Some Three-dimensional Self-similar Knots

Robert W. Fathauer
Proceedings of Bridges 2010: Mathematics, Music, Art, Architecture, Culture (2010)
Pages 103–110


We present a selection of three-dimensional self-similar knots. The most successful method we have found for creating these is to use a geometric basis consisting of a starting surface in which one or more smaller copies of the surface are embedded. Points along the surfaces are connected by strands outside, between, and inside the surfaces to form a starting knot. Self similarity is introduced by an iterative process in which the contents of the smaller surface(s) are replaced with the contents of the larger surface. A distinction is drawn between thickened two-dimensional knots and true three-dimensional knots, with examples of each type. Self similarity allows for a combination of complexity and order, leading to novel and interesting sculptural forms.