Is there an analytical formula for determining the location of the attachment points of the bulbs on the main cardioid? I was told there is an exact parametrization of the boundary of the main cardioid involving exp(2 pi i theta) but I could not find the formula.
I'm particularly interested in finding the exact location of the attachment points of the bulbs 3 (the largest bulbs on the side of the main cardioid).
To elaborate a bit more on my comments:
Given a hyperbolic component $U\subset M$ of the Mandelbrot set -- i.e. a connected component of the interior of $M$ such that for all $c\in U$ the polynomial $p_c(z)=z^2+c$ has an attracting cycle -- the length of attracting cycle of $p_c(z)$ is constant for all $c\in U$. Now if $U$ is an hyperbolic component of period $n$ one can show that the multiplier map $\mu:U\rightarrow \mathbb{D}$ given by $c\mapsto (p^n)'(Z(c))$ where $Z( c)$ is a point in the attracting $n$-cycle of $p_c(z)$ is an isomorphism. In fact this map can be extend continuously to an injective map over the boundary of $U$.
We then define the root of a hyperbolic component $U$ to be $\mu^{-1}(1)$. Note that this is necessarily a point in the boundary of $U$, and that if $\mu^{-1}(1)=c_1$ then the polynomial $p_{c_1}(z)=z^2+c_1$ has a parabolic cycle (of length possibly less then n) of multiplier one.
There are two different shapes of hyperbolic components primitive components - ones that have a cusp likes the main cardioid - and satellite components ones that don't have a cusp like the basilica component. Similarly, we call the root of primitive component a primitive root and likewise for the root of a satellite component. These satellite roots are the points of attachment, which are what you appear to be interested in.
For example, if you are interested in the basilica component -- the hyperbolic component of period two (i.e. the large one to the left of the main cardioid) -- then the root is $-3/4$. If solve for the two cycles of $f(z)=z^2-3/4$ you'll notice that the solutions are degenerate; in that they are fixed points with the solutions being $z=-1/2$ and $z=3/2$. Calculating $(f^2)'(z)$ of each of these points we see that $(f^2)'(-1/2)=1$ and so as claimed $-3/4$ is ththe root of this component. If you are to look at a picture of the Mandelbrot set you'll see that this appears to be where the main cardioid meets the basilica component. Note this is an example where the root parameter has smaller cycle than its hyperbolic component.
Finally, this can be used to find the equation for the boundary of the main cardioid. The main cardioid is the hyperbolic component $U\subset M$ such that for all $c\in U$ the polynomial $p_c(z)=z^2+c$ has an attracting fixed point. Now the boundary of $U$ is $\mu^{-1}(\partial\mathbb{D})$. Using this one can then solve for the equation giving the boundary of $U$, which is $.5e^{i\theta}-(.5e^{i\theta})^2$.
I think this might be discussed in Complex Dynamics by Carleson and Gamelin, but I do not currently have a copy to double check.
in the parameterization where the main cardioid is a circle, the bulbs are attached at rational angles $\phi=2\pi m/n$: see R.L. Devaney, The Mandelbrot bulbs.
Actually the answer to my question seem to have been there also on Wikipedia already under the Mandelbrot Set! in the section "Main cardioid and period bulbs".
Also, R.L. Devaney's kind response to my question confirms this:
"Here is the parameterization. On the boundary of the main cardioid, the map $z^2 + c$ has a neutral fixed point, i.e., a fixed point for which the derivative is $e^{2 \pi i \theta}$. So the boundary is given by solving the two equations:
$$z^2 + c = z $$ (that give a fixed point) $$2z = e^{2 \pi i \theta}$$
Solving these equations yields:
$$c = z - z^2 = \frac{1}{2}e^{2 \pi i \theta} - \frac{1}{4}e^{4 \pi i \theta}$$
That is the parameterization of the boundary of the main cardioid. When $\theta$ is rational, you are at a point where a bulb just meets this boundary. "
One can find the exact location of the attachment points of the bulbs 3 ( and other ) with Newton method near centers of components
Interestingly, the solution referenced by DolphinDream (Devaney's) traces the cardioid by rotating a circle of 1/4 radius about a stationary circle of the same size centered on the origin and one begins with the point of the cusp being where the trace being where the rotating circle touches the stationary circle at the cusp. The angle at which the rotating circle touches the stationary circle is the function of $m$ and $n$:
$$\varphi = 2\pi\frac{m}{n}.$$
Another approach to calculating the positions of the bond points of the buds is to start with the cardioid:
$$r = \frac{1-\cos(\varphi)}{2}.$$
This is the Mandelbrot Set's main cardioid, albeit shifted so that the cusp appears at the origin. At this point the angle $\varphi$ for the bond point of bud $m/n$ can be measured from the cusp and is equal to:
$$\varphi = 2\pi\frac{m}{n}.$$
Given $r$ and $\varphi$, one can calculate $x$ and $y$, then shift $x$ so that the central, stationary circle mentioned above is centered at the origin and solve for $x$, $y$:
\begin{align*} x &= r\cos(\varphi) + 0.25\\ y &= r\sin(\varphi). \end{align*}
I believe you will find the solution equivalent to that of Devaney as referenced by DolphinDream, but whereas Devaney derived his solution from mathematical principles the solution I have given is merely empirical: it first suggested itself when prior to seeing Devaney's solution I noticed the bond point of bud 1/4 was "precisely" above the cusp, at 1/4 a full turn. As such I prefer Devaney's solution cited by DolphinDream, but the solution I have given may be easier to implement and I find some pleasure in their giving the same results
