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Welcome and Overview
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Self-Similarity Dimension
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Some Properties and Features of Fractals
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Exponent and Logarithm Review (Optional)
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Video (10:00) Exponent Properties
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Quiz 1: Exponent Properties
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Quiz 2: More Exponent Properties
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Video (5:50) Logarithm definition
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Quiz 3: Logarithm Basics
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Solution to Quiz 3 (5:00)
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Video (5:40) Logarithm properties
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Quiz 4: Logarithm properties
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Solution to Quiz 4 (2:01)
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Video (2:16) Using logarithms
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Quiz 5: Using logarithms
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Solution to Quiz 5 (4:55)
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Video (4:04) Logarithms: different bases
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Summary
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Homework (Optional)
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Test
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Introduction
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Fractals via Deterministic Iteration
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Adding a Bit of Randomness
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The Chaos Game
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Diffusion-Limited Aggregation
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Summary
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Programs and Resources
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Homework (Optional)
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Test
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Introduction
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Basic Box-Counting
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Box-Counting in Practice
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Video (8:39) Introduction: s must be small
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Video (15:08) Box-counting a circle
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Quiz 1: Box-counting the Koch curve
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Quiz 2: The dimension of the Koch curve
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Video (7:26) Log-log plots
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Video (11:31) Log-log plot for a square
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Quiz 3: Log-log plots
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Video (2:10) Solution to quiz 3
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Video (11:03) Log-log plots: practicalities and issues
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Thinking about Dimensions
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Summary
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Homework (Optional)
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Test
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Introduction
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An Initial Example: Word Frequency
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Other Distributions
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Properties of Power Laws
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When do Averages Exist?
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Interview: Nick Watkins
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Summary
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Homework (Optional)
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Introduction
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Cumulative Distributions and Rank/Frequency Plots
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Estimating the Exponent
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Other Long-tailed Distributions
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Summary
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Additional Resources
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Homework (Optional)
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Test
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Introduction
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Rich-get-Richer Models
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Combinations of Exponentials
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Multiplicative Processes
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Optimization
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Phase Transitions?
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Homework (Optional)
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Additional Resources
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Test
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Introduction
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Metabolic Scaling
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WBE Theory of Metabolic Scaling
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Interview: Van Savage
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Interview: Geoffrey West
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Additional Resources
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Homework (Optional)
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Test
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Introduction
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Empirical Results
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A Theory of Urban Scaling?
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Interview: Christa Brelsford
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Interview: Luis Bettencourt
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Homework (Optional)
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Test
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Additional Resources
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Summary
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Farewell
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2.7 Programs and Resources » Links to programs
This page contains links to a variety of programs that will generate fractals.
Programs Mentioned in the Videos:
- Snowflake Activity. Another program to make Koch curve like fractals.
- Fractal Terrain Generator. by Xueqiao Xu.
- Gasket Online, a program to make fractals using the chaos game. Instructions here.
- Aglorithms, a program by Brian Hayes that simulates a model very similar to diffusion limited aggregation. An article from American Scientist by Hayes about this model is here.
Other Programs to Explore
- Fractal Machine. A site to make Koch curve and related fractals. Instructions for how to use the program are here. (I found the interface a bit non-intuitive, but once I figured it out it was fun.)
- L-Systems Renderer. I'm not going to cover the details of L-systems. Nevertheless, this might be fun to play around with.
- Fractal terrain generator. A short bit of javascript that makes impressive landscapes.
- Recursive Drawing. We won't go over this in the course, but I found this while searching around and it looks like it could be a very fun tool to explore and experiment with.
If you find other software that you think is good, please let me know and I'll post it here. Thanks.