18: Introduction to Fractals
"Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." - Benoit Mandelbrot.Lecture outline: What are fractals? How is fractal different from usual Euclidean objects?
1. What are fractals?
Fractal geometry of nature
Self-similarity: small pieces are like the whole and inexhaustible richness of structure
Scale symmetry or invariance.
Fractals created through iteration;
Example of Sierpinski’s Triangle
Fractal vs. Euclidean dimensions.
Exact self-similarity: Cantor’s set; Koch’s snowflake.
Mandelbrot’s examination of noise in transmission lines.
Dimensions of fractal objects? How long is the coast of Britain?
Statistical self-similarity: Self-similarity of Ethernet.
Computer generated natural fractals.
Fractals in Art.
Fractals in Engineering.
2. Fractal statistics
Fractals vs. non-fractals
A non-fractal game: the usual coin-toss
A fractal game: St. Petersburg’s game (fractal coin-tossing)
Primary reference for this lecture:
“High Speed Networks and Internets: Performance and Quality of Service” by William Stallings; Chapter 9: “Self-Similar Traffic”
Secondary references for this lecture:
1. “Wide-Area Traffic: The Failure of Poisson Modeling” by Vern Paxson and Sally Floyd [link].
2. “Internet Traffic Measurement” by Carey Williamson [link].
3. “Self-similarity in World Wide Web traffic: evidence and possible causes” by Crovella et al. [link].
4. “On the Self-similar Nature of Ethernet Traffic” by Leland et al. [link].