Complexity Explorer Santa Few Institute

Fractals and Scaling

Lead instructor:

About the Course:

We will begin by viewing fractals as self-similar geometric objects such as trees, ferns, clouds, mountain ranges, and river basins.  Fractals are scale-free, in the sense that there is not a typical length or time scale that captures their features.  A tree, for example, is made up of branches, off of which are smaller branches, off of which are smaller branches, and so on.  Fractals thus look similar, regardless of the scale at which they are viewed.  Fractals are often characterized by their dimension.  You will learn what it means to say that an object is 1.6 dimensional and how to calculate the dimension for different types of fractals.

In addition to physical objects, fractals are used to describe distributions resulting from processes that unfold in space and/or time.  Earthquake severity, the frequency of words in texts, the sizes of cities, and the number of links to websites are all examples of quantities described by fractal distributions of this sort, known as power laws.  Phenomena described by such distributions are said to scale or exhibit scaling, because there is a statistical relationship that is constant across scales. 

We will look at power laws in some detail and will give an overview of modern statistical techniques for calculating power law exponents.   We will also look more generally at fat-tailed distributions, a class of distributions of which power laws are a subset.  Next we will turn our attention to learning about some of the many processes that can generate fractals.  Finally, we will critically examine some recent applications of fractals and scaling in natural and social systems, including metabolic scaling and urban scaling.  These are, arguably, among the most successful and surprising areas of application of fractals and scaling.  They are also areas of current scientific activity and debate.

This course is intended for anyone who is interested in an overview of how ideas from fractals and scaling are used to study complex systems.  The course will make use of basic algebra, but potentially difficult topics will be reviewed, and help is available in the course discussion form.  There will be optional units for more mathematically advanced students and pointers to additional resources for those who want to dig deeper.

Course Outline

1. Introduction to fractals. Self-similarity dimension. Review of logarithms and exponents.

2. Box-counting dimension. Further examples of fractals. Stochastic fractals.

3. Power laws and their relation to fractals. Rank-frequency plots. How to estimate power law exponents.

4. Empirical examples of power laws. Other long-tailed distributions: log normals and stretched exponentials. Implications of long tails.

5. Mechanisms for generating power laws. Rich-get-richer phenomena. Phase transitions. Other mechanisms.

6. Metabolic scaling. West-Brown-Enquist scaling theory.

7. Urban scaling.


About the Instructor(s):

David Feldman is Professor of Physics and Mathematics at College of the Atlantic.  From 2004-2009 he was a faculty member in the Santa Fe Institute's Complex Systems Summer School in Beijing, China.  He served as the school's co-director from 2006-2009.  Dave is the author of Chaos and Fractals: An Elementary Introduction (Oxford University Press, 2012), a textbook on chaos and fractals for students with a background in high school algebra.  He has thrice offered a MOOC on Chaos and Dynamical systems on the Complexity Explorer site, in addition to this MOOC. Dave was a U.S. Fulbright Lecturer in Rwanda in 2011-12.






Class Introduction
How to use Complexity Explorer
Enrolled students:


Course dates:

Always available


Some high school algebra

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56 Student


  1. Introduction to Fractals and Dimension
  2. Generating Fractals
  3. Box-Counting Dimension
  4. Introducing Power Laws
  5. Power Laws in Empirical Data
  6. Generating Power Laws
  7. Metabolic Scaling
  8. Urban Scaling
  9. Conclusion and Summary