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- PublisherSpringer
- Publication date2024
- ISBN 10 3031475100
- ISBN 13 9783031475108
- BindingHardcover
- LanguageEnglish
- Number of pages257
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Reshaping Convex Polyhedra
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Springer Nature Switzerland Apr 2024, 2024
ISBN 10: 3031475100
ISBN 13: 9783031475108
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Buch. Condition: Neu. This item is printed on demand - it takes 3-4 days longer - Neuware -The focus of this monograph is converting-reshaping-one 3D convex polyhedron to another via an operation the authors call 'tailoring.' A convex polyhedron is a gem-like shape composed of flat facets, the focus of study since Plato and Euclid. The tailoring operation snips off a corner (a 'vertex') of a polyhedron and sutures closed the hole. This is akin to Johannes Kepler's 'vertex truncation,' but differs in that the hole left by a truncated vertex is filled with new surface, whereas tailoring zips the hole closed. A powerful 'gluing' theorem of A.D. Alexandrov from 1950 guarantees that, after closing the hole, the result is a new convex polyhedron. Given two convex polyhedra P, and Q inside P, repeated tailoringallows P to be reshaped to Q. Rescaling any Q to fit inside P, the result is universal: any P can be reshaped to any Q. This is one of the main theorems in Part I, with unexpected theoretical consequences.Part II carries out a systematic study of 'vertex-merging,' a technique that can be viewed as a type of inverse operation to tailoring. Here the start is P which is gradually enlarged as much as possible, by inserting new surface along slits. In a sense, repeated vertex-merging reshapes P to be closer to planarity. One endpoint of such a process leads to P being cut up and 'pasted' inside a cylinder. Then rolling the cylinder on a plane achieves an unfolding of P. The underlying subtext is a question posed by Geoffrey Shephard in 1975 and already implied by drawings by Albrecht Dürer in the 15th century: whether every convex polyhedron can be unfolded to a planar 'net.' Toward this end, the authors initiate an exploration of convexity on convex polyhedra, a topic rarely studiedin the literature but with considerable promise for future development.This monograph uncovers new research directions and reveals connections among several, apparently distant, topics in geometry: Alexandrov's Gluing Theorem, shortest paths and cut loci, Cauchy's Arm Lemma, domes, quasigeodesics, convexity, and algorithms throughout. The interplay between these topics and the way the main ideas develop throughout the book could make the 'journey' worthwhile for students and researchers in geometry, even if not directly interested in specific topics. Parts of the material will be of interest and accessible even to undergraduates. Although the proof difficulty varies from simple to quite intricate, with some proofs spanning several chapters, many examples and 125 figures help ease the exposition and illustrate the concepts. 260 pp. Englisch. Seller Inventory # 9783031475108
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Reshaping Convex Polyhedra
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Springer, Berlin|Springer Nature Switzerland|Springer, 2024
ISBN 10: 3031475100
ISBN 13: 9783031475108
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Condition: New. Dieser Artikel ist ein Print on Demand Artikel und wird nach Ihrer Bestellung fuer Sie gedruckt. The focus of this monograph is converting-reshaping-one 3D convex polyhedron to another via an operation the authors call tailoring. A convex polyhedron is a gem-like shape composed of flat facets, the focus of study since Plato and Eucli. Seller Inventory # 1128556351
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Reshaping Convex Polyhedra
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ISBN 10: 3031475100
ISBN 13: 9783031475108
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Buch. Condition: Neu. Druck auf Anfrage Neuware - Printed after ordering - The focus of this monograph is converting-reshaping-one 3D convex polyhedron to another via an operation the authors call 'tailoring.' A convex polyhedron is a gem-like shape composed of flat facets, the focus of study since Plato and Euclid. The tailoring operation snips off a corner (a 'vertex') of a polyhedron and sutures closed the hole. This is akin to Johannes Kepler's 'vertex truncation,' but differs in that the hole left by a truncated vertex is filled with new surface, whereas tailoring zips the hole closed. A powerful 'gluing' theorem of A.D. Alexandrov from 1950 guarantees that, after closing the hole, the result is a new convex polyhedron. Given two convex polyhedra P, and Q inside P, repeated tailoringallows P to be reshaped to Q. Rescaling any Q to fit inside P, the result is universal: any P can be reshaped to any Q. This is one of the main theorems in Part I, with unexpected theoretical consequences.Part II carries out a systematic study of 'vertex-merging,' a technique that can be viewed as a type of inverse operation to tailoring. Here the start is P which is gradually enlarged as much as possible, by inserting new surface along slits. In a sense, repeated vertex-merging reshapes P to be closer to planarity. One endpoint of such a process leads to P being cut up and 'pasted' inside a cylinder. Then rolling the cylinder on a plane achieves an unfolding of P. The underlying subtext is a question posed by Geoffrey Shephard in 1975 and already implied by drawings by Albrecht Dürer in the 15th century: whether every convex polyhedron can be unfolded to a planar 'net.' Toward this end, the authors initiate an exploration of convexity on convex polyhedra, a topic rarely studiedin the literature but with considerable promise for future development.This monograph uncovers new research directions and reveals connections among several, apparently distant, topics in geometry: Alexandrov's Gluing Theorem, shortest paths and cut loci, Cauchy's Arm Lemma, domes, quasigeodesics, convexity, and algorithms throughout. The interplay between these topics and the way the main ideas develop throughout the book could make the 'journey' worthwhile for students and researchers in geometry, even if not directly interested in specific topics. Parts of the material will be of interest and accessible even to undergraduates. Although the proof difficulty varies from simple to quite intricate, with some proofs spanning several chapters, many examples and 125 figures help ease the exposition and illustrate the concepts. Seller Inventory # 9783031475108
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