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Rotation — 2012-25-03

Note: I changed this from a page into a post, and deleted the page it came from. This content is from 2010-01-17.

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}.

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The Roots of Polynomials — 2009-6-12

Inspired by the beauty of these pictures I made Mathematica notebook to compute the roots of all the polynomials with coefficients of ±1 (ie. the group Z2). They are colored by which degree they are: red is the highest degree computed (13) and the lowest degree computed (1) and the rainbow is spanned evenly. The roots are plotted in the complex plane in the usual way: the real part is plotted horizontally and the imaginary part vertically. The colors correspond to the different degrees of polynomials. For example, the roots of the largest-degree polynomials might be colored yellow but the roots of the next-largest-degree polynomials might be colored green. The coloring scheme is not the same across all of the pictures that follow, but the different degrees are always distinguished by color.

The set is quite beautiful:

All roots of all Z2 polynomials up to degree 13.

Noting that ±1 are the two square roots of 1, I generalized this idea to writing down all the polynomials with coefficients that are other roots of unity. For example, this image is the set of all roots of polynomials whose coefficients are cube roots of unity, which is the third cyclic group, C3:

Click for a high-res, with a different coloring.

You can see in this set the similarities to the first set, Z2 (or C2). There are the sharp parts, and the leafy parts. Whereas the first set was 180-degree rotationally symmetric, the second set is 120-degree rotationally symmetric. This is a direct result of the group property of the coefficients.

The sets have another interesting property: they have two other symmetries. First, there’s the other obvious one, which is symmetry across the real (horizontal) axis. But there’s another one: the sets are inverse symmetric. What I mean is that if r is in the set, then so is 1/r. This isn’t obvious from the beginning, but if you see the picture, then you realize it might be true, and then it is relatively straightforward to show.

Here are the roots of the polynomials whose coefficients are in C4:

All of these images are linked to a higher-resolution version.

Here are the roots of the polynomials whose coefficients are in C5:

I found something quite interesting in this one. In the high-res version, on the negative real axis just on the inside of the structure there’s a little star surrounded by a black circle. The circle, in turn, is surrounded by alternating patches of green and orange, with five of each color. There are analogous structures in the other sets, but they are harder to see in the lower sets. There is, of course, an analogous structure (by the inverse symmetry) on the outer band near the end of the negative real axis. These structures are shown here (at slightly different scales):

Another interesting feature can be seen in the less dense outer-star picture: the holes near the outer edge (in the lower picture, that is to the left) are pentagonal. This too is a generic feature, except for the higher-degree sets they become hexagons, heptagons, etc. until they start looking like circles. Of course, by inverse symmetry there are holes on the right in the upper picture that are pentagonal, except at this size the holes are the size of pixels, so you can just barely convince yourself to see them.

The full list of images I’ve generated so far is here. The file naming convention is this: The first label is the group, the second is the highest degree polynomial computed, and the third the size of the picture. For example, C4-9-8000 corresponds to the polynomials up to degree 9 whose coefficients are in the group C4, in an 8000-by-8000 pixel image. If an image name starts with a p, it was computed using a faster parallel algorithm that I’m finally satisfied with.

I’ve experimented with choosing coefficients that belong to other symmetry-inspired structures, such as the meson octet and baryon decuplet by using the hypercharge as the imaginary parts of the coefficients and the third component of isospin as the real parts. Those pictures oddly do not come out in a symmetric way, and can be seen through the list above.

The roots were found using Mathematica’s NSolve and stored to 10-decimal point precision. They were plotted also using Mathematica. If you are interested in seeing my Mathematica notebook, please email me. At this point the difficulty mainly lies in the rendering of images at a reasonable resolution to explore the structures. For instance, I have computed the C3 polynomials to degree 14 (with more on the way) but Mathematica seems unable to plot images larger than 12000 pixels square.

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