This volume describes the status of fractal imaging research and looks to future directions. It will be useful to researchers in the areas of fractal image compression, analysis, and synthesis, iterated function systems, and fractals in education. In particular it includes a vision for the future of these areas. It aims to provide an efficient means by which researchers can look back over the last decade at what has been achieved, and look forward towards second-generation fractal imaging. The articles in themselves are not meant to be detailed reviews or expositions, but to serve as signposts to the state of the art in their areas. What is important is what they mention and what tools and ideas are seen now to be relevant to the future. The contributors, a number of whom have been involved since the start, are active in fractal imaging, and provide a well-informed viewpoint on both the status and the future. Most were invited participants at a meeting on Fractals in Multimedia held at the IMA in January 2001. Some goals of the mini-symposium, shared with this volume, were to demonstrate that the fractal viewpoint leads to a broad collection of useful mathematical tools, common themes, new ways of looking at and thinking about existing algorithms and applications in multimedia, and to consider future developments. This book should be useful to commercial and university researchers in the rapidly evolving field of digital imaging, specifically, chief information officers, professors, software engineers, and graduate students in the mathematical sciences. While much of the content is quite technical, it contains pointers to the state-of-the-art and the future in fractal imaging.

Fractals for the Classroom breaks new ground as it brings an exciting branch of mathematics into the classroom. The book is a collection of independent chapters on the major concepts related to the science and mathematics of fractals. Written at the mathematical level of an advanced secondary student, Fractals for the Classroom includes many fascinating insights for the classroom teacher and integrates illustrations from a wide variety of applications with an enjoyable text to help bring the concepts alive and make them understandable to the average reader. This book will have a tremendous impact upon teachers, students, and the mathematics education of the general public. With the forthcoming companion materials, including four books on strategic classroom activities and lessons with interactive computer software, this package will be unparalleled.

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, cellular automata, L-systems, percolation and strange attractors. This new edition has been thoroughly revised throughout. The appendices of the original edition were taken out since more recent publications cover this material in more depth. Instead of the focussed computer programs in BASIC, the authors provide 10 interactive JAVA-applets for this second edition.

This book provides an overview of current research in the fascinating, interdisciplinary field of computer science and sports. It includes papers from the 11th International Symposium on Computer Science in Sport (IACSS 2017), which took place in Constance, Germany, on September 6-9, 2017. The papers represent the state of the art in utilizing the latest developments in computer science to support coaches and athletes. The book covers a broad range of topics, reflecting the diversity of the field. It presents three categories of papers: those on concepts in informatics like modeling, virtual reality, simulation; those describing applications of computer science in sports like running, volleyball, water polo, and football; and contributions discussing the impact of computer science in sports federations and universities.

Fractals for the Classroom breaks new ground as it brings an exciting branch of mathematics into the classroom. The book is a collection of independent chapters on the major concepts related to the science and mathematics of fractals. Written at the mathematical level of an advanced secondary student, Fractals for the Classroom includes many fascinating insights for the classroom teacher and integrates illustrations from a wide variety of applications with an enjoyable text to help bring the concepts alive and make them understandable to the average reader. This book will have a tremendous impact upon teachers, students, and the mathematics education of the general public. With the forthcoming companion materials, including four books on strategic classroom activities and lessons with interactive computer software, this package will be unparalleled.

This book is based on notes for the course Fractals: lntroduction, Basics and Perspectives given by MichaelF. Barnsley, RobertL. Devaney, Heinz-Otto Peit gen, Dietmar Saupe and Richard F. Voss. The course was chaired by Heinz-Otto Peitgen and was part of the SIGGRAPH '87 (Anaheim, California) course pro gram. Though the five chapters of this book have emerged from those courses we have tried to make this book a coherent and uniformly styled presentation as much as possible. It is the first book which discusses fractals solely from the point of view of computer graphics. Though fundamental concepts and algo rithms are not introduced and discussed in mathematical rigor we have made a serious attempt to justify and motivate wherever it appeared to be desirable. Ba sic algorithms are typically presented in pseudo-code or a description so close to code that a reader who is familiar with elementary computer graphics should find no problem to get started. Mandelbrot's fractal geometry provides both a description and a mathemat ical model for many of the seemingly complex forms and patterns in nature and the sciences. Fractals have blossomed enormously in the past few years and have helped reconnect pure mathematics research with both natural sciences and computing. Computer graphics has played an essential role both in its de velopment and rapidly growing popularity. Conversely, fractal geometry now plays an important role in the rendering, modelling and animation of natural phenomena and fantastic shapes in computer graphics."

Fractals for the Classroom breaks new ground as it brings an exciting branch of mathematics into the classroom. The book is a collection of independent chapters on the major concepts related to the science and mathematics of fractals. Written at the mathematical level of an advanced secondary student, Fractals for the Classroom includes many fascinating insights for the classroom teacher and integrates illustrations from a wide variety of applications with an enjoyable text to help bring the concepts alive and make them understandable to the average reader. This book will have a tremendous impact upon teachers, students, and the mathematics education of the general public. With the forthcoming companion materials, including four books on strategic classroom activities and lessons with interactive computer software, this package will be unparalleled.

This IMA Volume in Mathematics and its Applications FRACTALS IN MULTIMEDIA is a result of a very successful three-day minisymposium on the same title. The event was an integral part of the IMA annual program on Mathemat ics in Multimedia, 2000-2001. We would like to thank Michael F. Barnsley (Department of Mathematics and Statistics, University of Melbourne), Di etmar Saupe (Institut fUr Informatik, UniversiUit Leipzig), and Edward R. Vrscay (Department of Applied Mathematics, University of Waterloo) for their excellent work as organizers of the meeting and for editing the proceedings. We take this opportunity to thank the National Science Foundation for their support of the IMA. Series Editors Douglas N. Arnold, Director of the IMA Fadil Santosa, Deputy Director of the IMA v PREFACE This volume grew out of a meeting on Fractals in Multimedia held at the IMA in January 2001. The meeting was an exciting and intense one, focused on fractal image compression, analysis, and synthesis, iterated function systems and fractals in education. The central concerns of the meeting were to establish within these areas where we are now and to develop a vision for the future."

Written by the award winning authors of Chaos and Fractals (0-387-97903-4), this work introduces the reader to iterated function systems through a lively, interactive approach. This well-written, clearly illustrated book explores the history and the unlimited potential of fractals, while developing a basic mathematical understanding and appreciation for the topics.

The same factors that motivated the writing of our first volume of strategic activities on fractals continued to encourage the assembly of additional activities for this second volume. Fractals provide a setting wherein students can enjoy hands-on experiences that involve important mathematical content connected to a wide range of physical and social phenomena. The striking graphic images, unexpected geometric properties, and fascinating numerical processes offer unparalleled opportunity for enthusiastic student inquiry. Students sense the vigor present in the growing and highly integrative discipline of fractal geom etry as they are introduced to mathematical developments that have occurred during the last half of the twentieth century. Few branches of mathematics and computer science offer such a contem porary portrayal of the wonderment available in careful analysis, in the amazing dialogue between numeric and geometric processes, and in the energetic interaction between mathematics and other disciplines. Fractals continue to supply an uncommon setting for animated teaching and learn ing activities that focus upon fundamental mathematical concepts, connections, problem-solving techniques, and many other major topics of elementary and advanced mathematics. It remains our hope that, through this second volume of strategic activities, readers will find their enjoyment of mathematics heightened and their appreciation for the dynamics of the world in creased. We want experiences with fractals to enliven curiosity and to stretch the imagination."

Die Realitiit ist vielleicht das reinste Chaos. Georg Christoph Lichtenberg 1953 erkannte ich, daj3 die gerade Linie zum Untergang der Menschheitfiihrt. Aber die gerade Linie ist zur absoluten Tyrannei geworden. Die gerade Linie ist der Fluch unserer Zivilisation. Heute erleben wir den Triumph der rationalen Technik, und wiihrenddessen befinden wir uns gleichzeitig vor dem Nichts. Friedensreich Hundertwasser Dieses Buch ist weder ein typisches Mathematikbuch noch ein iibliches popularwis- senschaftliches Buch. Vielmehr war beabsichtigt, eine Art Lesebuch vorzulegen, das es auch Laien erlaubt, ohne den Ballast zu vieler technisch-mathematischer Notationen, einen soliden Einblick in die Welt der aktuellen Chaostheorie und der fraktalen Geo- metrie zu gewinnen. Dieser erste Band konzentriert sich dabei mehr auf geometrische Phanomene, wahrend der zweite Band Chaos - Bausteine der Ordnung sich vor allem auf dynamische Phanomene stiitzt. Seit Ende der siebziger Jahre lauft eine Welle durch Mathematik und Naturwissen- schaften, die in ihrer Kraft, Kreativitat und Weitraumigkeit langst ein interdisziplinares Ereignis ersten Ranges geworden ist: Chaos und Fraktale. Dies ist umso bemerkenswer- ter, als sich die Chaostheorie und die fraktale Geometrie eigentlich in keiner Hinsicht mit den groBartigen Entwiirfen dieses Jahrhunderts, wie etwa der Quantentheorie oder der Relativitatstheorie, messen k6nnen. Chaostheorie und fraktale Geometrie haben Naturwissenschaftler und Mathematiker mit einer Reihe von Dberraschungen konfrontiert, deren Konsequenzen im Verhaltnis zu den Angeboten einer sich oft omnipotent gebenden Wissenschaft und Technik zugleich emiichternd und dramatisch sind: - Zahlreiche Phanomene sind trotz strengem naturgesetzlichem Deterrninismus prin- zipiell nicht prognostizierbar.

In den letzten Jahren hat sich der Workshop "Bildverarbeitung fur die Medizin" durch erfolgreiche Veranstaltungen etabliert. In ubersichtlicher Form das breite Spektrum universitarer und industrieller Anwendung und Forschung dargestellt. Dieser Band enthalt ca. 100 Beitrage u.a. zu folgenden Themen: Atlanten und anatomische Modelle; Bildanalyse; computergestutzte Chirurgie, Therapie und OP-Planung; Bildrekonstruktion; Visualisierung und 3D-Interaktion; Registrierung und Landmarkenfindung; Simulation; Mustererkennung; Segmentierung; Bildfilterung, Kompression und Korrektur."

Das vorliegende Arbeitsbuch ist Teil eines Paketes von verschiedenen Materialien, die das Ziel haben, das Thema Chaos und Fraktale in den mathematisch-naturwissenschaftlichen Unterricht einzufuhren. Ein weiteres Anliegen besteht darin, das mentale Bild von Mathematik im Bewusstsein der Schuler attraktiver zu gestalten. Mathematik ist die Antwort des Menschen auf die Komplexitat der Welt. Mathematik ist die Ordnungsmacht im Dschungel der Phanomene. Deshalb ist Mathematik le bendig, frisch und aktuell. Deshalb gibt es zwischen einzelnen Teilgebieten und Ergebnissen der Mathematik immer wieder uberraschende Querverbindungen, die oft das Verstandnis einer Sache erst wirklich erhellen. Und deshalb bietet es sich an, durch entdeckendes, explorierendes Lernen die Anziehungskraft dieser Eigenschaften der Mathematik im Unterricht auszunutzen. Chaos und Fraktale bieten hierfur eine besondere neue Chance. Beide sind jung und aktuell und belegen so ohne weiteres, dass Mathematik lebt. Fur beide gilt, dass einige ihrer schrittmachenden Entdeckungen nicht ohne Hilfe von Computern moeglich gewesen waren. Damit rucken faszinie rende Computerexperimente naturlich in den Mittelpunkt. Beide sind hochgradig interdisziplinar. Dieses heisst, dass gehaltvolle Anwendungen nicht erst muhsam konstruiert werden mussen. Beide behandeln Themen, die von sich aus wirken. Tatsachlich durchlaufen seit Ende der siebziger Jahre Mathematik und Naturwissenschaften eine Welle, die in ihrer Kraft, Kreativitat und Weitraumigkeit langst ein interdisziplinares Ereignis er sten Ranges geworden ist. Das andauernde Interesse innerhalb und ausserhalb der Wissenschaften ist in einer aufruttelnden Betroffenheit begrundet, die eine radikale Wende in dem uberkommenen naturwissenschaftlichen Weltbild und manchen uberdehnten Interpretationen ankundigt.

Das Buch gibt einen Querschnitt durch das Spektrum der Anwendungen der Computergraphik in Mathematik und Naturwissenschaften. Dabei werden auch neue computergraphische Algorithmen, komplexe Anwenderpakete und sogar HardwarelAsungen diskutiert. Die BeitrAge in diesem Band, die auf VortrAgen der ersten Bremer Computer-Graphik-Tage vom Juli 1988 basieren, sind entsprechend vielfAltig: Sie reichen von der Berechnung und Visualisierung mathematischer Objekte und Fraktale A1/4ber Probleme des Medical Imaging bei der Darstellung volumetrischer Daten, Untersuchungen von OberflAchentexturen fA1/4r Bildsynthese bis hin zur Diskussion verschiedener AnsAtze der Parallelisierung computergraphischer Algorithmen. Bei der Darstellung neuer LAsungswege fA1/4r komplexe Probleme der Visualisierung wurde besonderer Wert auf die praktischen Details gelegt, so daA der interessierte Leser diese LAsungen auch in einer eigenen Implementierung nachvollziehen kann.