Gareth Andrew


Plouffe Rays - Part 1

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.

Other Interesting Patterns

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.

  • Which patterns are "stable" - i.e. can we define some series of patterns which converges to a limit
  • What values of k and n give rise to each type of patttern.
  • Within each type of pattern can be identify extra properties such as radii, which values of x
  • What about rational k, non-rational k?
  • What happens if we use a function other than f(x) = kx mod n.