Rotation

I was searching for some hopefully-pretty generalizations of the sets that yield these pictures. For a brief overview, the story goes something like this:

  • Pick a set of complex numbers.
  • Write down all polynomials up to a certain degree whose coefficients come from that set.
  • Find all the roots of all of those polynomials.
  • Plot them in the complex plane

The set picked for the images linked above lay on the unit circle and are evenly spaced apart.

Just messing around, I asked myself, “What if we had two circles instead of one?” And so I picked my set of complex numbers to be two rings evenly spaced, one with a radius equal to the Golden Ratio (phi=1.61803…) and one with a radius 1/phi. I first generalized the C4 group. That is, I let my set be equal to {phi, i phi, -phi, -i phi, 1/phi, i/phi, -1/phi, -i/phi}. The first four are on the outer ring, the second four on the inner ring. Going to degree 6, one winds up with the following image:
Two four-fold symmetric rings
which really deserves to be examined in more detail. This should be compared to the image of C4. Don’t take the comparison too seriously–the set shown above is plotted from -4 to 4, whereas the linked image of just C4 is plotted from -2 to 2, and with a bigger pixel size for each root.

The next simple generalization occurred to me: rotate one ring with respect to the other. Arbitrarily, I fixed the position of the outer ring, and let the inner ring rotate (that is, the coefficients on the inner ring were all picked up a phase e^{i theta}, where theta is the angle of the rotation). Then, you get a different set of roots (and thus a different picture) for each value of theta.

How best to display them? Make a movie! In the movies below, the angle of rotation grows linearly as a function of time. You can watch the sets as they evolve with theta. When there are suddenly more points plotted, that corresponds to adding an additional degree of polynomials. For example, the first video below starts out with polynomials of degree 1 and adds degrees through degree 6.

The coefficients in this video are phi e^{2 pi i n /2} and (1/phi)e^{(2 pi i n /2) + i theta}.

The coefficients in this video are phi e^{2 pi i n /3} and (1/phi)e^{(2 pi i n /3) + i theta}.

The coefficients in this video are phi e^{2 pi i n /4} and (1/phi)e^{(2 pi i n /4) + i theta}.

The coefficients in this video are phi e^{2 pi i n /5} and (1/phi)e^{(2 pi i n /5) + i theta}.