Jeffrey S. Ely

“Newton Without Newton”

Large Format Print on Photographic Paper, 24" x 20", 2007.

This 3-dimensional image depicts the modulus of the Newton process when applied four times to the 5-th degree polynomial, f(s) = (s-q)5 - 1, where q = 0.7 + 0.3i and s is the complex variable, x + iy.

More concretely, this 3-dimensional (x,y,z) surface is

z = g(x,y) = |N(N(N(N(s))))|

where N(s) = s - f(s)/f'(s) is the Newton operator and || denotes the absolute value (modulus) of a complex number.

The surface has been cropped by discarding any point whose z-value is greater than 2, allowing us to peer inside some of the poles of the surface, especially the large center one. The surface is grey except for those points whose z-values closely match the moduli of one of the five roots, hence the 5 different colors.

Jeffrey S. Ely, Associate Professor of Computer Science, Mathematical Sciences Department, Lewis and Clark College, Portland, OR 97219

"I am interested in art that illuminates mathematical ideas and in mathematical ideas that make the art possible. This particular piece depicts Newton's method as a 3-dimensional surface. Ironically, my attempt to ray-trace the surface with a solver based on Newton's method failed to reliably converge, so I abandoned ray-tracing altogether in favor of constructing the surface as a large particle system. hence the title, "Newton Without Newton". I wrote the program to first principles in the "C" programming language. The computer calculated 50 billion points over three and a half days to produce this 7200 by 6000 pixel image. "