Holiday Special: Beauty of Polynomials Highlighted on Scientific American Website

January 5, 2010

In case you missed it during the holidays, Scientific American's display on the beauty of mathematics is still on view.

Posted by John Baez (University of California, Riverside), the website offers a collection of seven color images of polynomial roots by Dan Christensen (University of Western Ontario) and Sam Derbyshire (University of Warwick).

Christensen and Derbyshire plotted the roots of families of single-variable polynomials, imposing constraints on the polynomials' degrees and coefficients. The horizontal axis in their plots is the real numbers; the vertical axis is the imaginary numbers. A real root (such as –1) falls on the horizontal axis; an imaginary root (such as 2i) falls on the vertical axis. The rest of the imaginary numbers—those with both real and imaginary components—fills out the quadrants of any graph.

Plotted en masse, intricate and intriguing patterns emerge. In one example, Christensen plotted the roots of every polynomial whose degree is six or less and whose coefficients are integers between –4 and 4.

Thomas Leathrum (Jacksonville State University) described the slide show as a "different take on the fractal you get from looking at Newton's Method convergence to roots of unity in the complex plane, but it's interesting nonetheless."

Source: Scientific American (December 28, 2009)

Slideshow available here.