An Invitation To Fractal Geometry
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An Invitation To Fractal Geometry
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Author : Michel L. Lapidus
language : en
Publisher: American Mathematical Society
Release Date : 2024-12-31
An Invitation To Fractal Geometry written by Michel L. Lapidus and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-12-31 with Mathematics categories.
This book offers a comprehensive exploration of fractal dimensions, self-similarity, and fractal curves. Aimed at undergraduate and graduate students, postdocs, mathematicians, and scientists across disciplines, this text requires minimal prerequisites beyond a solid foundation in undergraduate mathematics. While fractal geometry may seem esoteric, this book demystifies it by providing a thorough introduction to its mathematical underpinnings and applications. Complete proofs are provided for most of the key results, and exercises of different levels of difficulty are proposed throughout the book. Key topics covered include the Hausdorff metric, Hausdorff measure, and fractal dimensions such as Hausdorff and Minkowski dimensions. The text meticulously constructs and analyzes Hausdorff measure, offering readers a deep understanding of its properties. Through emblematic examples like the Cantor set, the Sierpinski gasket, the Koch snowflake curve, and the Weierstrass curve, readers are introduced to self-similar sets and their construction via the iteration of contraction mappings. The book also sets the stage for the advanced theory of complex dimensions and fractal drums by gently introducing it via a variety of classical examples, including well-known fractal curves. By intertwining historical context with rigorous mathematical exposition, this book serves as both a stand-alone resource and a gateway to deeper explorations in fractal geometry.
An Invitation To Fractal Geometry
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Author : Michel L. Lapidus
language : en
Publisher: American Mathematical Society
Release Date : 2024-12-30
An Invitation To Fractal Geometry written by Michel L. Lapidus and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-12-30 with Mathematics categories.
This book offers a comprehensive exploration of fractal dimensions, self-similarity, and fractal curves. Aimed at undergraduate and graduate students, postdocs, mathematicians, and scientists across disciplines, this text requires minimal prerequisites beyond a solid foundation in undergraduate mathematics. While fractal geometry may seem esoteric, this book demystifies it by providing a thorough introduction to its mathematical underpinnings and applications. Complete proofs are provided for most of the key results, and exercises of different levels of difficulty are proposed throughout the book. Key topics covered include the Hausdorff metric, Hausdorff measure, and fractal dimensions such as Hausdorff and Minkowski dimensions. The text meticulously constructs and analyzes Hausdorff measure, offering readers a deep understanding of its properties. Through emblematic examples like the Cantor set, the Sierpinski gasket, the Koch snowflake curve, and the Weierstrass curve, readers are introduced to self-similar sets and their construction via the iteration of contraction mappings. The book also sets the stage for the advanced theory of complex dimensions and fractal drums by gently introducing it via a variety of classical examples, including well-known fractal curves. By intertwining historical context with rigorous mathematical exposition, this book serves as both a stand-alone resource and a gateway to deeper explorations in fractal geometry.
Fractals Graphics And Mathematics Education
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Author : Michael Frame
language : en
Publisher: Cambridge University Press
Release Date : 2002-06-20
Fractals Graphics And Mathematics Education written by Michael Frame and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002-06-20 with Computers categories.
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Fractal Zeta Functions And Fractal Drums
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Author : Michel L. Lapidus
language : en
Publisher: Springer
Release Date : 2017-06-07
Fractal Zeta Functions And Fractal Drums written by Michel L. Lapidus and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-06-07 with Mathematics categories.
This monograph gives a state-of-the-art and accessible treatment of a new general higher-dimensional theory of complex dimensions, valid for arbitrary bounded subsets of Euclidean spaces, as well as for their natural generalization, relative fractal drums. It provides a significant extension of the existing theory of zeta functions for fractal strings to fractal sets and arbitrary bounded sets in Euclidean spaces of any dimension. Two new classes of fractal zeta functions are introduced, namely, the distance and tube zeta functions of bounded sets, and their key properties are investigated. The theory is developed step-by-step at a slow pace, and every step is well motivated by numerous examples, historical remarks and comments, relating the objects under investigation to other concepts. Special emphasis is placed on the study of complex dimensions of bounded sets and their connections with the notions of Minkowski content and Minkowski measurability, as well as on fractal tube formulas. It is shown for the first time that essential singularities of fractal zeta functions can naturally emerge for various classes of fractal sets and have a significant geometric effect. The theory developed in this book leads naturally to a new definition of fractality, expressed in terms of the existence of underlying geometric oscillations or, equivalently, in terms of the existence of nonreal complex dimensions. The connections to previous extensive work of the first author and his collaborators on geometric zeta functions of fractal strings are clearly explained. Many concepts are discussed for the first time, making the book a rich source of new thoughts and ideas to be developed further. The book contains a large number of open problems and describes many possible directions for further research. The beginning chapters may be used as a part of a course on fractal geometry. The primary readership is aimed at graduate students and researchers working in Fractal Geometry and other related fields, such as Complex Analysis, Dynamical Systems, Geometric Measure Theory, Harmonic Analysis, Mathematical Physics, Analytic Number Theory and the Spectral Theory of Elliptic Differential Operators. The book should be accessible to nonexperts and newcomers to the field.
The Geometry Of Fractals A New Perspective
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Author : Pasquale De Marco
language : en
Publisher: Pasquale De Marco
Release Date : 2025-07-07
The Geometry Of Fractals A New Perspective written by Pasquale De Marco and has been published by Pasquale De Marco this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-07-07 with Science categories.
Embark on a captivating journey into the realm of fractals, where intricate patterns and self-similarity unveil the hidden beauty and complexity of the natural world. This comprehensive guide delves into the mathematical foundations, applications, and cultural significance of fractals, offering a fresh perspective on these fascinating geometric structures. From the delicate branching of snowflakes to the mesmerizing spirals of seashells, fractals are found in abundance in nature, revealing a level of order within apparent chaos. Botanists, geologists, and meteorologists alike have discovered the fractal patterns that govern the growth of plants, the formation of mountains, and the behavior of clouds. This book takes you on an exploration of the history of fractals, tracing their origins from early mathematical investigations to the groundbreaking work of Benoit Mandelbrot, who brought fractals to the forefront of scientific study. Discover the mathematical concepts that underlie fractal geometry, including set theory, topology, and measure theory. Beyond their mathematical elegance, fractals have found practical applications in diverse fields. Computer scientists utilize fractals to generate realistic images and animations, while engineers harness their properties for signal processing and data compression. Fractals have also inspired new forms of art and music, captivating audiences with their intricate patterns and dynamic forms. This book is not just a scientific treatise; it is an invitation to appreciate the beauty and wonder of fractals. Whether you are a mathematician, a scientist, an artist, or simply someone curious about the world around you, you will find something to engage and inspire you within these pages. **Unveil the Secrets of Fractals:** * Explore the mathematical underpinnings of fractal geometry, including set theory, topology, and measure theory. * Delve into the history of fractals, from early investigations to the groundbreaking work of Benoit Mandelbrot. * Discover the diverse applications of fractals in fields such as computer graphics, image processing, and signal processing. * Appreciate the cultural significance of fractals in art, music, and literature. * Gain a deeper understanding of the beauty and complexity of the natural world through the lens of fractals. If you like this book, write a review on google books!
A Random Walk Through Fractal Dimensions
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Author : Brian H. Kaye
language : en
Publisher: John Wiley & Sons
Release Date : 2008-07-11
A Random Walk Through Fractal Dimensions written by Brian H. Kaye and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-07-11 with Technology & Engineering categories.
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
Visual Explorations Unlocking The Beauty And Complexity Of Fractals
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Author : Pasquale De Marco
language : en
Publisher: Pasquale De Marco
Release Date : 2025-03-22
Visual Explorations Unlocking The Beauty And Complexity Of Fractals written by Pasquale De Marco and has been published by Pasquale De Marco this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-03-22 with Mathematics categories.
Embark on an enthralling journey into the captivating world of fractals, where intricate patterns and mesmerizing forms reveal the profound interconnectedness of nature, art, and science. This comprehensive guide unveils the secrets of fractal geometry, delving into the mathematical principles that govern these seemingly complex structures. Discover the elegance of self-similarity, where patterns repeat themselves across scales, creating an infinite tapestry of detail. Explore the diverse manifestations of fractals in the natural world, from the branching patterns of ferns to the swirling eddies of water, from the delicate tracery of snowflakes to the vastness of galaxies. Witness the beauty of fractal art and music, and gain insights into the applications of fractals in computer graphics, engineering, biology, and medicine. Beyond their scientific and artistic significance, fractals also hold deep philosophical implications. Question your notions of order and chaos, finitude and infinity, as you contemplate the intricate dance of mathematics and nature. Discover how fractals challenge our understanding of reality and offer a glimpse into the interconnectedness of all things. Written in an engaging and accessible style, **Visual Explorations: Unlocking the Beauty and Complexity of Fractals** is an invitation to expand your horizons and see the world in a new light. Whether you are a scientist, an artist, a philosopher, or simply a curious mind seeking to unravel the mysteries of the universe, this book promises an extraordinary journey into the heart of complexity. If you like this book, write a review!
Lumen Naturae
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Author : Matilde Marcolli
language : en
Publisher: MIT Press
Release Date : 2020-05-26
Lumen Naturae written by Matilde Marcolli and has been published by MIT Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-05-26 with Mathematics categories.
Exploring common themes in modern art, mathematics, and science, including the concept of space, the notion of randomness, and the shape of the cosmos. This is a book about art—and a book about mathematics and physics. In Lumen Naturae (the title refers to a purely immanent, non-supernatural form of enlightenment), mathematical physicist Matilde Marcolli explores common themes in modern art and modern science—the concept of space, the notion of randomness, the shape of the cosmos, and other puzzles of the universe—while mapping convergences with the work of such artists as Paul Cezanne, Mark Rothko, Sol LeWitt, and Lee Krasner. Her account, focusing on questions she has investigated in her own scientific work, is illustrated by more than two hundred color images of artworks by modern and contemporary artists. Thus Marcolli finds in still life paintings broad and deep philosophical reflections on space and time, and connects notions of space in mathematics to works by Paul Klee, Salvador Dalí, and others. She considers the relation of entropy and art and how notions of entropy have been expressed by such artists as Hans Arp and Fernand Léger; and traces the evolution of randomness as a mode of artistic expression. She analyzes the relation between graphical illustration and scientific text, and offers her own watercolor-decorated mathematical notebooks. Throughout, she balances discussions of science with explorations of art, using one to inform the other. (She employs some formal notation, which can easily be skipped by general readers.) Marcolli is not simply explaining art to scientists and science to artists; she charts unexpected interdependencies that illuminate the universe.
Portals Unveiled
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Author : Barrett Williams
language : en
Publisher: Barrett Williams
Release Date : 2024-11-23
Portals Unveiled written by Barrett Williams and has been published by Barrett Williams this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-11-23 with Science categories.
**Portals Unveiled Journey Through the Infinite Cosmos** Step into a world where the boundaries of reality and imagination blur, where science and philosophy intermingle to challenge the very fabric of existence. "Portals Unveiled" invites you to explore the thrilling landscapes of parallel dimensions, revealing the mysteries of the multiverse through a captivating blend of science, mathematics, art, and technology. Begin your odyssey with an introduction to the multiverse theory, tracing its origins and diving into historical and cultural references that have shaped our understanding of parallel worlds. Delve into the enigmatic realm of quantum mechanics and chaos theory, where mathematics becomes the key to deciphering the infinite complexities of our universe. Discover the mesmerizing patterns of fractals that permeate nature, guided by the pioneering insights of Benoit Mandelbrot. Explore the concept of a fractal cosmos and venture into the scientific and philosophical realms where the notion of portals bridges fiction and reality. How does the human mind perceive these extraordinary dimensions? Journey through the intricate connections between neuroscience, altered states of consciousness, and the mind’s potential as a gateway to new realities. Engage with the paradoxes and conundrums that challenge conventional logic, and witness how these ideas influence both art and ethics in society. "Portals Unveiled" doesn't merely ponder theoretical concepts; it also illuminates practical applications shaping the future of technology and exploration. Virtual realities and fractal designs inspire innovations across industries, while ethical and societal considerations provoke deep reflection on the impact of exploring other dimensions. Embark on this enthralling quest, and let "Portals Unveiled" expand your horizons, challenge your perceptions, and inspire curiosity about the interconnected universe we inhabit. As you turn each page, prepare to question, learn, and imagine what lies beyond the known world. Embrace the unknown and envision a future where the mysteries of the multiverse are finally unveiled.
Fractal Worlds
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Author : Michael Frame
language : en
Publisher: Yale University Press
Release Date : 2016-01-01
Fractal Worlds written by Michael Frame and has been published by Yale University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-01-01 with Mathematics categories.
In this essential primer, mathematician Michael Frame, a close collaborator with Benoit Mandelbrot, the founder of fractal geometry, and poet Amelia Urry explore the amazing world of fractals as they appear in nature, art, medicine, and technology