subject

Fractals and Scaling

 with  David Feldman
Class Central Course Rank
#2 in Subjects > Mathematics

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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.

Syllabus

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.

19 Student
reviews
Cost Free Online Course
Subject Mathematics
Institution Santa Fe Institute
Language English
Certificates Certificate Available
Hours 2-4 hours a week
Calendar 7 weeks long
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19 reviews

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6 out of 6 people found the following review useful
a year ago
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Anonymous completed this course.
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 Read More
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).

-steve

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4 out of 4 people found the following review useful
a year ago
Brian Callaghan completed this course.
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!
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3 out of 3 people found the following review useful
a year ago
Felix Cosmin Mocanu completed this course.
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.
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2 out of 2 people found the following review useful
a year ago
Aliaksandr Bely completed this course, spending 3 hours a week on it and found the course difficulty to be easy.
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.
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a month ago
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Anonymous is taking this course right now.
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 Read More
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.
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a year ago
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Anonymous partially completed this course.
I started to take the course but for reasons of time I could not finish it I am very sorry, but the time was taking the course was very interesting, I hope I can take back to thank you very much Dave was very interesting. An apology for not being able to finish thanks to SFI muchs
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3 weeks ago
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Anonymous partially completed this course.
This is one of the best online course I ever had till now. Dave is simply awesome. Earlier, I was having trouble understanding Nassim Taleb's underlying mathematical philosophy which was based on Fat tails distributions. This course alleviated that trouble.
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a year ago
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Anonymous completed this course.
Great Course. Learned a lot from it. I found fractals very interesting, and I believe I can definitely apply that in my Artificial Intelligence project. There's lots of statistics involved, but they are not really difficult.
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a year ago
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Anonymous completed this course.
Good introductory course. Highly recommended for anyone interested in overview of fractals. Perhaps some additional information on generation of fractals in mathematical constructs, their basins of attraction can be added.
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a year ago
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Anonymous is taking this course right now.
One of the best instructors I have heard in the last........many years!!

Real pleasure of learning,

Clear explanations,

The part if the fractal counting is a bit too long, but you know it

Thank you
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a year ago
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Anonymous completed this course.
Paul did a phenomenal job introducing new topics clearly, at good pace and with great examples. I also appreciate also the supplemental material and guest interview.
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a year ago
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Anonymous completed this course.
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a year ago
Francesca Gorrieri completed this course.
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2 weeks ago
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Javier partially completed this course.
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a year ago
Gaetano Pagani completed this course.
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a year ago
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Anonymous completed this course.
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a year ago
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6 months ago
Simon Crase completed this course.
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9 months ago
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Joseph Ramaswami completed this course.
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