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.
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)!
See the SoMaS information page for this module at http://maths.dept.shef.ac.uk/maths/module_info.php?module_code=PMA443.
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.
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: @sheffield.ac.uk
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.
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.
School of Mathematics and Statistics (SoMaS)