Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture (2013)

Pages 295–302

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.

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