Creating and Modifying Images Using Newton's Method for Solving Equations

Stanley Spencer
Proceedings of Bridges 2010: Mathematics, Music, Art, Architecture, Culture (2010)
Pages 183–190 Regular Papers


The pictures associated with Newton's Method of finding the roots of an equation are familiar to mathematicians and non-mathematicians. The main purpose of this paper is to look at ways by which a recognisable image can be produced or modified using the same mathematical process. This paper looks at several ways that an arbitrary pixel image can be iteratively transformed using Newton type algorithms. The standard approach of letting the pixels evolve towards attractors was interesting, but a second approach, one that inverted the process so that attractors became expellers, looks fruitful for creating interesting images. The process was designed to allow equations with up to a hundred roots which created challenges but also opportunities for meaningful outcomes.