In recent years there has been an explosion of interest in the mathematics of fractals – sets whose geometry we cannot easily describe in classical terms. There’s no simple definition, but all fractals have a highly intricate structure. Many fractals contain copies of themselves at many different scales, and computer pictures have shown that such sets (which are often very beautiful) are an outstanding representation of shapes of natural phenomena such as clouds, mountains and ferns. This module examines the theory of fractals and their geometry and examples of its application.
What you will study
The theory of fractal geometry provides a general framework for the study of sets that had been thought to be exceptional oddities. This is an active area of research and both the theory and applications of fractal geometry are still being developed.
The module is based on the set book Fractal Geometry: Mathematical Foundations and Applications (Third edition) by K. J. Falconer (Wiley), which is in two parts.
Part I has eight chapters dealing with the general theory of fractals and their geometry.
Part II looks at examples of fractals to which the theory of Part I can be applied. These examples are drawn from a wide variety of areas of mathematics and physics.
The module begins with an introductory chapter covering the necessary background material. Next we study the material in chapters two to four of the book, which introduce appropriate definitions of dimension and methods for calculating such dimensions.
The second half of the module looks at applications including data compression, examples from number theory, dynamical systems and Julia sets.
You will learn
Successful study of this module should enhance your skills in understanding complex mathematical texts, constructing solutions to problems logically and communicating mathematical ideas clearly.
Entry
You must have passed (or be studying in parallel) one of the following modules:
Preparatory work
All teaching is in English and your proficiency in the English language should be adequate for the level of study you wish to take. We strongly recommend that students have achieved an IELTS (International English Language Testing System) score of at least 7. To assess your English language skills in relation to your proposed studies you can visit the IELTS website.
Study materials
What's included
You’ll have access to a module website, which includes:
- a week-by-week study planner
- course-specific module materials
- audio and video content
- relevant computer software and associated guidance
- assessment details and submission section
- online tutorial access
- access to student forums.
You’ll also be provided with course notes covering the content of the module, including explanations, exercises and solutions to aid your understanding of the concepts and associated skills and techniques that are contained in the set book. You will need to obtain your own copy of the set book, and only the set book as printed by the publisher will be permitted in the examination, and not a version you have printed yourself.
Teaching and assessment
Support from your tutor
Throughout your module studies, you’ll get help and support from your assigned module tutor. They’ll help you by:
- Marking your assignments (TMAs) and providing detailed feedback for you to improve.
- Guiding you to additional learning resources.
- Providing individual guidance, whether that’s for general study skills or specific module content.
The module has a dedicated and moderated forum where you can join in online discussions with your fellow students. There are also online module-wide tutorials. While these tutorials won’t be compulsory for you to complete the module, you’re strongly encouraged to take part. If you want to participate, you’ll likely need a headset with a microphone.
Assessment
The assessment details can be found in the facts box.
Future availability
Fractal geometry (M835) starts every other year – in October.
This page describes the module that will start in October 2025.
We expect it to start for the last time in October 2029.