What are some good resources to start learning about fractals?

I am an undergraduate mathematics major looking for online resources to learn more about fractals and fractal geometry. I have only a basic knowledge of fractals and their properties, so I am only looking for introductory resources at this point.

$\begingroup$ Start by asking a smaller & slightly rotated question on math.stackexchange.com. $\endgroup$

$\begingroup$ Learn basic analysis and then look at Kenneth Falconers book, Fractal Geometry. $\endgroup$

$\begingroup$ Whatever else you read, you have to read Mandelbrot's book The Fractical Geometry of Nature, the seminal work on the topic. (You may find it quite annoying - I did - but you should still read it $\ddot$.) $\endgroup$

1 Answer 1

As the asker is an undergraduate mathematics major, I will give the reading recommendations that I typically give to first-year graduate students who are interested in joining our research group, which focuses on fractal geometry, analysis on fractals, and related topics. Note that these are not the online resources that the asker requested—while there are a lot of places on the internet where fractals are discussed, I would think that an undergraduate should have the mathematical maturity to start reading on a topic more directly.

• Mandelbrot, Benoit B., The fractal geometry of nature. Rev. ed. of “Fractals”, 1977, San Francisco: W. H. Freeman and Company. 461 p. (1982). ZBL0504.28001. Skim Mandelbrot's text. This text is a general introduction to the idea of fractals from a fairly non-rigorous point of view. The emphasis is on broad exposition and pretty pictures, which is good for getting an overview of the field (as it stood 40 years ago). From the point of view of the history of the study of fractals, this is the work that introduced fractals to a wide audience.
• Edgar, Gerald A., Measure, topology, and fractal geometry, Undergraduate Texts in Mathematics. New York etc.: Springer-Verlag. ix, 230 p. DM 58.00/hbk (1990). ZBL0727.28003. Edgar's book is a well-written introduction to the tools needed to rigorously study fractals. This is a text that you should read in more detail—the topology and measure theory introduced are part of a standard mathematics curriculum (every undergraduate mathematics major should take topology, and measure theory is generally taught in the first year of graduate studies, if not before). The text is relatively gentle (as these things go), and the exercises are good. My only complaint is a change from the first edition to the second: in the first edition, every proof ended with a smiley face.
• Hutchinson, John E., Fractals and self similarity, Indiana Univ. Math. J. 30, 713-747 (1981). ZBL0598.28011. This article is typically the first thing that I recommend for people who want to get serious about studying fractals. It is a seminal work in the field, and introduces many of the tools that a student is going to need to understand (self-similarity as described by iterated function systems; techniques for computing dimension; etc). An added advantage is that Hutchinson is quite readable.
• Falconer, Kenneth, Fractal geometry: mathematical foundations and applications, Chichester: John Wiley & Sons (ISBN 0-471-96777-7/pbk). xxii, 288 p. (1997). ZBL0871.28009. Falconer has written a number of important texts on fractals and fractal geometry. The one cited above is quite excellent. It is written at the level of an advanced undergraduate, but doesn't require too much background. Again, the best way to read this text is by working through the exercises. Falconer also has a more difficult text in the Cambridge "Tracts on Mathematics" series, as well as a very short introduction to fractals from Oxford University Press, both of which are worth having.

$\begingroup$ @zwettlerj: See also the references listed here. Most of these would probably be best for you after getting through most of Edgar's 1990 book, although the notes by Pollicott and the MS thesis of Worth could probably be profitably read in part now, unless you haven't gotten out of the elementary calculus sequence yet. Regarding Falconer's book (1990 1st edition, 2003 2nd edition), (continued) $\endgroup$

$\begingroup$ I've found his 1985 (but you'll want a $\geq$ 1986 corrected reprint) The Geometry of Fractal Sets easier to follow than his "Fractal Geometry" book because his "Fractal Geometry" book leaves a lot of details to the reader and it covers a lot of mathematical topics that a beginner might not know much about, but the 1985 book requires a bit more mathematical maturity. Incidentally, for others who might be interested, there is an error in the statement of Exercise 1.4 (pp. 18-19) in Falconer's 1985 book. $\endgroup$

$\begingroup$ (+1) I forgot to upvote this nice answer 45 minutes ago, being too busy writing my two earlier comments! $\endgroup$