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Santa Fe Institute

Fractals and Scaling

Santa Fe Institute via Complexity Explorer


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.


1. Introduction to fractals. Self-similarity dimension. Review of logarithams 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. Mechanism for generating power laws. Rich-get-richer phenomena. Phase transitions. Other mechanisms.

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

7. Urban scaling.

Taught by

David Feldman


4.8 rating, based on 56 Class Central reviews

Start your review of Fractals and Scaling

  • Anonymous
    The course was very well taught by the instructor, which provided a nice overview on the subject. However, I still missed some "real -life examples" during the course, yet the instructor discussed two nice applications, i.e., metabolic and urban sca…
  • Anonymous
    The content is presented in a clear manner. It is obvious that the instructor has a lot of experience teaching. He is able to breakdown concepts into smaller, understandable components. Overall, this is a great class. I am really happy I chanced upon it.
  • Anonymous
    If you are interested in or are pursuing computer science or digital visual arts, I highly recommend taking this course. This course totally beats that one-day lesson on fractals in algebra I had in middle school. Even though the professor doesn't a…
  • Anonymous
    Dave is an awesome teacher. He explains things very well, and relates these explanations back to everyday life situations in ways that make it easy to understand. I would highly recommend this course to anyone with an interest in complex systems, fractals, and power laws.
  • Anonymous
    For lack of my own personal time I was not able to finish the course but the course itself has ample time to be finished. The homework is great, the easy questions make sure you get the basics down and the harder questions are challenging and within…
  • Elizabeth Castillo
    As a social scientist who wanted to learn more about fractals, this class was perfect. Dave makes the mathematics very accessible. He speaks in plain English and has a charming way of breaking complex ideas down into simple building blocks. There are tutorials and optional exercises to help you learn things you might be rusty on. The homework and practice exams prepare you very well for the quizzes. The videos are mostly between 5-10 minutes so you can take the course in bite-size pieces as your time allows. Dave also provides resources for each module to continue your learning after the course. Highly recommended!
  • Anonymous
    I highly recommend this course.
    The structure of the course and the presentation are superb.
    For example, the clever balance of detailed explanations and of general explanations of a few topics, provides quite an interesting (and useful) perspective of the material.
    Exciting, interesting, thought-provoking fun on topics that are relevant to a wide range of disciplines (and growing).
  • Anonymous
    I really enjoyed this course. The material was interesting and the explanations were clear. The instructor was engaging. So I feel like an ingrate complaining, especially about something that is not within hos control. But I hope you'll find a way to adjust for a presenter's left handedness. Watching the back of his hand as he wrote on the whiteboard got to be frustrating after a while.

    Thank you so much for offering ordinary people the opportunity to learn such cool stuff!
  • Anonymous
    Nice course but this instructor is basically following his textbook written half a decade ago, from old papers, nothing much new to be honest. Yet a good course though, but I think instructors should risk more and treat students with more respect, we could just read his textbook.
  • Brian Callaghan
    This was a great class as in introduction to fractals and scaling. The metabolic scaling section was fascinating, and has encouraged me to do more in-depth reading and research on the subject. I'm definitely looking forward to the next Complexity Explorer course!
  • Profile image for Felix Cosmin Mocanu
    Felix Cosmin Mocanu
    Good course, really peaked my interest in the topics presented, good introduction! Could use some more hands on computational exercises to connect better with the material but was otherwise a very good course.
  • Very nice course! Quite easy without going deep into math behind, without any computational assignments. But topic is rather interesting and explanations were very clear and with good examples. You can watch it on 1.5x speed easily.
  • Profile image for Mr Lazy
    Mr Lazy
    Very good to learn very much how to make sure you have been working with you and your family and friends only help you with your company and the other YouTube channel badhu apyu chhe
  • Blake Toscani
    I don't speak english very well, but i completed the course without any difficult. Is necessary a basic mathematics formation.
  • Anonymous
    The instructor explains things really well and the course content is engaging along with various applications of fractals and online resources. I really appreciate that the homework is divided among beginner to advanced level. Somehow I wish that the advanced one were deeper, though.
  • James Wen
    I wasn't sure I'd have the time to take this course but I had taken another one taught by Dave Feldman last summer (Dynamical Systems and Chaos) and it was great. I had free time then but, having just started a job, I was more concerned about my ab…
  • I loved the course! for me linking something statistical to fractals was a revelation. I see myself applying some of what I've learnt in my work. This series of lectures are applied in nature rather than purely theoretical. The level is fairly easy for a non mathematician. Modeling any data before feeding it to a statistical blackbox like an SVM or a neural network is an imperative but almost overlooked aspect of data science. This course is necessary in that modelling toolbox.
  • Anonymous
    Professor Feldman is terrific at making the complex understandable. He's organized and the course materials and supplementary resources are excellent. He handles the occasional snafu with aplomb and humor. My favorite was the time there was a loud explosion followed by a power outage during recording. He carried on with the lesson seamlessly all the while hoping his computer wasn't " fried."
    Thank you, Professor Feldman for a great learning experience.
    Linda Alario
  • Anonymous
    This is an amazing course for anyone that wants to learn about this topic and does not have the time to go through all the mathematical rigour behind it. The lecturer has a natural skill to explain in a very simple way things that may seem very complicated in some textbooks or courses. You will have fun following these lectures and learn while you can hold a cup of coffee in one hand and a pocket calculator in the other to compute dimensions of fractional objects.
  • Flavio Massimo Saldicco
    This is another fantastic course from Prof. Feldman (the other one being Introduction to Dynamical Systems and Chaos), offered by the Santa Fe Institute. Prof. Feldman is able to explain difficult topics in a simple way, with plenty of practical examples taken from the real world and a bit of humour. Warmly recommended if you are interested in Scaling, even if you don't have a lot of mathematical background.

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