"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.
| 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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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11. Laboratory Exercises is a collection of field-tested extended hands-on activities that illustrate
many of the topics on these pages. |
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12. Lesson Plans is a collection of lesson plans for high school and middle school
classes. |
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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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