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1992 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Chaos and Fractals

New Frontiers of Science

verfasst von: Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe

Verlag: Springer New York

Enthalten in: Professional Book Archive

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Über dieses Buch

For almost 15 years chaos and fractals have been riding a wave that has enveloped many areas of mathematics and the natural sciences in its power, creativity and expanse. Traveling far beyond the traditional bounds of mathematics and science to the distant shores of popular culture, this wave captures the attention and enthusiasm of a worldwide audience. The fourteen chapters of this book cover the central ideas and concepts of chaos and fractals as well as many related topics including: the Mandelbrot Set, Julia Sets, Cellulair Automata, L- systems, Percolation and Strange Attractors. Each chapter is closed by a "Program of the Chapter" which provides computer code for a central experiment. Two appendices complement the book. The first, by Yuval Fisher, discusses the details and ideas of fractal images and compression; the second, by Carl J.G. Evertsz and Benoit Mandelbrot, introduces the foundations and implications of multifractals.

Inhaltsverzeichnis

Frontmatter
Introduction
Causality Principle, Deterministic Laws and Chaos
Abstract
For many, chaos theory already belongs to the greatest achievements in the natural sciences in this century. Indeed, it can be claimed that very few developments in natural science have awakened so much public interest. Here and there, we even hear of changing images of reality or of a revolution in the natural sciences.
Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe
Chapter 1. The Backbone of Fractals: Feedback and the Iterator
Abstract
When we think about fractals as images, forms or structures we usually perceive them as static objects. This is a legitimate initial standpoint in many cases, as for example if we deal with natural structures like the ones in figures 1.1 and 1.2.
Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe
Chapter 2. Classical Fractals and Self-Similarity
Abstract
Mandelbrot is often characterized as the father of fractal geometry. Some people, however, remark that many of the fractals and their descriptions go back to classical mathematics and mathematicians of the past like Georg Cantor (1872), Giuseppe Peano (1890), David Hilbert (1891), Helge von Koch (1904), Waclaw Sierpinski (1916), Gaston Julia (1918), or Felix Hausdorff (1919), to name just a few. Yes, indeed, it is true that the creations of these mathematicians played a key role in Mandelbrot’s concept of a new geometry. But at the same time it is true that they did not think of their creations as conceptual steps towards a new perception or a new geometry of nature. Rather, what we know so well as the Cantor set, the Koch curve, the Peano curve, the Hilbert curve and the Sierpinski gasket, were regarded as exceptional objects, as counter examples, as ‘mathematical monsters’. Maybe this is a bit overemphasized. Indeed, many of the early fractals arose in the attempt to fully explore the mathematical content and limits of fundamental notions (e.g. ‘continuous’ or ‘curve’). The Cantor set, the Sierpinski carpet and the Menger sponge stand out in particular because of their deep roots and essential role in the development of early topology.
Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe
Chapter 3. Limits and Self-Similarity
Abstract
Dyson is referring to mathematicians, like G. Cantor, D. Hilbert, and W. Sierpinski, who have been justly credited with having helped to lead mathematics out of its crisis at the turn of the century by building marvelous abstract foundations on which modern mathematics can now flourish safely. Without question, mathematics has changed during this century. What we see is an ever increasing dominance of the algebraic approach over the geometric. In their striving for absolute truth, mathematicians have developed new standards for determining the validity of mathematical arguments. In the process, many of the previously accepted methods have been abandoned as inappropriate. Geometric or visual arguments were increasingly forced out. While Newton’s Principia Mathematica, laying the fundamentals of modern mathematics, still made use of the strength of visual arguments, the new objectivity seems to require a dismissal of this approach. From this point of view, it is ironic that some of the constructions which Cantor, Hilbert, Sierpinski and others created to perfect their extremely abstract foundations simultaneously hold the clues to understanding the patterns of nature in a visual sense. The Cantor set, Hilbert curve, and Sierpinski gasket all give testimony to the delicacy and problems of modern set theory and at the same time, as Mandelbrot has taught us, are perfect models for the complexity of nature.
Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe
Chapter 4. Length, Area and Dimension: Measuring Complexity and Scaling Properties
Abstract
Geometry has always had two sides, and both together have played very important roles. There has been the analysis of patterns and forms on the one hand; and on the other, the measurement of patterns and forms. The incommensurability of the diagonal of a square was initially a problem of measuring length but soon moved to the very theoretical level of introducing irrational numbers. Attempts to compute the length of the circumference of the circle led to the discovery of the mysterious number π. Measuring the area enclosed between curves has, to a great extent, inspired the development of calculus.
Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe
Chapter 5. Encoding Images by Simple Transformations
Abstract
So far, we have discussed two extreme ends of fractal geometry. We have explored fractal monsters, such as the Cantor set, the Koch curve, and the Sierpinski gasket; and we have argued that there are many fractals in natural structures and patterns, such as coastlines, blood vessel systems, and cauliflowers. We have discussed features, such as self-similarity, scaling properties, and fractal dimensions shared by those natural structures and the monsters; but we have not yet seen that they are close relatives in the sense that maybe a cauliflower is just a ‘mutant’ of a Sierpinski gasket, and a fern is just a Koch curve ‘let loose’. Or phrased as a question, is there a framework in which a natural structure, such as a cauliflower, and an artificial structure, such as a Sierpinski gasket, are just examples of one unifying approach; and if so, what is it? Believe it or not, there is such a theory, and this chapter is devoted to it. It goes back to Mandelbrot’s book, The Fractal Geometry of Nature, and a beautiful paper by the Australian mathematician Hutchinson.2 Barnsley and Berger have extended these ideas and advocated the point of view that they are very promising for the encoding of images.3 In fact, this will be the focus of the appendix on image compression.
Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe
Chapter 6. The Chaos Game: How Randomness Creates Deterministic Shapes
Abstract
Our idea of randomness, especially with regard to images, is that structures or patterns which are created randomly look more or less arbitrary. Maybe there is some characteristic structure, but if so, it is probably not very interesting — like a box of nails poured out onto a table.
Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe
Chapter 7. Recursive Structures: Growing of Fractals and Plants
Abstract
The historical constructions of fractals by Cantor, Sierpinski, von Koch, Peano, etc., have been labeled as ‘mathematical monsters’. Their purpose had been mainly to provide certain counterexamples, for example, showing that there are curves that go through all points in a square. Today a different point of view has emerged due to the ground-breaking achievements of Mandelbrot. Those strange creations from the turn of the century are anything but exceptional counterexamples; their features are in fact typical of nature. Consequently, fractals are becoming essential components in the modeling and simulation of nature. Certainly, there is a great difference between the basic fractals shown in this book and their counterparts in nature: mountains, rivers, trees, etc. Surely, the artificial fractal mountains produced today in computer graphics already look stunningly real. But on the other hand they still lack something we would certainly feel while actually camping in the real mountains. Maybe it is the (intentional) disregarding of all developmental processes in the fractal models which is one of the factors responsible for this shortcoming.
Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe
Chapter 8. Pascal’s Triangle: Cellular Automata and Attractors
Abstract
Being introduced to the Pascal triangle for the first time, one might think that this mathematical object was a rather innocent one. Surprisingly it has attracted the attention of innumerable scientists and amateur scientists over many centuries. One of the earliest mentions (long before Pascal’s name became associated with it) is in a Chinese document from around 1303.1 Boris A. Bondarenko,2 in his beautiful recently published book, counts several hundred publications which have been devoted to the Pascal triangle and related problems just over the last two hundred years. Prominent mathematicians as well as popular science writers such as Ian Stewart,3 Evgeni B. Dynkin and Wladimir A. Uspenski,4 and Stephen Wolframs have devoted articles to the marvelous relationship between elementary number theory and the geometrical patterns found in the Pascal triangle. In chapter 2 we introduced one example: the relation between the Pascal triangle and the Sierpinski gasket.
Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe
Chapter 9. Irregular Shapes: Randomness in Fractal Constructions
Abstract
Self-similarity seems to be one of the fundamental geometrical construction principles in nature. For millions of years evolution has shaped organisms based on the survival of the fittest. In many plants and also organs of animals, this has led to fractal branching structures. For example, in a tree the branching structure allows the capture of a maximum amount of sun light by the leaves; the blood vessel system in a lung is similarly branched so that a maximum amount of oxygen can be assimilated. Although the self-similarity in these objects is not strict, we can identify the building blocks of the structure — the branches at different levels.
Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe
Chapter 10. Deterministic Chaos: Sensitivity, Mixing, and Periodic Points
Abstract
Mathematical research in chaos can be traced back at least to 1890, when Henri Poincaré studied the stability of the solar system. He asked if the planets would continue on indefinitely in roughly their present orbits, or might one of them wander off into eternal darkness or crash into the sun. He did not find an answer to his question, but he did create a new analytical method, the geometry of dynamics. Today his ideas have grown into the subject called topology, which is the geometry of continuous deformation. Poincaré made the first discovery of chaos in the orbital motion of three bodies which mutually exert gravitational forces on each other.
Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe
Chapter 11. Order and Chaos: Period-Doubling and its Chaotic Mirror
Abstract
Chaos theory began at the end of last century with some great initial ideas, concepts and results of the monumental French mathematician Henri Poincaré. Also the more recent path of the theory has many fascinating success stories. Probably the most beautiful and important one is the theme of this chapter. It is known as the route from order into chaos,or Feigenbaum’s universality.
Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe
Chapter 12. Strange Attractors: The Locus of Chaos
Abstract
Having discussed the phenomena of chaos and the routes leading to it in ‘simple’ one-dimensional settings, we continue with the exposition of chaos in dynamical systems of two or more dimensions. This is the relevant case for models in the natural sciences since very rarely can processes be described by only one single state variable. One of the main players in this context is the notion of strange attractors.
Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe
Chapter 13. Julia Sets: Fractal Basin Boundaries
Abstract
The goal of this chapter is to demonstrate how genuine mathematical research experiments open a door to a seemingly inexhaustible new reservoir of fantastic shapes and images. Their aesthetic appeal stems from structures which are beyond imagination and yet, at the same time, look strangely familiar. The ideas we present here are part of a world wide interest in so called complex dynamical systems. They deal with chaos and order, both in competition and coexistence. They show the transition from one condition to the other and how magnificently complex the transitional region generally is. One of the things many dynamical systems have in common is the competition of several centers for the domination of the plane. A single boundary between territories is seldom the result of this contest. Usually, an unending filigree entanglement and unceasing bargaining for even the smallest areas results. We studied the quadratic iterator in chapters 1, 10 and 11 and learned that it is the most prominent and important paradigm for chaos in deterministic dynamical systems. Now we will see that it is also a source of fantastic fractals. In fact the most exciting discovery in recent experimental mathematics, i.e., the Mandelbrot set, is an offspring of these studies. Now, about 10 years after Adrien Douady and John Hamal Hubbard started their research on the Mandelbrot set, many beautiful truths have been gained about this ‘most complex object mathematics has ever seen’. Almost all of this progress stems from their work.
Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe
Chapter 14. The Mandelbrot Set: Ordering the Julia Sets
Abstract
The Mandelbrot set is certainly the most popular fractal, probably the most popular object of contemporary mathematics at all. Some people claim that it is not only the most beautiful but also the most complex object which has been seen, i.e., made visible. Since Mandelbrot made his extraordinary experiment in 1979, it has been duplicated by tens of thousands of amateur scientists around the world.2 They all like to delve into the unlimited variety of pictures which can develop on a computer screen. Sometimes many hours are required for their generation; but this is the price you have to pay for the adventure of finding something new and fantastic where nobody has looked before.
Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe
Backmatter
Metadaten
Titel
Chaos and Fractals
verfasst von
Heinz-Otto Peitgen
Hartmut Jürgens
Dietmar Saupe
Copyright-Jahr
1992
Verlag
Springer New York
Electronic ISBN
978-1-4757-4740-9
Print ISBN
978-1-4757-4742-3
DOI
https://doi.org/10.1007/978-1-4757-4740-9