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:

$$r = \frac{\cos(\varphi)-1}{2}.$$

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 bind 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.