Strange fractal patterns emerge when you plot the complex roots of high order polynomials. This picture shows all the roots for all possible combinations of 18th order polynomials with coefficients of ±1. You can easily find the roots using Mathematica’s Root function:
(* runtime: 34 seconds *)
n = 12; m = 275; image = Table[0.0, {m}, {m}];
Do[Do[z = N[Root[Sum[(2Mod[Floor[(t - 1)/2^i], 2] - 1) #^(n - i), {i, 0, n}], root]];
{j,i} = Round[m({Re[z], Im[z]}/1.5 + 1)/2];
If[0 < i <= m && 0 < j <= m, image[[i, j]]++], {root, 1, n}], {t, 1, 2^n}];
ListDensityPlot[image, Mesh -> False, Frame -> False, PlotRange -> {0, 4}]
Links
- Constrained Coefficients – beautiful plots by Loki Jörgenson
- description – by John Baez
- Integer Coefficients – slightly different plots by Dan Christensen
- Root Finding Algorithms – Wikipedia
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Another very pretty rendering from you. Very interesting how the middle ring dissolves into more organic looking frills at thr border.
Fantastic!
Well I tried making that same image with some matlab code. So far i’ve only made ‘bottle cap’ plots instead of this fractal (i’m only doing a subset of this fractal at the moment).
I plan on asking my abstract algebra teacher about symmetries of polynomial roots of ℂ. That might result in saved time and computation. I plan on doing the final image in the summer. Hopefully I get similar results.
Very neat way to look at art. Most people tend to focus on their perceptions rather than the mathematics. Beautiful.
Very nice graphic and great blog. I’m suddenly interested in graphs of complex roots for lower order equations to build a base. Would love to know if you have some samples. Or I guess I’ll have to plot them myself. 🙂