Introduction

Fractals are mathematical structures created through repeated patterns called iteration. Unlike ordinary geometric shapes, fractals often display a property called self-similarity, where smaller parts resemble the whole structure when zooming into specific sections of the fractal. Fractals appear not only in mathematics but also in nature, including coastlines, trees, lightning, human biology and snowflakes. This project investigates how simple mathematical rules can generate highly complex patterns and explores the mathematical properties of several famous fractals. 

A fractal in mathematics is a class of complex geometric shapes that commonly have “fractional dimensions”, a concept first introduced by the mathematician Felix Hausdorff in 1918. Fractals are distinct from simple Euclidean geometry, such as the square, the circle, the sphere, and so forth, having dimensions that are fractions rather than whole numbers. They are capable of describing many irregularly shaped objects or spatially nonuniform phenomena in nature, such as coastlines and mountain ranges. The term fractal, derived from the Latin word fractus (“fragmented,” or “broken”), was coined by the Polish-born mathematician Benoit B. Mandelbrot. 

The Hausdorff fractal dimension is a method used to calculate the dimension of a complex shape and has also been a fast-to-calculate method to estimate complexity of fractal shapes...