Fractal Knots Created by Iterative Substitution

Robert W. Fathauer
Bridges Donostia: Mathematics, Music, Art, Architecture, Culture (2007)
Pages 335–342


A widely-applicable method for iterating knots is described. This method relies on substitution of portions of a knot with smaller copies of the entire knot. A starting knot is first arranged as a patch of tiles that contains individual tiles similar in shape to the overall patch. Iterative substitution leads to the creation of complex knots that are often esthetically pleasing, particularly for knots possessing a high degree of symmetry. The iteration process is designed to allow repetition ad infinitum; i.e., an infinite number of iterations leads to a unicursal fractal that is, therefore, a (wild) knot.