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To see objects that live in the fourth dimension we humans would need to add a fourth dimension to our three-dimensional vision. An example of such an object that lives in the fourth dimension is a hyper-sphere or “3-sphere”. The quest to imagine the elusive 3-sphere has deep historical roots: medieval poet Dante Alighieri, in his circa 1300 AD Divine Comedy, used a 3-sphere to convey his allegorical vision of the Christian afterlife. In 1917, Albert Einstein visualized the universe, at each instant in time, as a 3-sphere. He described his representation as “…the place where the reader’s imagination boggles. Nobody can imagine this thing.” Over time, however, our understanding of the concept of dimension evolved. By 2003, a researcher had successfully rendered into human vision the structure of a 4-web (think of an every increasingly-dense spider’s web). In this text Stephen Lipscomb takes his innovative dimension theory research a step further, using the 4-web to reveal a new partial image of a 3-sphere. Illustrations support the reader’s understanding of the mathematics behind this process. Lipscomb describes a computer program that can produce partial images of a 3-sphere and suggests methods of discerning other fourth-dimensional objects that may serve as the basis for future artwork.  Reviews The author’s notion of fractal-based computer art is fascinating-a clear expression of our technological age. With the color plates in this book and the available DVD animation the reader will not only substantiate this, but will also gain an intuitive sense about the nature of fractals and about the structure and origin of the 4-web. A.D. Parks, Ph.D., Principal Scientist, Head of Quantum Physics Group, Naval Surface Warfare Center, Dahlgren Virginia Using numerous illustrations, the author discusses the idea of a fourth dimension. The new feature here is his use of an object that up until recently lived only in the fourth dimension. This book should become useful, educational, and widely-read. Gerald Edgar, Professor (Emeritus) of Mathematics, The Ohio State University  I have read many books, but only a couple has been as suggestive in terms of connections between mathematics, art, and physics as this book. It will be exceptionally well received. John E. Gray, Senior Member of IEEE, Lead physicist (over 130 publications)  An accessible yet rigorous treatment of recent mathematical research, this book is particularly valuable since its author developed these concepts originally. J. Larry Lehman, Professor of Mathematics, University of Mary Washington.

Hyperspace. --- Art --- Geometry in art. --- Mathematics. --- Algebraic configurations in hyperspace --- Space of more than three dimensions --- Geometry --- Geometry, Differential --- Geometry, Non-Euclidean --- Mathematics --- Space and time --- Foundations --- Topology. --- Mathematics, general. --- Mathematics in the Humanities and Social Sciences. --- Mathematics in Art and Architecture. --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Math --- Science --- Social sciences. --- Behavioral sciences --- Human sciences --- Sciences, Social --- Social science --- Social studies --- Civilization

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To see objects that live in the fourth dimension we humans would need to add a fourth dimension to our three-dimensional vision. An example of such an object that lives in the fourth dimension is a hyper-sphere or “3-sphere”. The quest to imagine the elusive 3-sphere has deep historical roots: medieval poet Dante Alighieri, in his circa 1300 AD Divine Comedy, used a 3-sphere to convey his allegorical vision of the Christian afterlife. In 1917, Albert Einstein visualized the universe, at each instant in time, as a 3-sphere. He described his representation as “…the place where the reader’s imagination boggles. Nobody can imagine this thing.” Over time, however, our understanding of the concept of dimension evolved. By 2003, a researcher had successfully rendered into human vision the structure of a 4-web (think of an every increasingly-dense spider’s web). In this text Stephen Lipscomb takes his innovative dimension theory research a step further, using the 4-web to reveal a new partial image of a 3-sphere. Illustrations support the reader’s understanding of the mathematics behind this process. Lipscomb describes a computer program that can produce partial images of a 3-sphere and suggests methods of discerning other fourth-dimensional objects that may serve as the basis for future artwork.  Reviews The author’s notion of fractal-based computer art is fascinating-a clear expression of our technological age. With the color plates in this book and the available DVD animation the reader will not only substantiate this, but will also gain an intuitive sense about the nature of fractals and about the structure and origin of the 4-web. A.D. Parks, Ph.D., Principal Scientist, Head of Quantum Physics Group, Naval Surface Warfare Center, Dahlgren Virginia Using numerous illustrations, the author discusses the idea of a fourth dimension. The new feature here is his use of an object that up until recently lived only in the fourth dimension. This book should become useful, educational, and widely-read. Gerald Edgar, Professor (Emeritus) of Mathematics, The Ohio State University  I have read many books, but only a couple has been as suggestive in terms of connections between mathematics, art, and physics as this book. It will be exceptionally well received. John E. Gray, Senior Member of IEEE, Lead physicist (over 130 publications)  An accessible yet rigorous treatment of recent mathematical research, this book is particularly valuable since its author developed these concepts originally. J. Larry Lehman, Professor of Mathematics, University of Mary Washington.

Topology --- Mathematics --- wiskunde --- topologie

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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). While the classical theory is now well documented in several books, this is the first book to unify the modern theory (1960 - 2007). Like the classical theory, the modern theory fundamentally involves the unit interval. By the 1970s, the author of this monograph generalized Cantor's 1883 construction (identify adjacent-endpoints in Cantor's set) of the unit interval, obtaining for any given weight a one-dimensional metric space that contains rationals and irrationals as counterparts to those in the unit interval. Following the development of fractal geometry during the 1980s, these new spaces turned out to be the first examples of attractors of infinite iterated function systems generalized fractals.  The use of graphics to illustrate the fractal view of these spaces is a unique feature of this monograph. In addition, this book provides historical context for related research that includes imbedding theorems, graph theory, and closed imbeddings. This monograph will be useful to topologists, to mathematicians working in fractal geometry, and to historians of mathematics. It can also serve as a text for graduate seminars or self-study the interested reader will find many relevant open problems that will motivate further research into these topics.

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Sediment transport --- Stream measurements --- Alaska

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Hydrology --- Hydrology --- Hydrology --- Hydrology. --- Idaho