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].