The Z2 Polynomials
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 set is quite beautiful:
All roots of all Z2 polynomials up to degree 13.
All roots of all Z2 polynomials up to degree 18.
Higher res, with an alternative coloring.
All roots of all C3 polynomials up to degree 9.
Here is the full list of generated images so far
The set exhibits two obvious symmetries: left-to-right and top-to-bottom. It has an additional symmetry: inside-to-outside. That is, the picture is roughly an annulus around |z|=1. If a root is r then so is 1/r, so that the inner edge and outer edge get swapped, and the spacey texture on the outside gets traded for the dense texture inside.
Comments
2 Responses to “The Z2 Polynomials”
APOD
My browser (Firefox 3) says
The image “http://eberkowitz.com/images/polyRoots/Z2-18-12000.png” cannot be displayed, because it contains errors.
Except for that one, all very nice; thank you!
I’m not sure why—it renders properly here. It is quite a large file… perhaps try downloading it instead of rendering it in firefox?