eberkowitz.com http://eberkowitz.com Personal Site of Evan Berkowitz Fri, 11 Dec 2009 02:39:03 +0000 http://wordpress.org/?v=2.9.1 en hourly 1 The Z2 Polynomials http://eberkowitz.com/2009/12/06/11/ http://eberkowitz.com/2009/12/06/11/#comments Sun, 06 Dec 2009 20:31:28 +0000 Evan http://eberkowitz.com/?p=11 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:

The roots of all Z2 polynomials up to degree 13
All roots of all Z2 polynomials up to degree 13.


The roots of all Z2 polynomials up to degree 18
All roots of all Z2 polynomials up to degree 18.
Higher res, with an alternative coloring.


The roots of all C3 polynomials up to degree 9
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.

]]> http://eberkowitz.com/2009/12/06/11/feed/ 2 A fresh look http://eberkowitz.com/2009/05/27/a-fresh-look/ http://eberkowitz.com/2009/05/27/a-fresh-look/#comments Wed, 27 May 2009 03:19:41 +0000 Evan http://eberkowitz.com/?p=3 Well, since I own a domain I may as well use it.

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