Fractals and Scaling Online Course

In this course, College of the Atlantic Professor David Feldman explores fractals and scaling. 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.

The course begins 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.  

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.

The course will then delve into power laws in some detail and will give an overview of modern statistical techniques for calculating power law exponents, followed by a more general look at fat-tailed distributions, a class of distributions of which power laws are a subset.  Next the course will introduce some of the many processes that can generate fractals.  Finally, metabolic scaling and urban scaling will be discussed.  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.

Instructor: David Feldman