# Fractal Geometry

 Yale University Michael Frame, Benoit Mandelbrot (1924-2010), and Nial Neger

 An Escheresque fractal by Peter Raedschelders.

"I find the ideas in the fractals, both as a body of knowledge and as a metaphor, an incredibly important way of looking at the world." Vice President and Nobel Laureate Al Gore, New York Times, Wednesday, June 21, 2000, discussing some of the "big think" questions that intrigue him.

 This is a collection of pages meant to support a first course in fractal geometry for students without especially strong mathematical preparation, or any particular interest in science. Each of the topics contains examples of fractals in the arts, humanities, or social sciences; these and other examples are collected in the panorama. Fractal geometry is a new way of looking at the world; we have been surrounded by natural patterns, unsuspected but easily recognized after only an hour's training.

 1. Introduction to Fractals and IFS is an introduction to some basic geometry of fractal sets, with emphasis on the Iterated Function System (IFS) formalism for generating fractals. In addition, we explore the application of IFS to detect patterns, and also several examples of architectural fractals. 2. Natural Fractals and Dimensions presents a method of measuring the complexity of fractals. Generalizing the familiar notion of Euclidean dimension, fractal dimension can be computed from experimental data. These computations have design consequences in such areas as antennas and fiber optics. 3. The Mandelbrot Set and Julia Sets is remarkable deconstruction of the notions of simplicity and complexity: a single quadratic equation contains infinitely detailed worlds of baroque splendor that pose mathematical questions unanswered even today. Yet the algorithm to generate these pictures can be understood by anyone familiar with basic arithmetic. 4. Cellular Automata and Fractal Evolution, or how to build a world in a computer. These simple worlds can generate fractals, and exhibit wonderfully complicated dynamics. The biological paradigm can be extended to evolve populations of computer programs, and we are led, perhaps, to fractal aspects of evolution. 5. Random Fractals and the Stock Market extends the geometrical fractals studied so far to fractals involving some elements of randomness. After examples from biology, physics, and astronomy, we apply these ideas to the stock market. Do we uncover useful information? Wait and see. 6. Chaos is type of dynamical behavior most commonly characterized by sensitivity to initial conditions: tiny changes can grow to huge effects. Inevitible uncertainties in our knowledge of the initial conditions grow to overwhelm long-term prediction. Yet we shall see chaos has engineering and medical applications. 7. Multifractals generalizes the notion of fractals as objects to fractals as measures. We can examine the distribution of resources in a region, compute the dimension of the parts with the same amount, and plot dimension as a function of amount. This gives a single picture embracing the entire range of complexity. 8. Fractal Trees is a short analysis of dimensions of several aspects of mathematical (not realistic) fractal trees. Yet even this simple problem has some surprises. 9. Circle Inversions is Iterated Function Systems when the affine transformations are replaced by inversions in circles. The loss of IFS linearity gives rise to new families of pictures, and to new mathematical problems. 10. Panorama of Fractals and Their Uses is a growing web document, a catalogue of some applications that we have found interesting. You are invited to share your favorites with us. 11. Laboratory Exercises is a collection of field-tested extended hands-on activities that illustrate many of the topics on these pages. 12. Lesson Plans is a collection of lesson plans for high school and middle school classes. 13. Software is a collection of Java applets to study fractals. In addition, there are limited collections of Macintosh software, PC software, and Mathematica notebooks.

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Comments or questions should be directed to michael.frame@yale.edu

 This material is based upon work supported by the National Science Foundation under Grant No. 0203203. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.