Complexity Explorer Santa Few Institute

Fractals and Scaling (2023)

Lead instructor:


When does the course start?  Whenever you want. It is self-paced and available on demand.

Who is the instructor?  Dave Feldman, Professor of Physics and Mathematics, College of the Atlantic



How much does it cost?  Nothing.  The course is completely free.

How is the course funded?  The course is funded by the Santa Fe Institue (SFI) through a grant from the John Templeton Foundation and by donations from users.  In order to support future courses, we will be asking for small, voluntary donations to cover the costs of development and delivery.



Who is the intended audience? 

This course is intended for anyone with a modest background who wants to gain an understanding of fractals and scaling, particularly those interested in some of the recent work on scaling in biological and social systems.

What are the prerequisites?  This class will use some math at the level of basic high school algebra.  Students taking this course should have understood basic algebra at some point in their lives.  It's ok if you feel like you've forgotten most of it; it will come back quicker than you might expect, I'll review topics along the way, and you can get help in the discussion forms.  Logarithms will be used throughout, but there will be an optional unit reviewing logarithms for those who need it.  The topics in this course are broad and deep enough that it should be of interest to people with a wide range of math backgrounds. 

How does the math level compare to other Complexity Explorer courses?  This course will be more mathematically involved than Melanie Mitchell's Introduction to Complexity.  On balance, the level of math should be a bit less than what I used in my other course, Introduction to Dynamical Systems and Chaos.



How does the course work?  Each unit consists of a series of short videos, with each video corresponding to subtopics of the unit's main topic. The course website leads you through the videos in order, allowing you to skip or repeat videos as you desire. You can watch these videos at your own pace and in any order you desire; once posted, they will remain available throughout the course. The videos are interspersed with short exercises and quizzes, designed to test your understanding of the material covered in the previous video.  At the end of most units there is a test (graded automatically), as well as optional, ungraded homework.



How is the course graded? As described above, your grade will be based on the end-of-unit tests. Your
total course score will be your average score over these tests. Of course, since the course is not for credit, your scores are meant to be for your own tracking of your progress in the course. They will not be seen by anyone but you, and possibly our course team. 



How well do I need to do to receive a certificate, and will the certificate list my grade? You need to have completed all of the end-of-unit tests, with an average score of 60% or greater (i.e., averaged over all tests), to receive a certificate of successful completion. Your total course score will be the average of your test scores. Note:  You don't need to get 60% or greater on each test, only on the average over all tests. The certificate will not list your total score; it will simply say that you have successfully completed this course.

Can I get university credit for this course? No, not at this time. It is possible that in the future we may be
able to partner with colleges and universities so as to offer our courses for credit, but there is currently no
mechanism for this.

Will I get any kind of certificate? Everyone who successfully finishes the course will receive a certificate
of completion from the Santa Fe Institute.



Is there a required textbook? No textbook is required. The lectures will stand on their own, but will be complemented by some adidtional resources to be posted on the supplementary materials section of this site.



In what ways am I allowed to use these resources?  All the materials on this site are available for your use for any non-commercial purpose. All materials (videos, code, write-ups, etc.) are covered by the Creative Commons Attribution-NonCommercial-ShareAlike License ( This states that you may copy, distribute, and transmit the work under the condition that you give attribution to, and your use is for non-commercial purposes.



Are subtitles available?  The Complexity Explorer Project has an on-going project in which users volunteer to create subtitles in different languages.  If subtitles are available for a given video they can be accessed in the following way.  First, start and pause the video.  A tool bar will now be visible along the bottom of the video.  Second, click on the gear-shaped button located directly to the right of the “CC” button.  A list of available languages will be shown in the middle drop-down menu located in the box the opens after hitting the gear-shaped button.  Select the language you would like to use for subtitles and click on it.  

How do I download and use subtitles offline? You can download videos and available subtitles to watch offline if you wish. Information on downloading videos is located below under technical requirements.  In order to make the subtitles you download play with the video, you will need to go through a few steps.  Our suggested method is detailed in this help document.  Click the link to download the instructional pdf.  

Can I download a plain text transcript of the video? For any video that has subtitles available, there will also be a plain text transcript (in .txt format) available for download, for each subtitle language available.  When you click on Subtitles & Transcripts you will be given all of the language options available, and you can choose to download either the subtitle or the transcript, or both.  



Do I have to enroll to take the course? Yes, you need to enroll in order to access any of the course
materials. However, enrollment is easy, quick, and free.

How do I enroll?  Enrollment is open from December 1st 2018. 

Can I enroll after the course begins? Yes, but to receive a grade and a certificate you must register and complete the exams before the end of the course.



How much time does the course require? To complete the course in the given 10-week period, you
should expect to spend 1–2 hours per week watching videos and taking quizzes and exams, and 2–4 hours
per week on homework, for a total of 3–6 hours per week.  That said, the time spent on the course will vary depending on your math background and how many of the optional assigments you complete.



What are the rules on collaboration with other people? You are free, and encouraged, to discuss
anything with anyone.  The course website hosts an online forum for students to discuss the course
material, homework, etc. However, we ask that the end-of-unit tests be taken entirely on your own, without
collaboration with others or help from the Web. Of course, we are relying on the honor system for our
students to abide by these rules.

Is there a forum for discussion of course topics?  Yes. The course website hosts a forum in which course
participants can post questions, answers, and otherwise discuss the course materials. Questions posted to this forum will be answered by the instructor, teaching assistant, and/or other students

Will there be any other kind of social networking for participants?  We hope that students can use the course forum to set-up local "Meetups" for course participants who would like to meet in person.



How do I get the videos to play at a faster rate (e.g., 2x)? Our videos are streamed through
YouTube. You can opt in on YouTube for their html5 player, which allows you to speed up or slow down
videos. To opt in, go to

Can I download the videos directly, rather than watching them via YouTube? Yes, just click on the
"Download" button that appears above the video screen on the page for each video.



I forgot my password or something is wrong with the site, what should I do?  Please send an email to



What if I have more questions?  Please address other questions to