Mitered Fractal Trees: Constructions and Properties

Tom Verhoeff and Koos Verhoeff
Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture (2012)
Pages 25–32 Regular papers


Tree-like structures, that is, branching structures without cycles, are attractive for artful expression. Especially interesting are fractal trees, where each subtree is a scaled and possibly otherwise transformed version of the entire tree. Such trees can be rendered in 3D by using beams with a polygonal cross section for the trunk and the branches. The challenge is to connect the beams at the branching points in such a way that the beam edges nicely meet. This is related to the miter joint, but does not necessarily involve ternary miter joints.

In this article, we explore a parameterized family of fractal trees that can be rendered with polygonal beams whose edges meet properly at the branching points. We present various constructions and analyze their mathematical properties. Some of these trees have been constructed as artwork in wood and bronze.