How to prove the shape of the "Flower of Venus"

Question

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?

Answer 1

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.

Answer 2

Here's the insane approach ... Let's find the equation of the Flower curve!

The curve is the envelope of line segments determined by the two planet-points, which we can parameterize as $$P = p \,( \cos 8 t, \sin 8 t ) \qquad Q = q\, ( \cos 13 t, \sin 13 t )$$ where $p$ and $q$ are the radii of the orbits. The line $PQ$ is then $$f(t) := x \left( p \sin 8 t - q \sin 13 t \right) + y ( - p \cos 8 t + q \cos 13 t ) + p q \sin 5 t \tag{1}$$ so that $$f^\prime(t) := x \left( 8 p \cos 8 t - 13 q \cos 13 t \right) + y \left( 8 p \sin 8 t - 13 q \sin 13 t \right) + 5 p q \cos 5 t \tag{2}$$ To derive the equation of the envelope, "all we have to do" is eliminate $t$ from $f(t)$ and $f^\prime(t)$. With lots of classical envelopes (cardioids, astroids, parabolas, etc), the elimination step is fairly straightforward. Here, with $t$ trapped inside the various sines and cosines, things are a bit complicated. It helps a bit to rewrite the equations in terms of the complex exponential, $\omega := \exp(it)$; it'll help further to define $z := x + i y$ and $\bar{z} := x - i y$. So, we make the substitutions $$\cos kt = \frac12(\omega^k+\omega^{-k}) \qquad \sin kt = \frac1{2i}(\omega^k-\omega^{-k}) \qquad x = \frac12(z+\bar{z}) \qquad y = \frac1{2i}(z-\bar{z})$$

Then $(1)$ and $(2)$ become, after clearing denominators, $$\omega^{26} q \bar{z} - \omega^{21} p \bar{z} - \omega^{18} p q + \omega^8 p q + \omega^5 p z - q z \tag{3}$$ $$13 \omega^{26} q \bar{z} - 8 \omega^{21} p \bar{z} - 5 \omega^{18} p q - 5 \omega^8 p q - 8 \omega^5 p z + 13 q z \tag{4}$$ and now "all we have to do" is eliminate $\omega$. Again, this is conceptually straightforward: we merely invoke the method of resultants, which Mathematica graciously implements as its Resultant[] function. However, the computational complexity of resultant-finding is related to the sum of the degrees of the polynomials ---here 52--- which causes Resultant[] to bog down on my computer. Happily, I was able to get a faster implementation by having Mathematica construct the Sylvester Matrix for the polynomials, and take its determinant (which is equivalent to the resultant). The result(ant) is :

$$\begin{align} &2814749767106560000000000 p^{36} q^{42} \\ + &\cdots \;\text{($760$ terms)}\; \cdots \\ + &18258084432456195379316013815625 p^{26} q^{16} z^{13} \bar{z}^{23} \end{align}\tag{5}$$

Now, instead of restoring $z$ and $\bar{z}$ back to $x$ and $y$ form, I can use to the more-appropriate complex polar form: $$z = r \exp(i\theta) \qquad \bar{z} = r\exp(-i\theta)$$ to get $$\begin{align} &15879378388503914086400000 \;\left(e^{15i\theta}+e^{-15i\theta} \right)\; p^{39} q^{24} r^{15} \\ +&\phantom{1}3 \left(e^{10i\theta}+e^{-10i\theta} \right)\;p^{26} q^{16} r^{10} (3289829448089600000000000 p^{12} q^{14} + \cdots) \\ -&12 \left(e^{5i\theta}+e^{-5i\theta} \right)\; p^{13} q^8 r^5 (1179648000000000000000 p^{26} q^{26} + \cdots) \\ -&\phantom{1}2\;(1407374883553280000000000 p^{36} q^{42} + \cdots) \\ \end{align} \tag{6}$$ Conveniently, the complex exponentials pair-up to form cosines (with a factor of $2$), so that the polar equation of the Flower of Venus has the form

$$\begin{align} 0 \;=\;\; &31758756777007828172800000 \;\cos 15\theta\; p^{39} q^{24} r^{15} \\ +&\phantom{1}6 \cos 10\theta \;p^{26} q^{16} r^{10} (3289829448089600000000000 p^{12} q^{14} + \cdots) \\ -&24 \cos 5\theta \; p^{13} q^8 r^5 (1179648000000000000000 p^{26} q^{26} + \cdots) \\ -&\phantom{1}2\;(1407374883553280000000000 p^{36} q^{42} + \cdots) \\ \end{align}\tag{$\star$}$$

Since the arguments of the cosines are all multiples of $5\theta$, the curve must have $5$-fold rotational symmetry. Easy-peasy! $\square$

Of course, this ad hoc approach doesn't provide any insights at all into why the $5$-fold symmetry appears, what happens with other "ratio differences", etc. I really just wanted to see for myself how crazy the Flower curve's equation is. I suspect that there must be a way to anticipate the symmetry from $(1)$ and $(2)$ (or $(3)$ and $(4)$) without having to actually calculate the resultant; if so, then (presumably) that strategy should apply to arbitrary ratio differences.