3 - Excursions Through a Forest of Golden Fractal Trees

Summary

Abstract. This paper presents an exploration of various features of the four self-contacting symmetric binary fractal trees that scale according to the golden ratio. This excursion demonstrates some beautiful connections in math by providing a wealth of interesting exercises involving geometry, trigonometry, infinite series, the Fibonacci sequence, self-similarity and symmetry.

Introduction

This paper presents an exploration of various features of the four self-contacting symmetric binary trees that scale according to the golden ratio. We begin with an introduction to background material. This includes relevant definitions, notations and results regarding symmetric binary fractal trees; various aspects of the golden ratio; and connections between fractals and the golden ratio. The main part of the paper consists of four subsections, with each subsection discussing a particular ‘golden tree’. Each tree possesses remarkable symmetries. Classical geometrical objects such as the pentagon and decagon make appearances, as do fractal objects such as a golden Cantor set and a golden Koch curve. We find new representations of well-known facts about the golden ratio by using the scaling nature of the trees.

Symmetric Binary Fractal Trees

Fractal trees were first introduced by Mandelbrot in “The Fractal Geometry of Nature” [8]. In general, fractal trees are compact connected subsets of ℝⁿ (for some n ≥ 2) that exhibit some kind of branching pattern at arbitrary levels. The class of symmetric binary fractal trees was more recently studied by Mandelbrot and Frame [9].