Unfolding Symmetric Fractal Trees

Bernat Espigulé Pons
Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture (2013)
Pages 295–302 Regular Papers


This work shows how the angles and ratios of side to diagonal in the regular polygons generate interesting nested motifs by branching a canonical trunk recursively. The resulting fractal trees add new material to the theory of proportions, and may prove useful to other fields such as tessellations, knots and graphs. I call these families of symmetric fractal trees harmonic fractal trees because their limiting elements, i.e., when the polygon is a circle, have the overtones or harmonics of a vibrating string 1/2, 1/3, 1/4, ... as their scaling branch ratios. The term harmonic is also used here to distinguish them from other types of self-contacting symmetric fractal trees that don't have a constantly connected tip set under a three-dimensional unfolding process. Binary harmonic trees represent well-known Lévy and Koch curves, while higher-order harmonic trees provide new families of generalized fractal curves. The maps of the harmonic fractal trees are provided as well as the underlying parametric equations.