Introduction

What are fractals?

Fractals are everywhere! If you don’t believe me, just take a look outside your window. From the shapes of trees and bushes to the jagged profiles of mountains to the irregular coastlines, many features of our natural world seem to be modeled by fractal geometry. But what exactly is a fractal? As you will learn in this module, a fractal is an object that displays self-similarity at every level. That is, when you zoom in on one section, it resembles the whole image. This self-similarity doesn’t have to be exact; in fact many fractals show some variation or randomness. Below is a video illustrating how the Mandelbrot set, a well-known fractal, displays self-similarity.

https://www.youtube.com/embed/G_GBwuYuOOs

While some fractals (like the Mandelbrot set) could pass for works of art, the true beauty of fractals is in how such intricate designs and patterns can result from very elementary generating formulas or rules. In this module, you will learn how to create fractal patterns such as the Mandelbrot set using a simple formula such as:

[latex]z_{n+1} = z_n^2 + c[/latex]

Of course there are many details that still need to be explained, such as the relationship between fractals and complex numbers. The values of [latex]c[/latex], [latex]z_n[/latex] and [latex]z_{n+1}[/latex] in the above formula are supposed to be complex numbers, that is, numbers that include the imaginary unit, [latex]i = \sqrt{-1}[/latex]. The imaginary number [latex]i[/latex] is something completely different than any number you have ever seen. In fact, [latex]i[/latex] does not show up on the number line at all! Instead, as you will soon discover, the imaginary unit lives on its own separate number line, called the imaginary axis, which is perpendicular to the usual number line (or real axis). The Mandelbrot set itself is made up of the complex numbers that satisfy a certain rule related to a simple equation. The resulting picture is amazing, and just gets more and more fascinating as you zoom in!

A Mandelbrot set in the Complex plane.

Learning Objectives

Generate fractals given an initiator and generation rule

Generate a fractal with random variation
Calculate Fractal Dimension using scaling relation

Complex Numbers

Identify and make arithmetic calculations with imaginary numbers
Plot complex numbers on the complex plane
Define a recursive sequence that will generate a fractal in the complex plane
Determine whether a complex number is part of the Mandlebrot set

Licenses & Attributions

CC licensed content, Original

Why It Matters: Fractals. Authored by: Lumen Learning. License: CC BY: Attribution.

CC licensed content, Shared previously

Mandelbrot Set. Authored by: Lars H. Rohwedder. Located at: https://commons.wikimedia.org/wiki/File:Mandelbrot_Set_in_Complex_Plane.png. License: Public Domain: No Known Copyright.

מדריכי לימוד > Mathematics for the Liberal Arts