c(z) &= z \\ f_c'(z) &= e^{it}, \end{align} for some $t$. Taking into account the fact that $f_c(z) = z^2+c$, we get \begin{align} z^2+c &= z \\ 2z &= e^{it}. \end{align} This system of equations can be solved for $z$ and $c$ in terms of $t$. It's not even particularly hard. The second equation yields $z=e^{it}/2$. This can be plugged into the first equation to get $$c = e^{it}/2 - e^{2it}/4.$$ This happens to be the parametric representation of a cardioid and is exactly the formula used to generate the image above.

To obtain parametrizations of higher order bulbs, we can do similar things with (for example) \begin{align} f_c(f_c(z)) &= z \\ (f_c\circ f_c)'(z) &= e^{it} \end{align} or \begin{align} \left(c+z^2\right)^2+c &= z \\ 4 z\left(c+z^2\right) &= e^{i t}. \end{align}

This system can be solved for $c$ to obtain $c=-1+e^{it}/4$, which describes the period two disk attached to the left of the main cardioid.

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Similar things can be done for higher order bifurcation locuses. For $f_c(z)=z^3+c$, for example, we obtain for the main body: \begin{align} z^3 + c &= z \\ 3z^2 &= e^{i t}. \end{align}

This can be solved for $c$ to yield $$c = \frac{e^{{3 i t}/{2}}-3 e^{{i t}/{2}}}{3\sqrt{3}}.$$ I used that parametrization to generate the following image: