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In this book, the authors take the viewpoint that number theory and fractal geometry can be fruitfully combined. They study, in particular, the vibrations of fractal strings (one-dimensional drums with fractal boundary) and zeros of zeta-functions.

In earlier publications on fractal and spectral geometry, the Riemann Hypothesis was studied and this hinted at the notion of complex dimension as a means to describe certain geometric properties of a fractal, such as its fractal (Minkowski) dimension or the oscillations in the volume of its tubular neighborhoods. This notion of complex dimension is now precisely defined in this book.

A central problem in contemporary mathematics-often expressed as "Can one hear the shape of a drum?" -- consists in describing the relationship between the shape (geometry) of a drum and its sound (its spectrum). In the case of fractal strings, the complex dimensions provide a unified description of the oscillations in the geometry and the spectrum. This description is provided by an explicit formula -- an analytical tool, originally developed for the proof of the Prime Number Theorem, which is extended here to apply to the zeta-functions associated with fractals.

The context of vibrating fractal strings enables the authors to put the Riemann Hypothesis in a geometric setting. This famous conjecture states that the zeros r in the critical strip 0 {\leq} Re {\rho} {\leq} 1 of the Riemann zeta-function all lie on the critical line Re {\rho} = . Here, this conjecture becomes an inverse spectral problem, and its interpretation in the language of fractal strings, which have complex dimensions with real part between 0 and 1, is "One can hear if a fractal string is Minkowski measurable provided that its fractal dimension is not ".

In the more restricted context of fractal Cantor strings, the complex dimensions of which form an infinite vertical arithmetic progression, the inverse spectral problem gets an affirmative answer. The number-theoretical interpretation of this insight is that the Riemann zeta-function does not have an infinite vertical arithmetic progression of zeros. This result is generalized to apply to many other zeta-functions.

This highly original, self-contained monograph will appeal to geometers, fractalists, mathematical physicists, and number theorists, as well as to graduate students in these fields. ... Read more

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Fractal sets serve as excellent models in physics, chemistry, biology and other fields. However, the actual applications of these models are often unknown even to specialists in fractal theories. The communication of recent results utilizing fractal concepts in experimental and industrial engineering research was thus a major challenge to the above conference. The resulting proceedings which are published here are organized around five main themes: materials processing, flow, natural sciences, signal and image analysis and mathematical methods. Almost 50 papers are included in this volume. Besides the classical topics, the book introduces an important session on fractal signal and image analysis, a subject which is rapidly gaining recognition in computer science and engineering for its promising industrial applications. ... Read more

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Fractal analysis research is expanding into a variety of engineering domains. The strong potential of this work is now beginning to be seen in important applications in real industrial situations. Recent research progress has already led to new developments in domains such as signal processing and chemical engineering, and the major advances in fractal theory that underlie such developments are detailed here. New domains of applications are also presented, among them environmental science and rough surface analysis. Sections include multifractal analysis, iterated function systems, random processes, network traffic analysis, fractals and waves, image compression, and applications in physics. Fractals in Engineering emphasizes the connection between fractal analysis research and applications to industry. It is an important volume that illustrates the scientific and industrial value of this exciting field. ... Read more

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Written for mathematicians, engineers, researchers in experimental science, and anyone interested in fractals, this book presents the fundamentals of curve analysis with a new and clear introduction to fractal dimension. It explains the geometrical and analytical properties of trajectories, aggregate contours, geographical coastlines, profiles of rough surfaces, and other curves of finite and fractal length. The approach is through precise definitions from which properties are deduced and applications and computational methods are derived. Written without the traditional heavy symbolism of mathematics texts, this book requires two years of calculus as a prerequisite to understanding. This text also contains material appropriate for graduate coursework in curve analysis and/or fractal dimension. ... Read more

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Owing to the rapid emergence and growth of techniques in the engineering application of fractals, it has become necessary to gather the most recent advances on a regular basis. This book is a continuation of the first volume - published in 1997 - but contains interesting developments. A major point is that mathematics has become more and more involved in the definition and use of fractal models. It seems that the time of the qualitative observation of fractal phenomena has gone. Now the main models are strongly based upon theoretical arguments. Fractals: Theory and Applications in Engineering is a multidisciplinary book which should interest every scientist working in areas connected to fractals. ... Read more

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The nine review articles in this book give a fairly completesurvey of theapplications of fractals to astronomical andastrophysical problems. Thebook is self-contained andwritten with the particular aim of teachinggraduatestudents how to apply fractal geometry to new problems.After ageneral introduction to the basic concepts,applications are given to astrophysical dynamical systems,to strange attractors of stellar pulsations, to the solargranular pattern, and to both the large-scalestructure ofthe universe - a central theme of the book - and toastronomical data analysis. ... Read more

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This volume contains the proceedings of the first IFIP conference on fractals. The mathematical concept of fractal was coined by B.B. Mandelbrot in the early seventies to characterize objects with self-similar structures at all levels. Now the field of fractals is being subject to intensive research and an extensive growth of applications in many sciences. This first IFIP conference, intended as a start of periodic conferences on fractals, enables a productive cross fertilization of ideas among specialists in fractals and in information processing. In addition to the volume's 44 colour illustrations, on page v is a rare view of fractals and the Portuguese architecture at the time of the navigators, produced by Master Lima de Freitas, a leading Portuguese artist and member of the conference organizing committee. The design illustrates the symbiosis of mathematics and human artifacts represented by the real world. ... Read more

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Proceedings of the International Conference on Fractals held in Denver, U.S.A. Topics include: Crystal pattern formation, Fractal metrology, Fluid flow analysis, Estimation of multifractal measures and more. ... Read more

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This text will be a valuable tool and reference work for all those interested in fractal forms and their properties. The past decade has seen an almost exponential amount of work done in the field of non-integral dimensions. Upon publication of Mandelbrot's Objets Fractals (1975), a great number of researchers understood the interest of fractals. Many tried, by means of the theory, to set up immediately useable tools, including the author, who opted for an experimental approach and developed instrumentation to explore the widest possible range in both time and space (time resolution of the laser spectroscopy of biomolecules). In this spirit, these expert researchers organized monthly ``Hausdorff seminars on the notion of non-integral dimension and its applications'' to tackle the problems confronting practical scientists. This work captures, with many illustrative examples, the main themes of each session. ... Read more

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Fractal geometry has become popular in the last 15 years,its applicationscan be found in technology, science, oreven arts. Fractal methods andformalism are seen today as ageneral, abstract, but nevertheless practical instrument forthe description of nature in a wide sense. But itwasComputer Graphics which made possible the increasingpopularity offractals several years ago, and long aftertheir mathematical formulation.The two disciplines aretightly linked.The book contains the scientificcontributions presentedin an international workshop in the "Computer GraphicsCenter" in Darmstadt, Germany. Thetarget of the workshop wasto present the wide spectrumof interrelationships and interactions between FractalGeometry and Computer Graphics. The topics vary fromfundamentals and new theoretical results to variousapplications and systems development. All contributions areoriginal, unpublished papers.The presentations have beendiscussed in two working groups; the discussion results,together with actual trends and topics of future research,are reported in the last section.The topics of the book are divides into foursections:Fundamentals, Computer Graphics and Optical Simulation,Simulation of Natural Phenomena, Image Processing and ImageAnalysis. ... Read more

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Provides an Intuitive View of Various Ecological Patterns and Processes

Ecologists sometimes have a less-than-rigorous background in quantitative methods, yet research within this broad field is becoming increasingly mathematical. Written in a step-by-step fashion, Fractals and Multifractals in Ecology and Aquatic Science provides scientists with a basic understanding of fractals and multifractals and the techniques for utilizing them when analyzing ecological phenomenon.

With illustrations, tables, and graphs on virtually every page – several in color – this book is a comprehensive source of state-of-the-art ecological scaling and multiscaling methods at temporal and spatial scales, respectfully ranging from seconds to months and from millimeters to thousands of kilometers. It illustrates most of the data analysis techniques with real case studies often based on original findings. It also incorporates descriptions of current and new numerical techniques to analyze and deepen understanding of ecological situations and their solutions.

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Fractals are the visual depictions of mathematical equations, the same equations, in fact, that describe natural phenomena such as coastlines, plant shapes and weather patterns. A computer program assigns a color to each point in the image based on the answers to a chosen equation, which then results in abstract fractal shapes. Dozens of variables are manipulated in order to create these fascinating images.
The Fractal Cosmos 2011 Wall Calendar features the organic designs of Alice Kelley, who says, Fractals are an intuitive glimpse into the infinite order that comprises the natural world, as well as being proof that math is beautiful.
The interior pages of this calendar are printed on 100% recycled, 50% post-consumer paper, processed chlorine free and printed with soy based inks. ... Read more

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"This delightful book illustrates beautifully the paradigm shift inphysics from writing equations and solving them to computer modeling andexperimentation." -- Greg Chaitin, author of The Limits ofMathematics

In this book Gary William Flake develops in depth the simple idea thatrecurrent rules can produce rich and complicated behaviors.Distinguishing "agents" (e.g., molecules, cells, animals, and species)from their interactions (e.g., chemical reactions, immune systemresponses, sexual reproduction, and evolution), Flake argues that it isthe computational properties of interactions that account for much ofwhat we think of as "beautiful" and "interesting." From this basicthesis, Flake explores what he considers to be today's four mostinteresting computational topics: fractals, chaos, complex systems, andadaptation.

Each of the book's parts can be read independently, enabling even thecasual reader to understand and work with the basic equations andprograms. Yet the parts are bound together by the theme of the computeras a laboratory and a metaphor for understanding the universe. Theinspired reader will experiment further with the ideas presented tocreate fractal landscapes, chaotic systems, artificial life forms,genetic algorithms, and artificial neural networks. ... Read more

Customer Reviews (17)

Not fun
Your eyes will glaze over. Fascinating subjects, but I don't feel they were explained very well.

A Waaaay Cool Book!
IMHO this book should be part of every US high school or undergradate Science/Math curriculum, and would be worth twice the price.The author's enthusiasm is infectious, his writing style very clear, and his material well cited.He also maintains a website with free software downloads that illustrate the many mind expanding (w/o drugs! (-:) concepts discussed. Although thanks to magazines like Wired (minus its aggressive leftist politics and more aggressive BB censoring)the sciences have become "cooler." Nevertheless, there is still A LOT of work to do, and damage to be undone, from academically inferior and unenthusiastic so called math/science "teachers" in US schools. I don't usually rate books five stars, but I am rating this one a big FIVE stars.

An enjoyable read
Granted you can find most of this info elsewhere but still this is a great read. Well written, a nice collection of material, and downloadable source code. I found it to be a very inspiring book.

Great book
This is an excellent book.I've been reading it for weeks.The chapters are not long, but the content is amazing. The author combines explanations and equations in a format that is demonstrative and repeatible.This is a very good book to study yielding the understanding necessary to penetrate many other advanced books on complexity theory. The author starts by examining whole numbers and real number problems.Next, he examine Godel's incomplete theory of predicate logic showing that no formal language is complete.Next, he examines fractals, self-similar patterns, low ordered with high compression and high order with low order compression, L-systems, and Juliet and Mandel-brot fractals. Fractals open up an emmense study into the complex pattern from simple rules and recursion. Next, he examines equations of strange attraction, chaos, and demonstrates stability behavior within complexity. Next, he looks at small universes created by running CA.NNs and GAs are examined in the last chapter. I was excited to write down many of the authors processes and run them using OpenGL and C.I believe this book to be an excellent book for college students. The material is easy to understand and the content very demonstratible.Cause and effect are very cohesive in this book.Even though the book seems simple, it covers an vast amount of topics necessary to understanding AI and AL.

Good first book on the subject of simulating natural phenomena
This is a good introductory textbook for college undergraduate mathematics and computer science students that attempts to combine the theory of computation with some mathematical concepts and in the end, manages to model virtual life by explaining basic concepts in chaos, adaptation, fractals, and complex systems. There are better books on all of these subjects, but few others do such a good job of tying together key concepts from each discipline into the one theme of this book. However, there is only enough room to outline the included subjects rather than investigate them thoroughly.

Also, the mathematics is elementary enough to be accessible to a mathematically mature high school student. The mathematics is concisely explained as it is needed, with just a page or two for each of calculus, linear algebra, affine transformations, complex numbers, vector calculus, and matrix algebra. Thus, the included mathematics makes a better refresher than a tutorial for the novice even though the author states in the preface that he wrote this book for a younger version of himself. This book teaches its subject matter mainly by demonstrating concepts through simulations that are expressed in dozens of programs which illustrate the points being made. Instructions on using the programs are scattered throughout the book. The source code is available for download on the web, along with selected excerpts from the book.

I would recommend this as a first book for those interested in simulating natural concepts, but it should not be your last if your goal is to truly grasp the concepts presented and produce simulations of your own. However, an even better book on this subject is "Mathematical Models in Biology", although it is an advanced text. A very accessible book that is also more advanced than this text is "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering". It clearly explains the mathematics while tying it into key concepts in nature. "Chaos and Fractals" by Peitgen is a good book on the subject for the layperson with a fascination for mathematics presented in some depth. The book also has various Java programs that illustrate key concepts. ... Read more

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This volume considers river basins and drainage networks in the light of their scaling and multi-scaling properties, and the dynamics responsible for their development. ... Read more

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Good overview of subject, lots of references
Easy to understand analysis of river systems.This is a good text for a graduate-level course.It is also a valuable reference for any geologist or hydrologist involved in work that deals with modern or ancient fluvial systems.CJD ... Read more

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Offering a new version of FRACTINT that allows users to generate built-in fractals or devise original patterns, this book-and-disk package also includes FDESIGN, a powerful new fractal-generating program that allows users to create natural shapes. Original. ... Read more

Customer Reviews (1)

A cool book that I wish the authors would update!
I really loved this book when it came out, and wish the authors would get together to update both the material in the book and the software that came with it to run on 'modern' computing equipment, (this was written when DOS was still king!)The book provided a really good, hands-on approach to fractal theory.It is a little light on application, but wasn't meant to have any of that, so I can forgive and forget.Moving down through to the borders of a Mandlebrot or Julia set in 16bit colour was amazing, and I would love to see what it would look like in the 16 million + colours available on computers today! ... Read more

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This is a one-stop introduction to the methods of ergodic theory applied to holomorphic iteration. The authors begin with introductory chapters presenting the necessary tools from ergodic theory thermodynamical formalism, and then focus on recent developments in the field of 1-dimensional holomorphic iterations and underlying fractal sets, from the point of view of geometric measure theory and rigidity. Detailed proofs are included. Developed from university courses taught by the authors, this book is ideal for graduate students. Researchers will also find it a valuable source of reference to a large and rapidly expanding field. It eases the reader into the subject and provides a vital springboard for those beginning their own research. Many helpful exercises are also included to aid understanding of the material presented and the authors provide links to further reading and related areas of research. ... Read more

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This book presents material on three topics, namely the amountof information involved in non-random functions, the amount ofinformation involved in non-probabilistic square matrices (i.e. whichare not quantum density matrices), and a new model of complex-valuedfractional Brownian motion of order n defined via random walks in thecomplex plane. These three subjects, which on the surface have nocommon features, are, in fact, direct consequences of the maximumentropy principle. Moreover, information on non-random functions andcomplex fractional Brownian motion are directly related to fractals.Thus, a unified framework is constructed which encompasses informationwith and without probability, quantum information of square matriceswith and without probabilistic meaning, and fractals in the complexplane. This volume also features many applications.
Audience: This work is intended for theoretical and mathematicalphysicists, but also for applied mathematicians, experimentalphysicists, communication engineers, electrical engineers,practitioners in pattern recognition and computer vision, controlsystems engineers, and theoretical biologists. ... Read more

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