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.
 A: 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.


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*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.
