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This book focuses on the practical applications of fractal programming. Readers will learn how to implement the various logarithms, how to generate images using fractal and procedural techniques, and what practical uses there are for these techniques. Contains eight pages of full-color fractals and includes a disk with source code to create the images presented. ... Read more

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Third and final volume focuses upon the images of fractals that for many people have provided a compelling lure into an investigation of the intricate properties embedded within them. ... Read more

Customer Reviews (2)

Fractals for the Classroom, Book I
I was needing the book quickly and the seller sent it priority without charging me extra.Very Nice!!I used the book to do professional development with some high school teachers.

Chaos and fractals unleashed
If the words fractals, chaos and dynamical systems are attractive fixed points for your mind, go ahead. This is a book about this stuff. The three authors really unleash those topics, using numerous intriguing examples.The mathematical level is elementary. There is a good treatment bothmathematical and heuristic of the discrete dynamical system generated bythe logistic equation, this icon of the chaos and fractals theory. At eachchapter's end there is a program written in old-fashioned Basic language,but the skeleton may be recycled in, say, Mathematica or Matlab. ... Read more

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Great book 'bout fractals
simple but complete, this book is a very good introduction to the world of fractals

Random fractals
From the standpoint of deterministic dynamics and multifractals defined by a generating partition, I do not like this book. However, for affine fractals and methods like R/S analysis it provides a very nice introduction! Like Hull's book on options, stochastic calculus is unfortunately not formulated using Ito calculus. Also nice, the gambler's ruin is presented as an example of affine scaling. What is missing is an introductory discussion of multiaffine scaling, which is of interest for soft turbulence in fluids (an introduction to multiaffine scaling with simple examples can be found in the book on surface fluctuations by Barabasi and Stanley). ... Read more

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Published in cooperation with the National Council of Teachers of Mathematics, this text is based on lectures given to various communities of teachers and students. Written especially for teachers and intended for classroom use at high school and college levels, the book also can be used for independent study. ... Read more

Customer Reviews (1)

Fractals for the Classroom
Actually, this is a pretty good book, but if you are seeking a simple way to understand and begin to understand Fractals, this book still require a good amount of math knowledge.I have yet to find a cogent, simple book for the curious, but not math minded that is not so "dumbed" down to the point of just showing pretty pictures. ... Read more

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This book will allow you to travel through time and space. To facilitate your journey, the editor has scoured the four corners of the earth in a quest for unusual people and their fascinating patterns. From Mozambique, to Asia, to many European countries, the contributors to The Pattern Book include world-famous cancer researchers, little-known artists and eclectirc computer program-mers. Some of the patterns are ultramodern, while others are centuries old. Many of the patterns are drawn from the universe of mathematics. Computer recipes are scattered throughout.

Although the emphasis is on computer-generated patterns, the book is informal and the intended audience spans several fields. The emphasis is on the fun that the true pattern lover finds in doing, rather than in reading about the doing! The book is organized into three main parts: Representing Nature (for those patterns which describe or show real physical phenomena, e.g., visualizations of protein mo!tion, sea lilies, etc.), Mathematics and Symmetry (for those patterns which describe or show mathematical behavior, e.g. fractals), and Human Art (for those patterns which are artistic works of humans and made without the aid of a computer, e.g. Moslem tiling patterns.) ... Read more

Customer Reviews (2)

Better for Mathematicians
As an artist, I had hoped this book would have presented, on equal footing, mathematics and natural/artistic elements. I ordered it based upon a review that reflected such, but was a bit disappointed to find it was definitely more formulaic based.However, that having been said, there were a few good things I extracted from this book, just not as much as I would have liked based upon the review. Would I recommend this book to a fellow artist? Likely not. To a math/science colleague...definitely. I think Gombrich is a better bet for the artist.

A natural first book for one interested in fractals
Fractals can be found everywhere, and Clifford Pickover has done a good job in selecting some of the best. The range of the collection is considerable, from "Wood Pattern" to "Trajectories of a Neural Network Quantizer in Rhythm Space." All possess a simplistic beauty that justifies the presence of the word art in the title.
Such creations, possessing enormous complexity from very simple origins, truly forces us to stretch our imaginations. All of the generation processes are very easy to understand, and some modifications for additional exploration are obvious.
One more in the collection of works devoted to fractals, this book ranks among the easiest to understand. A natural first book for those interested in fractals as objects of mathematical study or art.

Published in Journal of Recreational Mathematics, reprinted with permission. ... Read more

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The theory of fractals has been used to describe complicated shapes in nature, such as clouds or trees, by introducing a new quantity called the fractal dimension. This revolutionary theory has been applied in many fields and is being compared to Newton's theory of mechanics. This textbook begins with an elementary introduction to fractals, and goes on to deal with more sophisticated problems, identifying a number of examples of fractals in nature and explaining how to identify fractal patterns. Computer simulated fractals are also introduced with short programs for use on personal computers. For the more advanced reader, the text also deals with theoretical topics such as chaos, percolation, statistical models of turbulence and renormalization group methods. ... Read more

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This 1998 book describes the progress that had been made towards the development of a comprehensive understanding of the formation of complex, disorderly patterns under conditions far from equilibrium. The application of fractal geometry and scaling concepts to the quantitative description and understanding of structure formed under non-equilibrium conditions is described. Self-similar fractals, multi-fractals and scaling methods are discussed, with examples, to facilitate applications in the physical sciences. Computer simulations and experimental studies are emphasised, but the author also includes discussion of theoretical advances in the subject. Much of the book deals with diffusion-limited growth processes and the evolution of rough surfaces, although a broad range of other applications is also included. The techniques and topics will be relevant to graduate students and researchers in physics, chemistry, materials science, engineering and the earth sciences, interested in applying the ideas of fractals and scaling. ... Read more

Customer Reviews (1)

useful for material research
While fractals have proven of great use in computer graphics, this text instead looks at how to apply these ideas to physical observations. There is an emphasis on diffusion limited growth processes, as on solid surfaces, for example.

Fluid phenomena are also studied. Like the formation of viscous fingers between two fluids. Also of interest is how Meakin shows that long range persistence can arise out of localised interactions. Considerable space is devoted to studying diffusion limited aggregation, because of the importance of this model in many systems.

Overall, quite a worthy book for a materials researcher. The math and computer simulations is well developed but clearly understandable. ... Read more

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University of Washington, Seattle.Physiological research on fractals and chaos. Fractals are objects or processes with multiple-scale properties. For physiologists or mathematicians. ... Read more

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Customer Reviews (4)

Science and art, together again
I was aware of fractals for some time and after watching a NOVA film "Hunting the Hidden Dimension" (2008). I went hunting for some of the many books by Benoit Mandelbrot. You can imagine my surprise with I tripped over this book, "An Eye for Fractals" by Michael McGuire.

Stacked up against other fractal works, this book has more stunning pictures both in quantity and quality. Even though the formulas are a tad complex with number lines and complex planes, the ideas on how to make and see fractals are quite clear.

If we were not enamored with fractal the book would stand on its own as great monochrome depictions of nature and monuments. There is one picture by Ansel Adams and the rest are by M McGuire. If you did not see the titles I challenge you to find the Adams picture.

Every time I read this book I get a different insight into the world of fractals and believe you will also.

Fractals: Hunting the Hidden Dimension (Blu-ray) [Blu-ray]

A good primer on fractals and an excellent introduction to their appearance in nature
Looking around in nature, it is easy to find examples of fractals. Individual plants, forests, volcanic activity, wind and water erosion, the flow of water and the movement of plants with the wind can all be described by an appropriate fractal function. Most of the pages of this book are taken up with spectacular images of such natural fractals. A few of the pages are devoted to the mathematical background used to generate the images. It is not rigorous; it was the author's intention to only give some of the general ideas rather than a complete description. They are understandable, even if your math background stops at algebra. Although it was written in 1991, this book is still an excellent primer on what fractals are as well as the large number of ways that they appear in nature.

beautiful BW photographs
I just got this book from the library. It many well printed BW photographs. There is some math discussion on fractals but there is no discussion of computer software for generating fractals pictures. Still a great book.

A beautiful journey through math and nature
One of the only fractal books available for non-mathematicians, An Eye forFractals interweaves beautiful photographs of natural scenes withexplanations of their fractal geometric characteristics.It gives enoughinformation about fractals for a beginner to understand what they are andhow they apply to nature.Its photography makes it a perfect book to leaveout on the coffee table.

Fractal geometry is revolutionizing the way thatscientists perceive the universe and its underlying order.Anyone canbenefit greatly from this understanding by reading this book. ... Read more

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Fractal structures are found everywhere in nature, and as a consequence anomalous diffusion has far reaching implications in a host of phenomena. This book describes diffusion and transport in disordered media such as fractals, porous rocks and random resistor networks. Part I contains material of general interest to statistical physics: fractals, percolation theory, regular random walks and diffusion, continuous time random walks and Levy walks, and flights. Part II covers anomalous diffusion in fractals and disordered media, while Part III serves as an introduction to the kinetics of diffusion-limited reactions. Part IV discusses the problem of diffusion-limited coalescence in one dimension. This book will be of particular interest to researchers requiring a clear introduction to the field. It will also be a valuable source to graduate students studying in areas of physics, chemistry, and engineering. ... Read more

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A fantastic introduction to the field...
I am a graduate student in statistical physics with a primary interest in diffusion-limited reactions. I have found the above book to be invaluable, so much so that I'm disappointed that I didn't know about it earlier, as I'm sure it would have accelerated my understanding. It is, in my opinion, the best introduction to the field of diffusion-limited reactions that I'm aware of and certainly makes the research literature in this area much more accessible. I have also found the other sections of the book to be particularly illuminating and essential background reading, with Part I covering many of the principles of statistical physics. Each chapter contains a list of exercises to enhance understanding of the material and a list of "Open Challenges" which provoke new avenues of research. I personally think that the book references the research literature extremely well (indeed the authors have themselves made substantial contributions to the field) which makes it easy to pursue further details on areas of interest. I regard the book as essential for anyone wishing to develop an understanding of anomalous diffusion in disordered media and to be particularly accessible to graduate students and young researchers. It has certainly made it easier for me to make the transition from student to researcher in this area of statistical physics. ... Read more

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The first instance of pre-computer fractals was noted by the French mathematician Gaston Julia. He wondered what a complex polynomial function would look like, such as the ones named after him (in the form of z2 + c, where c is a complex constant with real and imaginary parts). The idea behind this formula is that one takes the x and y coordinates of a point z, and plug them into z in the form of x + i*y, where i is the square root of -1, square this number, and then add c, a constant. Then plug the resulting pair of real and imaginary numbers back into z, run the operation again, and keep doing that until the result is greater than some number. The number of times you have to run the equations to get out of an 'orbit' not specified here can be assigned a colour and then the pixel (x,y) gets turned that colour, unless those coordinates can't get out of their orbit, in which case they are made black. Later it was Benoit Mandelbrot who used computers to produce fractals. A basic property of fractals is that they contain a large degree of self similarity, i.e., they usually contain little copies within the original, and these copies also have infinite detail. That means the more you zoom in on a fractal, the more detail you get, and this keeps going on forever and ever.The well-written book 'Getting acquainted with fractals' by Gilbert Helmberg provides a mathematically oriented introduction to fractals, with a focus upon three types of fractals: fractals of curves, attractors for iterative function systems in the plane, and Julia sets. The presentation is on an undergraduate level, with an ample presentation of the corresponding mathematical background, e.g., linear algebra, calculus, algebra, geometry, topology, measure theory and complex analysis. The book contains over 100 color illustrations. ... Read more

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Fractal Geometry in Biological Systems was written by the leading experts in the field of mathematics and the biological sciences together. It is intended to inform researchers in the bringing about the fundamental nature of fractals and their widespread appearance in biological systems. The chapters explain how the presence of fractal geometry can be used in an analytical way to predict outcomes in systems, to generate hypotheses, and to help design experiments. The authors make the mathematics accessible to a wide audience and do not assume prior experience in this area. ... Read more

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Historically, for metric spaces the quest for universal spaces in dimension theory spanned approximately a century of mathematical research. The history breaks naturally into two periods - the classical (separable metric) and the modern (not-necessarily separable metric).

The classical theory is now well documented in several books. This monograph is the first book to unify the modern theory from 1960-2007. Like the classical theory, the modern theory fundamentally involves the unit interval.

Unique features include:
* The use of graphics to illustrate the fractal view of these spaces;
* Lucid coverage of a range of topics including point-set topology and mapping theory, fractal geometry, and algebraic topology;
* A final chapter contains surveys and provides historical context for related research that includes other imbedding theorems, graph theory, and closed imbeddings;
* Each chapter contains a comment section that provides historical context with references that serve as a bridge to the literature.

This monograph will be useful to topologists, to mathematicians working in fractal geometry, and to historians of mathematics. Being the first monograph to focus on the connection between generalized fractals and universal spaces in dimension theory, it will be a natural text for graduate seminars or self-study - the interested reader will find many relevant open problems which will create further research into these topics.

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Fractals are shapes in which an identical motif repeats itself on an ever diminishing scale. A coastline, for instance, is a fractal, with each bay or headland having its own smaller bays and headlands--as is a tree with a trunk that separates into two smaller side branches, which in their turn separate into side branches that are smaller still. No longer mathematical curiosities, fractals are now a vital subject of mathematical study, practical application, and popular interest. For readers interested in graphic design, computers, and science and mathematics in general, Hans Lauwerier provides an accessible introduction to fractals that makes only modest use of mathematical techniques. Lauwerier calls this volume a "book to work with." Readers with access to microcomputers can design new figures, as well as re-create famous examples. They can start with the final chapter, try out one of the programs described there (preferably in a compiled version such as TURBO BASIC), and consult the earlier chapters for whatever is needed to understand the fractals produced in this way. The first chapter, which builds on the relationship of binary number systems to the "tree fractal" described above, is the best place to start if one has no computer. There will be much to enjoy on the way, including the beautiful color illustrations. ... Read more

Customer Reviews (7)

Short, packed with information, math backround needed
If you want to read this book, there are a couple of recommendations from me:

First, review your math: number systems, mods, logarithms, trigonometry, functions.
Second, prepare yourself for a book that is packed with information in each line. Don't expect even a line skip unnecessary.
Third, if you want to make a full use of book, don't read it and put it aside. You have to bear with the author and work out the examples. These two facts, combined with your willing to analyze the code algorithms will make you learn the fractals -relatively- deeply.

The bonus fact is that the authour explains how to create your own fractals in the last chapter.

As "the cons" I can say that the turbo basic programs are outdated. They need a good revision, possibly a port to C, Java (or maybe Ruby for the fans). In my opinion, a clean C code would do the trick.

Finally here is the chapter list:

i. Preface
ii. Acknowledgements
iii. Introduction

1. Counting and Number Systems
2. Numbers and Points
3. Meanders and Fractals
4. Spirals, Trees and Stars
5. The Analysis of a Fractal
6. Chance in Fractals
7. Poincare, Julia, Mandelbrot
8. Making Your Own Fractals

Appendix A. Complex Numbers
Appendix B. Programs
Bibliography
Index

For people seeking to program Fractals or Chaos
This is a great book. Only until you work with it will you find how good it is. My favorite thing in this book is what the author calls contraction mirroring and is discussed in chapters 4,5,6,8.

A Classic of Fractals
In all my library of fractal books this one stands out as the most enlightening and the most useful. Hans Lauwerier is a master of Chaos and fractal theory. His method of analysis of IFS fractals is the best. He is just publishing a new book that should be rewarding as well.

A Excellent Introduction to Fractals
This book is nicely written, well-organized and beautifully illustrated. It introduces most of the standard topics with a minimum of math, for example, iterated function systems, chaos, Mandelbrot and Julia sets, and random fractals. Among introductory semi-formal treatments of fractals I have seen, it strikes the best balance between concision, simplicity, and mathematical detail.

However, this somewhat dated volume needs a revision to upgrade the code from Basic to, say, Java. When the book was first published, microcomputers were relatively weak. Consequently, the book makes a few digressions into some rather involved algorithms designed to minimize memory use. Of course, today's machines are much more powerful. It is a lot simpler to use recursion (although this uses up memory liberally) in the fractal programs.

Finally, I think that the geometry could be made conceptually cleaner by mentioning that a general similitude (of which a contraction mapping is one example) on the plane can be written as a composition of rotations, translations, reflections, and scalings.

For more substantial treatments of fractals that don't demand too much math background, see "Fractals Everywhere" by M. Barnsley and "Introduction to Fractals and Chaos" by R. Crownover. However, one should read Lauwerier's slim and elegant volume before and after studying these more advanced works--before, as an introduction, and after, as a delightful summary and "bird's eye view" of the subject.

Very nice book...short but packed full of information
This is a nice book that will start you on the wonderful world of fractals. Contains BASIC source code for you to try. Very informative, you'll learn about the history of fractals and shows you the many differentideas and mathematical insights about fractals. This is really a goodstarter book (though you need background in algebra and trigonometry tofollow the math equations). ... Read more

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Fractal Cities is the pioneering study of the development and use of fractal geometry for understanding and planning the physical form of cities, showing how this geometry enables cities to be simulated throughcomputer graphics. The book explains how the structure of cities evolve in ways which at first sight may appear irregular, but when understood in terms of fractals reveal a complex and diverse underlying order. The book includes numerous illustrations and 16 pages full-color plates of stunning computer graphics, along with explanations of how to construct them. The authors provide an accessible and thought-provoking introduction to fractal geometry, as well as an exciting visual understanding of the formof cities. This approach, bolstered by new insights into the complexity of social systems, provides one of the best introductions to fractal geometry available for non-mathematicians and social scientists.
Fractal Cities is useful as a textbook for courses on geographic information systems, urban geography, regional science, and fractal geometry. Planners and architects will find that many aspects of fractal geometry covered in this book are relevant to their own interests. Those involved in fractals and chaos, computer graphics, and systems theory will also find important methods and examples germane to their work.
Michael Batty is Director of the National Center for Geographic Information and analysis in the State University of New York at Buffalo, and has worked in planning theory and urban modeling. Paul Longley is a lecturer in geography at the University of Bristol, and is involved in the development of geographic information systems in urban policy analysis.

Richly illustrated, including 16 pages of full-color plates of brilliant computer graphics
Provides an introduction to fractal geometry for the non-mathematician and social scientist
Explains the influence of fractals on the evolution of the physical form of cities ... Read more

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New developments in computer science, biology, mathematics and physics offer possibilities to obtain deeper understanding of growth and forms of organisms. It is now possible to carry out simulation experiments in which the growth process can be simulated in virtual computer objects. In this book, methods from fractal geometry are applied to model growth forms. As a case study, a type of growth process is used which can be found among various taxonomic classes of organisms such as sponges and corals. The growth of these organisms is simulated with 2D and 3D geometrical objects. The models presented in the book provide a rendering method for natural objects which is based on the actual growth process. The models can be used, for example, to understand the amazing variety of forms to be found in a coral reef. Models which mimic the growth of forms and the environmental influence on the growth process are also useful for ecologists. A combination of simulation models and the actual growth forms can be used to detect the effects of slow changes in the environment. ... Read more

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An in-depth survey of the design and REALIZATIONS of miniaturized fractal microwave and RF filters

Engineers are continually searching for design methods that can satisfy the ever-increasing demand for miniaturization, accuracy, reliability, and fast development time. Design and Realizations of Miniaturized Fractal RF and Microwave Filters provides RF and microwave engineers and researchers, advanced graduate students, and wireless and telecommunication engineers with the knowledge and skills to design and realize miniaturized fractal microwave and RF filters. This book is an essential resource for the realization of portable and cellular phones, WiFi, 3G and 4G, and satellite networks.

The text focuses on the synthesis and fabrication of miniaturized fractal filters with symmetrical and asymmetrical frequency characteristics in the C, X and Ku bands, though applications to other frequency bands are considered. Readers will find helpful guidance on:

• Miniaturized filters in bilevel fashion

• Simplified methods for the synthesis of pseudo-elliptic electrical networks

• Methods for extracting coupling coefficients and external quality factors from simulations of the RF/microwave structure

• Methods for matching theoretical couplings to couplings of structure

Including studies of the real-world performance of fractal resonators and sensitivity analyses of suspended substrate realizations, this is a definitive resource for both practicing engineers and students who need timely insight on fractal resonators for compact and low-power microwave and RF applications. ... Read more

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This book covers analysis on fractals, a developing area of mathematics that focuses on the dynamical aspects of fractals, such as heat diffusion on fractals and the vibration of a material with fractal structure. The book provides a self-contained introduction to the subject, starting from the basic geometry of self-similar sets and going on to discuss recent results, including the properties of eigenvalues and eigenfunctions of the Laplacians, and the asymptotical behaviors of heat kernels on self-similar sets. Requiring only a basic knowledge of advanced analysis, general topology and measure theory, this book will be of value to graduate students and researchers in analysis and probability theory. It will also be useful as a supplementary text for graduate courses covering fractals. ... Read more

Customer Reviews (1)

Metrics, fractals, analysis...
Fractals make headlines from time to time[--are they everywhere?], and and they make beautiful color pictures; but they are also part of a substantial mathematical theory, even one with an exciting mathematical history. This lovely book presents the subject elegantly in a way that it can be taught to students. It starts with the basics, then systematically, step by step, it builds up the central results. Great for the classrom, or for selfstudy! In view of the many applications to geometric analysis, to PDE, and to statistics, it is likely that fractal geometry will soon be a standard math course taught in many (more) math departments. Example: By now it is widely recognized that the selfsimilarity aspects of the wavelet algorithms are key to their sucess. Compared to other books at the same level, for example, by Falconer, 1990, and the equally attractive one from 1985[Falconer: The geometry of fractal sets], Kigami's book stresses the analysis of the Laplace operator. Falconer covers the theory more generally, and his books have a slightly more potential theoretic bent. ... Read more

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Fractal geometry is revolutionizing the descriptive mathematics of applied materials systems. Rather than presenting a mathematical treatise, Brian Kaye demonstrates the power of fractal geometry in describing materials ranging from Swiss cheese to pyrolytic graphite. Written from a practical point of view, the author assiduously avoids the use of equations while introducing the reader to numerous interesting and challenging problems in subject areas ranging from geography to fine particle science. The second edition of this successful book provides up-to-date literature coverage of the use of fractal geometry in all areas of science.

From reviews of the first edition:

'...no stone is left unturned in the quest for applications of fractal geometry to fine particle problems....This book should provide hours of enjoyable reading to those wishing to become acquainted with the ideas of fractal geometry as applied to practical materials problems.' MRS Bulletin ... Read more