How to prove the shape of the "Flower of Venus" This is the gif that inspired this question. So, considering that Venus and Earth's orbits are coplanar concentric circles and that their orbits follow a 13:8 (difference of five) ratio, how can I prove this five-fold symmetry?
I've been playing around in GeoGebra and created symmetries with other numbers, for example, a 'ratio difference' of 9 creates a nine-fold symmetry as seen above.
What I want is to, using relatively simple math tools (I'm a high school student), analytically prove how and why this symmetry corresponds to the difference in the ratio. 
One of my ideas was to reach the equation of the line between the two planets and prove that it passes five (or whatever difference) times through the centre of the orbits, given that each time the trace passes over the centre, it creates one of the "loops" of the "cardioid". Another idea would be using complex numbers, but I'm not sure how to do that. 
The main thing I have no idea how to do analytically is how to consider the "speed" of the orbits and their difference. Should there be some form of Calculus going on?
Edit: I originally posted a question of the same topic in Astronomy Stack Exchange, but I was looking for a more technical answer.
Any ideas? 
 A: Suppose Earth and Venus are at their minimum distance at $t=0$. They will be back to the same position at $t=8$ years, the Earth having made $8$ revolutions around the Sun and Venus $13$. 
But they will reach a new position of minimum distance earlier than that. If $T_E=1$ and $T_V=8/13$ are their revolution periods (in years), then at time $t$ they have travelled a fraction $t/T_E$ and $t/T_V$ of a complete revolution, so they are at their minimum distance again if Venus has made one complete revolution more than Earth, that is if
$$
{t\over T_V}={t\over T_E}+1,
\quad\text{whence:}\quad
t={T_ET_V\over T_E-T_V}={8\over5}.
$$
It follows that, during the 8-year Venus-Earth cycle, the two planets reach their minimum distance 5 times: that explains the 5-fold symmetry of your plot.
EDIT.
Here's an animation showing how after $8/5\cdot 360°=576°$ Earth is again at minimum distance from Venus. During the same time Venus has travelled an angle 
$13/8\cdot576=936°=576°+360°$, thus making one full revolution more than Earth.

