PMA 443 Fractals 2009-10

by Dr. P. G. Dixon



picture of the Barnsley fern

Course summary and objectives

The concept of fractional dimension has been around for about 90 years, but the term `fractal' and the interest in fractals, both popular and scientific, date from the proliferation of microcomputers in the late 1970's. The first aim of this course is to develop an understanding of the classical theory of dimension (there are several competing definitions to consider) and its relation to the recent applications of fractals in science and technology.

However `fractals' are not just objects with fractional dimension. Most of the well-known fractals also possess `self-similarity': they are composed of several parts, each of which is a small-scale copy of the whole. Amazingly, specifying this self-similarity is enough to determine the fractal. This has important applications to the compression of image data (and was used by Microsoft in their first encyclopaedia on a CD). The proof, which is a beautiful application of the Contraction Mapping Theorem in an unusual setting, is another feature of the course.

The overall structure of the course is this: we begin by studying the construction of fractals as self-similar sets; then we study dimension theory; finally, the two strands are brought together in Hutchinson's Theorem which allows one to compute the dimension from the self-similarity data for a wide class of fractals.

Throughout, the chapters on fractals are interspersed with chapters on some basic analysis prerequisite to the present study and to the understanding of abstract analysis generally. Indeed, whilst the primary aim of this course is the study of dimension and self-similarity, an important secondary aim is to consolidate your knowledge of the basic ideas of abstract analysis. The sections on countability, metric spaces, compactness and Lipschitz maps should be viewed in this light.

Animated Sierpinski gasket; copyright P.G.Dixon 2000

Course organization

The course is two lectures per week in the second semester (Wednesdays at 10  in LT 9 and Thursdays at 4 in LT C)). Note the changed times since the level 3/4 Handbook was written.  Duplicated notes will be distributed during the course, after the corresponding lectures. Homework will be set each week with solutions posted online the following week.  Every two or three weeks, the homework will also be collected in for marking.  Students will not be expected to use computers or to have substantial knowledge of computing.


There will be no feedback (unless you hand in work)!   

Aims, Learning Outcomes, etc.

See the SoMaS information page for this module at

Recommended Books

K. J. Falconer, ``Fractal geometry: mathematical foundations and applications'' (Wiley, 1989), ISBN-13: 9780470848623, £34.95, ML 513.84(F), SLC 513.84(F), ASL 513(F).

Richard M. Crownover, `Introduction to fractals and chaos' (Jones and Bartlett, 1995) ISBN-13: 9780867204643, £39.99 (out-of-print?) [also useful for the Chaos course.].

D. Gulick,``Encounters with chaos'' (McGraw-Hill, 1992) ISBN-13: 9780070252035 ML (2 OWL copies, 1 SLC) 1 HICKS SLC, all 531.3(G) [also recommended for the Chaos course, but out of print.]

H. Lauwerier, ``Fractals: images of chaos'' (Penguin, 1991) £14.00 ISBN-13: 9780140144116, ML 513.84(L) (out-of-print?)

Link to Blackwell's reading lists site.

For a comprehensive list of books on fractals (to August 2007) compiled by Benoit Mandelbrot, see here.

Online Resources

  1. The search engine Yahoo has a page of, mainly popular, fractal sites, and Google has a page on Chaos and Fractals.
  2. Yuval Fisher (Institute for Nonlinear Science, University for California, San Diego) has a web page on fractal image compression here.
  3. A bibliography on fractal image compression is available the web site of Brendt Wohlberg, (University of Cape Town). Click here.
  4. David Wright's Dynamical Systems and Fractals Lecture Notes are useful, though more elementary than PMA343.
  5. The Dynamical Systems and Technology Project at Boston University has lots of exciting stuff. For example, Robert Devaney's article on The Chaos Game as an high-school level introduction to chaos and fractals is worth looking at, and you can play the Chaos Game interactively, to understand how the Siepinski Triangle can be reconstructed from its IFS.
  6. A Java applet for drawing plane filling curves.
  7. The Fractal Geometry page at Yale includes some good Java software.
  8. The Waterloo Fractal Coding and Analysis Group page.
  9. Michael Barnsley has an interesting article in the January 2010  issue of the Notices of the American Mathematical Society entitled "The Life and Survival of Mathematical Ideas". The article is to some extent a general discussion, as its title indicates, but consists mainly of a case study of Iterated Function Systems.   He gives a very condensed introduction to IFSs; easier to follow when you are half way through this module.  Incidentally, this issue of the AMS Notices is devoted to articles about Mathematics and the Arts; the front page for the issue is here.
  10. Another AMS Notices article, by Robert S. Strichartz, "Analysis on Fractals" discusses one direction of current research on fractals.  In SoMaS, Dr. Jonathan Jordan works in this area; his preprints and publications page is here.

Examples of fractals

  1. Jim Loy's page on the Koch curve.
  2. Wikipedia's image of the quadratic Koch curve construction.
  3. Nice pictures from Wikipedia of the Sierpinski gasket.
  4. A lot of info from the Springer Encyclopaedia of Mathematics
  5. The Sierpinski gasket in three colours.
  6. A Sierpinski carpet antenna.
  7. Various articles about fractal antennas: Military Antenna Design, Fractal Antenna Constructions, Fractal Antenna Systems Inc's homepage, a document from Coopers (page 81).
  8. Invisibility cloaks!
  9. An interactive Sierpinski carpet showing successive stages of the construction.
  10. The Menger sponge, a three-dimensional version of the Sierpinski carpet. 
  11. A photo of a Romanesco Broccoli, from Wikipedia.   And another and another.
  12. Paul Bourke's fractals page has a lot of nice examples.

Examples of fractals that are attractors of  IFSs with condensation

  1. Telegraph poles.
  2. Logarithmic spirals, which occur naturally e.g. in shells ancient and modern.
  3. Pythagoras tree, from Wikipedia.
  4. Interactive construction of a Pythagoras tree.


Applications of fractal dimension

  1. A recent example of fractals in Physics is reported in Physics News Update No.578 (27 Feb. 2002) "Fractal Carbon Nanopore Network".
  2. Another example, from Physics News Update Number 629 March 19, 2003 "Blood Vessel Networks".

Notes and problem sheets

The notes and problem sheets will be posted as the course progresses.    The Menger sponge, referred to in the last homework sheet, may be viewed here; for the solution you need to dissect it, like this.  A statement about non-examinable sections of the course will appear in due course.

Class size

17 students.

Lecturer's Office Hours, Telephone, E-mail.

Office hours: I shall not be having office hours but I shall be very happy to respond to queries by e-mail.
Telephone extension 23775 (prefix by 22 from outside the University and further prefix by 0114 from outside Sheffield).
E-mail: P.Dixon  with the usual extension:


One formal 2.5 hour exam. Format: 4 out of 5 questions, with a compulsory Question 1. 

In general you can expect the examination paper to test your knowledge of basic definitions, without
which you would not know what you were talking about, then the statements of key theorems, and then
the ability to solve problems and reconstruct proofs encountered in lectures. Recent past papers should be a good guide
to the type of questions to be set, except for small changes in the syllabus: Question 4 parts (ii) and (iv) of the 2008 paper are no longer in the syllabus; the 2009 paper is a good guide. 

Past Examination Papers

The past papers for  June 2008  and June 2009 are available in pdf  form. 

Solutions to past papers are available in PDF form (though without diagrams): June 2008, June 2009, June 2010


The following are links to the software used for demonstrations during the course. Use the right mouse button and `save link' option to save the software to your hard disk and then unzip it if necessary using PKUNZIP, WinZip or Zip Central.

We shall use FRACTINT, a very extensive program for drawing fractals, including Mandelbrot and Julia sets (from the Chaos course) as well as the Barnsley fern, Sierpinski gasket, etc.. Fractint 20.0 is a DOS program, but there is also an (earlier) Windows version WINFRACT version 18.21.

Finally, here is a Fractals icon for your Windows desktop.

Peter Dixon's home page

School of Mathematics and Statistics (SoMaS)