Standard Level Mathematics: Analysis & Approaches Internal Assessment
Mathematical Exploration of the Mandelbrot set
Candidate code: jjs506
May 2021
,Table of contents
2
, 1. Introduction
Mathematics has no generally accepted definition, however, it can be defined as the science of
numbers and their operations, interrelations, combinations, generalizations, and abstractions and of
space configurations and their structure, measurement, transformations, and generalizations
(Merriam-Webster, 2019). Many people have a preconceived notion that all maths is complicated and
that it involves many difficult equations and constructs. However, I find that notion to be incorrect.
Whilst a lot of maths is extremely complicated, many ideas can be expressed with very simple
concepts and equations. Maths can also be found in unexpected places, expressed through different
theories.
In nature, we often encounter many patterns, which can usually be modelled using mathematical
concepts. Some well-known examples include the Nautilus shell, or the Romanesco Green
Cauliflower, which can be accurately modelled using the Fibonacci sequence. Other mathematical
tools that can describe shapes in nature are fractals, but it is not their only purpose. One of the earliest
fractals to have been described is Koch’s Snowflake; a mathematical curve, which can be constructed
from a simple iterating geometric shape.
For my IA, I decided to investigate fractals, and in particular, the Mandelbrot Set. This is a
mathematical construct outside of the IB syllabus. I found the repetitive nature of the process really
fascinating during my research and I wanted to investigate how it works.
2. The Mandelbrot Set
2.1. What is the Mandelbrot Set
The Mandelbrot Set was first investigated by two French mathematicians, Pierre Fatou and Gaston
Julia, at the beginning of the 20th century (Horgan, 2009). It can be defined as a set of complex
numbers for which the function does not diverge when iterated at z=0 and remains bounded at
3
Mathematical Exploration of the Mandelbrot set
Candidate code: jjs506
May 2021
,Table of contents
2
, 1. Introduction
Mathematics has no generally accepted definition, however, it can be defined as the science of
numbers and their operations, interrelations, combinations, generalizations, and abstractions and of
space configurations and their structure, measurement, transformations, and generalizations
(Merriam-Webster, 2019). Many people have a preconceived notion that all maths is complicated and
that it involves many difficult equations and constructs. However, I find that notion to be incorrect.
Whilst a lot of maths is extremely complicated, many ideas can be expressed with very simple
concepts and equations. Maths can also be found in unexpected places, expressed through different
theories.
In nature, we often encounter many patterns, which can usually be modelled using mathematical
concepts. Some well-known examples include the Nautilus shell, or the Romanesco Green
Cauliflower, which can be accurately modelled using the Fibonacci sequence. Other mathematical
tools that can describe shapes in nature are fractals, but it is not their only purpose. One of the earliest
fractals to have been described is Koch’s Snowflake; a mathematical curve, which can be constructed
from a simple iterating geometric shape.
For my IA, I decided to investigate fractals, and in particular, the Mandelbrot Set. This is a
mathematical construct outside of the IB syllabus. I found the repetitive nature of the process really
fascinating during my research and I wanted to investigate how it works.
2. The Mandelbrot Set
2.1. What is the Mandelbrot Set
The Mandelbrot Set was first investigated by two French mathematicians, Pierre Fatou and Gaston
Julia, at the beginning of the 20th century (Horgan, 2009). It can be defined as a set of complex
numbers for which the function does not diverge when iterated at z=0 and remains bounded at
3
