I saw this post on Facebook illustrating the so-called "Dance of Earth and Venus" and thought it was pretty cool. Venus completes (close to) 13 revolutions for every 8 earth revolutions; I guess we could say that Venus' speed relative to Earth is 13/8. As they revolve around the sun, the line segment betweens them envelopes a curve with pleasant five-fold symmetry.

I was curious about what other types of patterns are possible so I created this visualization that allows you to play with speed ratio. Fractions like p/q, are nice. the Earth goes around q times while the interior planet goes around p times. Other numbers, like 1.23 or π (entered as pi) or the Golden Ratio (entered as phi) are interesting as well, but it's hard to say how long the animation will run for.

### Questions for consideration

Here are a few questions about the relationship between the images we see and the parameters we input.

- If we enter a fraction p/q for the relative speed, how many revolutions does the blue planet make before the planets are aligned in their initial position?
- What happens if we enter an irrational number for the relative speed?
- What happens if we enter a negative number for the relative speed?
- How is the symmetry we see related to the relative speed that we enter? Does your description work for positive and negative numbers?
- How does the inner radius affect the image you see?