Complexity Explorer Santa Few Institute

Fractals and Scaling (Fall, 2015)

Lead instructor:

This course is no longer in session.

Additional Reading (Still in Progress....)

non-technical books:

  • Mitchell, Melanie. Complexity: a guided tour. Oxford University Press, 2009.  A general-audience book about complex systems.  Chapter 7, 16, and 17 are particularly relevant to this course.

  • Falconer, Kenneth. Fractals: a very short introduction. Oxford University Press, 2013. A very nice introduction to fractals.  Uses just a bit of math. Highly recommended.

  • Gleick, James. Chaos: Making a new science. Random House, 1997.  Very readable, focuses on the scientists and mathematicians behind many of the key results in chaos.  Includes a discussion of fractals.

history and philosophy of science: these books and articles are mostly non-technical, but are scholarly pieces on the history and philosophy of science. 

  • In the Wake of Chaos: Unpredictable Order in Dynamical Systems.  Stephen H. Kellert.  University of Chicago Press.  1994.
  • Smith, Peter. Explaining chaos. Cambridge University Press, 1998.  Primarily about dynamical systems, but includes a chapter on fractals.

textbooks with more mathematical details:  I've listed these textbooks in approximate order from introductory to advanced.

  • Feldman, David P. Chaos and Fractals: An Elementary Introduction. Oxford University Press, 2012.  At roughly the same level as this class.

  • Flake, Gary William. The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptations. The MIT Press, 1998.  In a few places uses calculus.  A very clear and engaging introduction to chaos, fractals, and complex systems more generally.

  • Sayama, Hiroki, Introduction to the Modeling and Analysis of Complex Systems.  OpenSUNY Textbooks.  Looks like a great survey of a range of topics in complex systems, including dynamical systems.  A pdf is available here.

  • Kaplan, Daniel, and Leon Glass. Understanding Nonlinear Dynamics. Springer-Verlag, 1995.  Primarily about dynamical systems, but has a chapter on fractals.

  • Strogatz, Steven. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry and engineering. Westview Press, 2001. Includes a short chapter about fractals.

  • Peitgen, Heinz-Otto, Hartmut Jürgens, and Dietmar Saupe. Chaos and fractals: new frontiers of science. Springer, 2004.

  • Arnold, Barry. Pareto Distributions (second edition). CRC Press, 2015.


statistics and data analysis

  • James, A., and M. J. Plank. "On fitting power laws to ecological data." arXiv preprint arXiv:0712.0613 (2007).
  • Clauset, Aaron, Cosma Rohilla Shalizi, and Mark EJ Newman. "Power-law distributions in empirical data." SIAM review 51.4 (2009): 661-703.  jstor

allometric scaling

  • LaBarbera, Michael. "Analyzing body size as a factor in ecology and evolution." Annual review of ecology and systematics (1989): 97-117. jstor  pdf
  • Tung, K. K. Topics in mathematical modelling. Princeton University Press, 2007.  Chapter 2 is a clear and introduction to allometric scaling and also touches upon some rich-get-richer phenomena.  Math level is just a bit beyond this course.

power laws: reviews and generating mechanisms

  • Reed, William J., and Barry D. Hughes. "From gene families and genera to incomes and internet file sizes: Why power laws are so common in nature." Physical Review E 66.6 (2002): 067103.  pdf
  • Simkin, Mikhail V., and Vwani P. Roychowdhury. "Re-inventing willis." Physics Reports 502.1 (2011): 1-35.  A review of the many re-discoveries of mechanisms that generate power laws. "Scientists often re-invent things that were long known. Here we review these activities as related to the mechanism of producing power law distributions, originally proposed in 1922 by Yule to explain experimental data on the sizes of biological genera, collected by Willis." arXiv

  • Stumpf, Michael PH, and Mason A. Porter. "Critical truths about power laws." Science 335.6069 (2012): 665-666.  pdf

  • Watkins, Nicholas W., et al. "25 Years of Self-organized Criticality: Concepts and Controversies." Space Science Reviews (2015): 1-42. pdf

Simon--Mandlebrot exchange


lectures and documentaries

  • Raymond Flood.  Butterflies, Chaos and Fractals.  Museum of London.  17 September 2013.  link
  • Nic Stacey and Jim Al-Khalili.  The Secret Life of Chaos.  BBC 4.  January 2010. (with arabic(?) subtitles)
  • Jos Leys, Etienne Ghys, and Aurelien Alvarez, Chaos: A Mathematical Adventure.  A set of videos availalbe in 11 different languagues.