Fractals In Your Future - Educational Math Content - A Primer on the Theory and Applications of Fractals and Chaos

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Fractals In Your Future A Primer on the Theory and Applications of Fractals and Chaos
Written by Ron Lewis 204 Simon Lake Drive Box 175 Naughton, Ontario Canada P0M 2M0 (011) 1-705-692-4592 ronlewis@sympatico.ca Table of Contents
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Table of Contents Foreword Introduction: "...to See the World in a Grain of Sand..."; A Question of Scale Part A: Fractal Geometry Chapter 1: How Long is a Coastline? The Structured Walk Chapter 2: Snowflakes and Other Infinite Curves Chapter 3: Sierpinski Carpets and Other Almost Surfaces Chapter 4: Cantor Dust. Coin Tosses and Fuzzy Points Chapter 5: DLA Growth: Electrolytic Trees and Viscous Fingers Chapter 6: Pixel Rainbows and the Mandelbrosian Primal Egg Part B: Order in Random Chaos Chapter 7: Randomwalks and Bell Curves Chapter 8: Log-Normal Behaviour: Climbing, Sliding, Leaping, Walking, and Avoiding Oneself Chapter 9: Poisson Distributions: Chaos in the Candy Store and Other Rare Events Chapter 10: Buffon's Needle: PI Without a Circle Part C: Unexpected Chaos in Orderly Systems Chapter 11: Attractors and Strange Attractors Chapter 12: Adventures in Phase Space Chapter 13: The Rise and Fall of the Tent Caterpillar and Other Dynamical Systems Chapter 14: A Call to Order: Cascades of Bifurcations Part D: Appendix Chapter 15: Computer Alternatives for Analyzing Boundary Fractals: Equipace, Pixel Sausage, and Box Counting Chapter 16: Korcak Fractals and Other Shattering Experiences
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Foreword This book is an invitation to share the utter joy of fractals and chaos, two new concepts of the twentieth century, whose onset marks a revolution in human thought. The tremendous current interest in these subjects is manifest in the numerous articles devoted to them in popular magazines, professional journals and books, and more recently, television and conferences on education. It seems that anyone who has seen a fractal becomes instantly curious, even fascinated. And everyone identifies with chaos. The ranks of those who have never heard of either are shrinking dramatically. Not long ago anyone who knew you taught fractals and chaos used to ask: "What are they, anyway?" Now the question would more likely be: "Are they of any practical value?", with a hint that probably they are not. Such reactions are understandable at this point and time. Most people's first exposure to fractals is through pictures or movies of Mandelbrot sets, so naturally they form an impression that fractals are about beautiful computer images of infinite colour and structure. And they are. But fractals are much more! When Dr. Brian Kaye, renowned physicist in the field of fineparticles research at Laurentian University, who was then writing his book A Random Walk Through Fractal Dimensions, offered to involved high school students in programs he was developing on fractal geometry, I jumped at the opportunity. Actually, I took one small step to the telephone, which turned out to be a giant leap for my students and me. I still do not know how I convinced the first small group of gifted and talented students at Nickel District Secondary School that something very special, about which I knew virtually nothing, was about to happen in Dr. Kaye's lecture room. The next thing I can recall, we were seated with a professor of geology, a professor of astronomy, two engineers from the International Nickel Company, and some university students who were taking the course for credit. All of us were spell-bound. After a few weeks, we were still not sure exactly what a fractal was, mathematically, but several students had already immersed themselves in projects, observing, measuring, and growing fractals. Over the next six semesters, six projects produced by these students were to win first prizes at school or regional science fairs. Whatever fractals were, they were as real as they were fascinating. It was student enthusiasm for fractals which motivated me to channel my energy into this pilot project. The Sudbury Board of Education then granted me a sabbatical leave to put together a text and laboratory book that would allow others to share our experiences. As my students were aware, the experiments they were doing comprised new science, real research. Most of the 186 experiments of the book are creations of Dr. Kaye. The words are mine. Much of the data, and some of the diagrams and computer programs were produced by students.
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Introduction ". . . to See the World in a Grain of Sand . . . "; a Question of Scale In D'Arcy Thompson's classic book On Growth and Form he discusses how every creature has a right shape and scale for its type of existence. The fleas of a dog do not resemble the dog, and, for that matter, goose bumps do not resemble geese. Instead, fleas are suitably constructed for their own lifestyle. An elephant could not be as small as a mouse without fundamental changes in its physiology, amounting to becoming more mouse-like. Even in engineering, a bridge design could not be rescaled without significant modifications of proportions. Numerous examples hint at a universal law of scale, namely, a right size for everything. Recent developments declare that long standing mathematical models of nature's geometry were quite incomplete and inadequate. Specifically, most shapes found in nature are not circles, rectangles nor other simple curves from the classical geometry of Euclid (c.300 B.C.) whose theorems still find a place in today's geometry texts. In 1975 a profound new book changed the way we shall see nature. Using shapes that had been called "pathological" because of their maverick properties, the new geometry gives us fresh paradigms for understanding patterns that had formerly been portrayed as merely rough or irregular, but which include most natural forms. The English version of this book, which appeared in 1977 was Fractals: Form, Chance and Dimension, by Benoit B. Mandelbrot. His work is widely considered one great intellectual leaps in the history of thought. Since its release, many scientists, such as Brian H. Kaye of Laurentian University, have found that fractal geometry is not only a descriptive device but is also a powerful tool for prying secrets from a wide variety of systems and solving important problems in applied science. The list of known physical fractal systems is long and growing rapidly. Here are just a few examples: Brownian tracks, coastlines, clouds, cauliflower, clusters of galaxies or seismic events, craters on the moon, dust, felt filters, feathers, ferns, and forest fires, mountains, noise, oil deposits, rivers, rock fractures, soot, sneezes, and snowflakes, turbulent eddies, trees, the heart, the lung, the nervous system. By contrast we can see that a fractal object has not a classical Euclidean geometric shape. The fractal is more complex, but not less elegant. What is the definition of a fractal object? First, a fractal is rugged. The ruggedness of a fractured rock found on the surface of an inner planet or on some moon of a larger planet is a frozen image reflecting the complexity both of its internal structure and of forces acting at different scales simultaneously. It may not be possible to tell whether the picture below is a rendering of a landscape or a photomicrograph of a particle of moon dust. This dilemma is based on perhaps the most profound and startling property of a fractal, which is that on close inspection each part of the object resembles the object as a whole, often through several orders of magnitude. This property, called "self-similarity," is sometimes strikingly obvious, and usually at least more or less apparent; moreover, it is quite common in nature. Thirdly, fractals usually have "fractional dimensions," as we shall see! Fractals seem to be formed where complexity, such as the random motion of atoms, is constrained by simple forces. This is the duality of chaos and order.
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Chapter 1 How Long Is A Coastline? The Structured Walk Purpose 1) To recognize the "fractal" nature of rugged boundaries. 2) To introduce a manual method for analyzing the fractal structure of rugged boundaries. 3) To create a working definition of the concept of "fractal dimension." 4) To review a number of systems which can be considered as "boundary fractals" or "fractal profiles." Background: The Length of a Coastline Most physical systems of nature and many human artifacts are not regular geometric shapes of the standard geometry derived from Euclid. When Benoit B. Mandelbrot answered the now famous question "How long is the coastline of Great Britain?" He was pointing the way to the study of rugged systems which before his time were inaccessible to mathematics. The type of answers Mandelbrot suggests, "It all depends . . ." or "Infinite," are hardly what one would expect from a geometry book. They are, however, not only appropriate but also very revealing. The length of the coastline of Great Britain can be found in Encyclopedia Britannica (1960), as equivalent to 3760 km, although the three sides of the triangle which outline the coast total only about 1630 km. Evidently, the coastline is very rugged. But the length stated in the encyclopedia is meaningful only if we know how it was determined. This point turns out to be surprisingly crucial in the discussion, as we shall see. Now, consider a statistic from "The World Almanac and Book of Facts, 1994." Here we find that the length of the general outline of the United States seacoast is about 19 900 km, but specifies that the unit of measure was of latitude, which is roughly 56 km. Without this stipulation of the survey unit, however, the coastal length statistic would be meaningless. Even assuming that the scale of this measurement had been chosen appropriately for the apparent ruggedness of this coastline, so as not to miss much in stepping across prominent bays and peninsulas, the answer would be different if the scale of measurement were finer, say 0.5 km, and very much different again if measured with a metre stick. Of course the perimeter of a coastline would be larger at finer scales, because it includes rugged details of the coastline left out on coarser scales. Suppose that some students decided to enter a book of records by using dental floss to explore every nook and cranny of the coastline, visible without the aid of a microscope. But then, what of the microscopic details; can they be properly ignored? Or is any such discrepancy really worth considering? Again, consulting the Almanac, we find that the "shoreline" of the United States, including features of a scale of roughly 30 m, is an astounding 143 000 km! a) Assuming that naval maps could be even more detailed than this, how might the navy's estimate compare to the latter value? Perhaps you see what Mandelbrot was getting at. An ocean's breadth away from Great Britain, on the Atlantic coast of Canada, is found another charming island, called Cape Breton. Three polygons are used below to estimate its perimeter. Notice that the polygons differ in lengths of sides, but also in number of sides. If we use a graph to record the lengths of the coastline obtained using the different measuring sticks we find that the points fit a curve known mathematically as a hyperbola. This mathematical curve has one very remarkable property, namely that as you move from right to left along the curve it rises to infinity. In terms of measuring a coastline this means that as we attempt to examine the details of the coastline with ever finer resolution, the values we get for its perimeter will eventually surpass any estimate! Experiments will show that the perimeter of a rugged object tends to infinity, thus confirming Mandelbrot's concept that the length of the coastline is indeterminate; there is no definite numerical answer. [chapter 1 continues in the book]

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