Step 1. Draw a circle with n evenly spaced points around the circumference, label them from 0 to n - 1. Then draw a line from each point to twice the point, looping around the circle as required i.e. modulo n.
e.g. n = 12
Perhaps n isn't large enough to see the pattern, lets see what happens as we scale it up.
We see a lovely curve "forming" - a cardioid. We'll need to be a little more specific about what we mean by "forming" later.
What happens if we multiply by another number instead of two?
Now we can see we get a nice series of epicycloids, with the number of cusps one less than the multiplication factor.
These aren't the only interesting patterns. For small k as we increase n, the pattern seems to converge smoothly to a limit.
Larger values of k seem to be less stable and many other patterns begin to emerge. For example here is k=17 with n animated.
While a 16-cusped epicycloid can be seen emerging near the rim of the circle, we can also see various other patterns emerging in the center of the circle.
Over a series of posts I hope to explore these patterns and develop some Plouffe Ray theory.