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.
"