Geometry Books

Book SynopsisThis book provides a systematic presentation of the mathematical foundation of modern physics with applications particularly within classical mechanics and the theory of relativity. Written to be self-contained, this book provides complete and rigorous proofs of all the results presented within. Among the themes illustrated in the book are differentiable manifolds, differential forms, fiber bundles and differential geometry with non-trivial applications especially within the general theory of relativity. The emphasis is upon a systematic and logical construction of the mathematical foundations. It can be used as a textbook for a pure mathematics course in differential geometry, assuming the reader has a good understanding of basic analysis, linear algebra and point set topology. The book will also appeal to students of theoretical physics interested in the mathematical foundation of the theories.Table of ContentsIntroduction. Smooth manifolds and vector bundles. Vector fields and differential equations. Tensors. Differential forms. Integration on manifolds. Metric and symplectic structures. Lie groups. Group actions. Fibre bundles. Isometric immersions and the second fundamental form. Jet bundles. Appendix.

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Book SynopsisThis book is a comprehensive tool both for self-study and for use as a text in classical geometry. It explains the concepts that form the basis for computer-aided geometric design.Table of ContentsI: Some Linear Algebra 1. Linear Systems 2. Linear Spaces 3. Least Squares II: Images and Projections 4. Parallel Projections 5. Moving the Object 6. Perspective Drawings 7. The Mapping Matrix 8. Reconstruction III: Affine Geometry 9. Affine Space 10. The Barycentric Calculus 11. Affine Maps 12. Affine Figures 13. Quadrics in Affine Spaces 14. More on Affine Quadrics 15. Homothetic Pencils IV: Euclidean Geometry 16. The Euclidean Space 17. Some Euclidean Figures 18. Quadrics in Euclidean Space 19. Focal Properties V: Some Projective Geometry 20. The Projective Space 21. Projective Maps 22. Some Projective Figures 23. Projective Quadrics VI: Some Descriptive Geometry 24. Associated Projections 25. Penetrations VII: Basic Algebraic Geometry 26. Implicit Curves and Surfaces 27. Parametric Curves and Surfaces 28. Some Elimination Methods 29. Implicitization, Inversion and Intersection VIII: Differential Geometry 30. Curves 31. Curves on Surfaces 32. Surfaces

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Book SynopsisTwists, Tilings, and Tessellation describes the underlying principles and mathematics of the broad and exciting field of abstract and mathematical origami, most notably the field of origami tessellations. It contains folding instructions, underlying principles, mathematical concepts, and many beautiful photos of the latest work in this fast-expanding field.Table of ContentsIntroduction. Vertices. Periodicity. Simple Twists. Twist Tiles. Tilings. Primal-Dual Tessellations. Rigid Foldability. Spherical Vertices. 3D Vectors. Rotational Solids. Afterword

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Book SynopsisThe Handbook of Finite Translation Planes provides a comprehensive listing of all translation planes derived from a fundamental construction technique, an explanation of the classes of translation planes using both descriptions and construction methods, and thorough sketches of the major relevant theorems.From the methods of André to coordinate and linear algebra, the book unifies the numerous diverse approaches for analyzing finite translation planes. It pays particular attention to the processes that are used to study translation planes, including ovoid and Klein quadric projection, multiple derivation, hyper-regulus replacement, subregular lifting, conical distortion, and Hermitian sequences. In addition, the book demonstrates how the collineation group can affect the structure of the plane and what information can be obtained by imposing group theoretic conditions on the plane. The authors also examine semifield and division ring planes and introduce the geometries of two-dimensional translation planes.As a compendium of examples, processes, construction techniques, and models, the Handbook of Finite Translation Planes equips readers with precise information for finding a particular plane. It presents the classification results for translation planes and the general outlines of their proofs, offers a full review of all recognized construction techniques for translation planes, and illustrates known examples.Trade Review"The authors, who are the undisputed leaders in the subject, present the huge material in shorter but virtually independent chapters, each dedicated to a particular aspect, such as the connection between translation planes and quasifields... This book highly recommended for the very clear, rigorous and detailed expostion and cannot be missing in the library of any researcher in Geometry."-Bambina Larato, Zentralblatt MATH, 2008, 1136Table of ContentsPreface and Acknowledgments. An Overview. Translation Plane Structure Theory. Partial Spreads and Translation Nets. Partial Spreads and Generalizations. Quasifields. Derivation. Frequently Used Tools. Sharply Transitive Sets. SL(2, p) × SL(2, p)-Planes. Classical Semifields. Groups of Generalized Twisted Field Planes. Nuclear Fusion in Semifields. Cyclic Semifields. T-Cyclic GL(2, q)-Spreads. Cone Representation Theory. André Net Replacements and Ostrom-Wilke Generalizations. Foulser's ?-Planes. Regulus Lifts, Intersections over Extension Fields. Hyper-Reguli Arising from André Hyper-Reguli. Translation Planes with Large Homology Groups. Derived Generalized André Planes. The Classes of Generalized André Planes. C-System Nearfields. Subregular Spreads. Fano Configurations. Fano Configurations in Generalized André Planes. Planes with Many Elation Axes. Klein Quadric. Parallelisms. Transitive Parallelisms. Ovoids.Known Ovoids. Simple T-Extensions of Derivable Nets. Baer Groups on Parabolic Spreads. Algebraic Lifting. Semifield Planes of Orders q4, q6. Known Classes of Semifields. Methods of Oyama-Suetake Planes. Coupled Planes. Hyper-Reguli. Subgeometry Partitions. Groups on Multiple Hyper-Reguli. Hyper-Reguli of Dimension 3. Elation-Baer Incompatibility. Hering-Ostrom Elation Theorem. Baer-Elation Theory. Spreads Admitting Unimodular Sections-Foulser-Johnson Theorem. Spreads of Order q2-Groups of Order q2. Transversal Extensions. Indicator Sets. Geometries and Partitions. Maximal Partial Spreads. Sperner Spaces. Conical Flocks. Ostrom and Flock Derivation. Transitive Skeletons. BLT-Set Examples. Many Ostrom-Derivates. Infinite Classes of Flocks. Sporadic Flocks. Hyperbolic Fibrations. Spreads with Many Homologies. Nests of Reguli. Chains. Multiple Nests. A Few Remarks on Isomorphisms. Flag-Transitive Geometries. Quartic Groups in Translation Planes. Double Transitivity. Triangle Transitive Planes. Hiramine-Johnson-Draayer Theory. Bol Planes. 2/3-Transitive Axial Groups. Doubly Transitive Ovals and Unitals. Rank 3 Affine Planes. Transitive Extensions. Higher-Dimensional Flocks. j…j-Planes. Orthogonal Spreads. Symplectic Groups-The Basics. Symplectic Flag-Transitive Spreads. Symplectic Spreads. When Is a Spread Not Symplectic? When Is a Spread Symplectic? The Translation Dual of a Semifield. Unitals in Translation Planes. Hyperbolic Unital Groups. Transitive Parabolic Groups. Doubly Transitive Hyperbolic Unital Groups. Retraction. Multiple Spread Retraction. Transitive Baer Subgeometry Partitions. Geometric and Algebraic Lifting. Quasi-Subgeometry Partitions. Hyper-Regulus Partitions. Small-Order Translation Planes. Dual Translation Planes and Their Derivates. Affine Planes with Transitive Groups. Cartesian Group Planes-Coulter-Matthews. Planes Admitting PGL(3, q). Planes of Order = 25. Real Orthogonal Groups and Lattices. Aspects of Symplectic and Orthogonal Geometry. Fundamental Results on Groups. Atlas of Planes and Processes. Bibliography. Theorems. Models. General Index.

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Book SynopsisAlgebraic and Differential Topology presents in a clear, concise, and detailed manner the fundamentals of homology theory. It first defines the concept of a complex and its Betti groups, then discusses the topolgoical invariance of a Betti group. The book next presents various applications of homology theory, such as mapping of polyhedrons onto other polyhedrons as well as onto themselves. The third volume in L.S. Pontryagin's Selected Works, this book provides as many insights into algebraic topology for today's mathematician as it did when the author was making his initial endeavors into this field.Table of ContentsAlgebraic and Differential Topology presents in a clear, concise, and detailed manner the fundamentals of homology theory. It first defines the concept of a complex and its Betti groups, then discusses the topolgoical invariance of a Betti group. The book next presents various applications of homology theory, such as mapping of polyhedrons onto other polyhedrons as well as onto themselves. The third volume in L.S. Pontryagin's Selected Works, this book provides as many insights into algebraic topology for today's mathematician as it did when the author was making his initial endeavors into this field.

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Book SynopsisThis is the first comprehensive monograph to thoroughly investigate constant width bodies, which is a classic area of interest within convex geometry. It examines bodies of constant width from several points of view, and, in doing so, shows surprising connections between various areas of mathematics. Concise explanations and detailed proofs demonstrate the many interesting properties and applications of these bodies. Numerous instructive diagrams are provided throughout to illustrate these concepts.An introduction to convexity theory is first provided, and the basic properties of constant width bodies are then presented. The book then delves into a number of related topics, which include Constant width bodies in convexity (sections and projections, complete and reduced sets, mixed volumes, and further partial fields) Sets of constant width in non-Euclidean geometries (in real Banach spaces, and in hyperbolic, spherical, and further non-Euclidean spaces) The concept of constant width in analysis (using Fourier series, spherical integration, and other related methods) Sets of constant width in differential geometry (using systems of lines and discussing notions like curvature, evolutes, etc.) Bodies of constant width in topology (hyperspaces, transnormal manifolds, fiber bundles, and related topics) The notion of constant width in discrete geometry (referring to geometric inequalities, packings and coverings, etc.) Technical applications, such as film projectors, the square-hole drill, and rotary engines Bodies of Constant Width: An Introduction to Convex Geometry with Applications will be a valuable resource for graduate and advanced undergraduate students studying convex geometry and related fields. Additionally, it will appeal to any mathematicians with a general interest in geometry.Trade Review“The book … provides an excellent treatment of the theory of bodies of constant width and gives an encyclopedic account of the topic from multiple perspectives. It can be viewed as a reference source for mathematicians working in this field ... . The whole book is written in a clear and attractive way. Besides the main text, additional notes and impressive collections of exercises at the end of each chapter direct the reader to additional venues and references … .” (Valery P. Soltan, zbMATH 1468.52001, 2021)Table of ContentsIntroduction.- Convex Geometry.- Basic Properties of Bodies of Constant Width.- Figures of Constant Width.- Systems of Lines in the Plane.- Spindle Convexity.- Complete and Reduced Convex Bodies.- Examples and Constructions.- Sections of Bodies of Constant Width.- Bodies of Constant Width in Mikowski Spaces.- Bodies of Constant Width in Differential Geometry.- Mixed Volumes.- Bodies of Constant Width in Analysis.- Geometric Inequalities.- Bodies of Constant Width in Discrete Geometry.- Bodies of Constant Width in Topology.- Concepts Related to Constant Width.- Bodies of Constant Width in Art, Design, and Engineering.

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Book SynopsisContinuing the theme of the previous volumes, these seminar notes reflect general trends in the study of Geometric Aspects of Functional Analysis, understood in a broad sense. Two classical topics represented are the Concentration of Measure Phenomenon in the Local Theory of Banach Spaces, which has recently had triumphs in Random Matrix Theory, and the Central Limit Theorem, one of the earliest examples of regularity and order in high dimensions. Central to the text is the study of the Poincaré and log-Sobolev functional inequalities, their reverses, and other inequalities, in which a crucial role is often played by convexity assumptions such as Log-Concavity. The concept and properties of Entropy form an important subject, with Bourgain's slicing problem and its variants drawing much attention. Constructions related to Convexity Theory are proposed and revisited, as well as inequalities that go beyond the Brunn–Minkowski theory. One of the major current research directions addressed is the identification of lower-dimensional structures with remarkable properties in rather arbitrary high-dimensional objects. In addition to functional analytic results, connections to Computer Science and to Differential Geometry are also discussed. Table of Contents- Jean Bourgain: In Memoriam. - A Generalized Central Limit Conjecture for Convex Bodies. - The Lower Bound for Koldobsky’s Slicing Inequality via Random Rounding. - Two-Sided Estimates for Order Statistics of Log-Concave Random Vectors. - Further Investigations of Rényi Entropy Power Inequalities and an Entropic Characterization of s-Concave Densities. - Small Ball Probability for the Condition Number of Random Matrices. - Concentration of the Intrinsic Volumes of a Convex Body. - Two Remarks on Generalized Entropy Power Inequalities. - On the Geometry of Random Polytopes. - Reciprocals and Flowers in Convexity. - Moments of the Distance Between Independent Random Vectors. - The Alon–Milman Theorem for Non-symmetric Bodies. - An Interpolation Proof of Ehrhard’s Inequality. - Bounds on Dimension Reduction in the Nuclear Norm. - High-Dimensional Convex Sets Arising in Algebraic Geometry. - Polylog Dimensional Subspaces of lN/∞. - On a Formula for the Volume of Polytopes.

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Book SynopsisThis textbook teaches the transformations of plane Euclidean geometry through problems, offering a transformation-based perspective on problems that have appeared in recent years at mathematics competitions around the globe, as well as on some classical examples and theorems. It is based on the combined teaching experience of the authors (coaches of several Mathematical Olympiad teams in Brazil, Romania and the USA) and presents comprehensive theoretical discussions of isometries, homotheties and spiral similarities, and inversions, all illustrated by examples and followed by myriad problems left for the reader to solve. These problems were carefully selected and arranged to introduce students to the topics by gradually moving from basic to expert level. Most of them have appeared in competitions such as Mathematical Olympiads or in mathematical journals aimed at an audience interested in mathematics competitions, while some are fundamental facts of mathematics discussed in the framework of geometric transformations. The book offers a global view of the geometric content of today's mathematics competitions, bringing many new methods and ideas to the attention of the public.Talented high school and middle school students seeking to improve their problem-solving skills can benefit from this book, as well as high school and college instructors who want to add nonstandard questions to their courses. People who enjoy solving elementary math problems as a hobby will also enjoy this work.Trade Review“This book … is a nice addition to the literature. … for instructors teaching geometry courses in which these are a topic, this book should provide an excellent source of interesting examples and problems. The large number of solved problems should also make useful reading for people preparing for mathematical contests and Olympiads.” (Mark Hunacek, MAA Reviews, October 4, 2022)Table of ContentsIntroduction.- Part I: Problems - 1. Isometries.- 2. Homotheties and Spiral Similarities.- 3. Inversions.- 4. A Synthesis.- Part II: Hints - 5. Isometries.- 6. Homotheties and Spiral Similarities.- 7. Inversions.- 8. A Synthesis.- Part III: Solutions - 9. Isometries.- 10. Homotheties and Spiral Similarities.- 11. Inversions.- 12. A Synthesis.

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Book SynopsisThis book serves as an introductory asset for learning metric geometry by delivering an in-depth examination of key constructions and providing an analysis of universal spaces, injective spaces, the Gromov-Hausdorff convergence, and ultralimits. This book illustrates basic examples of domestic affairs of metric spaces, this includes Alexandrov geometry, geometric group theory, metric-measure spaces and optimal transport.Researchers in metric geometry will find this book appealing and helpful, in addition to graduate students in mathematics, and advanced undergraduate students in need of an introduction to metric geometry. Any previous knowledge of classical geometry, differential geometry, topology, and real analysis will be useful in understanding the presented topics. Table of Contents1 Definitions. - A. Metric spaces. -B. Topology. - C. Variations. -D. Maximal metric and gluing. - E. Completeness. -F. Compact spaces. - G. Proper spaces. -H. Geodesics. - I. Metric trees. - J. Length. - K. Length spaces. - 2 Universal spaces. - A. Embedding in a normed space. - B. Extension property. - C. Universality. - D. Uniqueness and homogeneity. - E. Remarks. - 3 Injective spaces. - A. Definition. - B. Admissible and extremal functions. - C. Equivalent conditions. - D. Space of extremal functions. - E. Injective envelope. - F. Remarks. - 4 Space of subsets. - A. Hausdorff distance. -B. Hausdorff convergence. - C. An application. -D. Remarks. - 5 Space of spaces. - A. Gromov–Hausdorff metric. - B. Approximations and almost isometries 49; C. Optimal realization 50; D. Convergence 51; E. Uniformly totally bonded families 52; F. Gromov selection theorem. - G. Universal ambient space. - H. Remarks. - 6 Ultralimits. - A. Faces of ultrafilters. - B. Ultralimits of points. - C. An illustration. - D. Ultralimits of spaces. - E. Ultrapower. - F. Tangent and asymptotic spaces. - G. Remarks. - Semisolutions. -Index. - Bibliography

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Book SynopsisThe book is the second volume of a collection which consists of surveys that focus on important topics in geometry which are at the heart of current research. The topics in the present volume include the conformal and the metric geometry of surfaces, Teichmüller spaces, immersed surfaces of prescribed extrinsic curvature in 3-dimensional manifolds, symplectic geometry, the metric theory of Grassmann spaces, homogeneous metric spaces, polytopes, the higher-dimensional Gauss–Bonnet formula, isoperimetry in finitely generated groups and Coxeter groups.Each chapter is intended for graduate students and researchers. Several chapters are based on lectures given by their authors to middle-advanced level students and young researchers. The whole book is intended to be an introduction to important topics in geometry.Table of Contents1 Athanase Papadopoulos: Introduction.- 2 Norbert A'Campo and Athanase Papadopoulos, Geometry of surfaces.- 3 Ken’ichiOhshika: Teichmuller spaces and their various metrics.- 4 Marc Troyanov: Double forms, curvature integrals and the Gauss-Bonnet formula.- 5 Graham Smith, Quaternions, Monge–Amp˜A¨re structures and κ-surfaces.- 6 Peter Kristel and Eric Schippers, Lagrangian Grassmannians of polarizations.- 7 Arpad Kurusa, Metric characterizations of projective-metric spaces.- 8 Arpad Kurusa, Supplement to “Metric Characterization of Projective-metric Spaces”.- 9 Boumediene Et-Taoui, Metric problems in projective and Grassmann spaces.- 10 Valeriı Berestovskiı and Yuriı Nikonorov, On the geometry of finite homogeneous subsets of Euclidean spaces.- 11 Gue-Seon Lee and Ludovic Marquis, Discrete Coxeter groups.- 12 Bruno Luiz Santos Correia and Marc Troyanov, Isoperimetry in Finitely Generated Groups.

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Book SynopsisThis monograph considers several well-known mathematical theorems and asks the question, “Why prove it again?” while examining alternative proofs. It explores the different rationales mathematicians may have for pursuing and presenting new proofs of previously established results, as well as how they judge whether two proofs of a given result are different. While a number of books have examined alternative proofs of individual theorems, this is the first that presents comparative case studies of other methods for a variety of different theorems.The author begins by laying out the criteria for distinguishing among proofs and enumerates reasons why new proofs have, for so long, played a prominent role in mathematical practice. He then outlines various purposes that alternative proofs may serve. Each chapter that follows provides a detailed case study of alternative proofs for particular theorems, including the Pythagorean Theorem, the Fundamental Theorem of Arithmetic, Desargues’ Theorem, the Prime Number Theorem, and the proof of the irreducibility of cyclotomic polynomials.Why Prove It Again? will appeal to a broad range of readers, including historians and philosophers of mathematics, students, and practicing mathematicians. Additionally, teachers will find it to be a useful source of alternative methods of presenting material to their students.Trade Review“The book motivates and introduces its topic well and successively argues for the claim that comparative studies or proofs are a worthwhile occupation. All chapters are accessible to a generally informed mathematical audience, most of them to mathematical laymen with a basic knowledge of number theory and geometry.” (Merlin Carl, Mathematical Reviews, April, 2016)“This book addresses the question of why mathematicians prove certain fundamental theorems again and again. … Each chapter is a historical account of how and why these theorems have been reproved several times throughout several centuries. The primary readers of this book will be historians or philosophers of mathematics … .” (M. Bona, Choice, Vol. 53 (6), February, 2016)“This is an impressive book, giving proofs, sketches, or ideas of proofs of a variety of fundamental theorems of mathematics, ranging from Pythagoras’s theorem, through the fundamental theorems of arithmetic and algebra, to the compactness theorem of first-order logic. … because of the many examples given, there should be something to suit everybody’s taste … .” (Jessica Carter, Philosophia Mathematica, February, 2016)Table of ContentsProofs in Mathematical Practice.- Motives for Finding Alternative Proofs.- Sums of Integers.- Quadratic Surds.- The Pythagorean Theorem.- The Fundamental Theorem of Arithmetic.- The Infinitude of the Primes.- The Fundamental Theorem of Algebra.- Desargues's Theorem.- The Prime Number Theorem.- The Irreducibility of the Cyclotomic Polynomials.- The Compactness of First-Order Languages.- Other Case Studies.

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Book SynopsisThis book describes several mathematical models of the primary visual cortex, referring them to a vast ensemble of experimental data and putting forward an original geometrical model for its functional architecture, that is, the highly specific organization of its neural connections. The book spells out the geometrical algorithms implemented by this functional architecture, or put another way, the “neurogeometry” immanent in visual perception. Focusing on the neural origins of our spatial representations, it demonstrates three things: firstly, the way the visual neurons filter the optical signal is closely related to a wavelet analysis; secondly, the contact structure of the 1-jets of the curves in the plane (the retinal plane here) is implemented by the cortical functional architecture; and lastly, the visual algorithms for integrating contours from what may be rather incomplete sensory data can be modelled by the sub-Riemannian geometry associated with this contact structure. As such, it provides readers with the first systematic interpretation of a number of important neurophysiological observations in a well-defined mathematical framework. The book’s neuromathematical exploration appeals to graduate students and researchers in integrative-functional-cognitive neuroscience with a good mathematical background, as well as those in applied mathematics with an interest in neurophysiology.Table of ContentsPreface.- Introdcution.- Receptive Fields and Profiles, and Wavelet Analysis.- Pinwheels of V1 Horizontal Connections and Contact Structure.-Transition to Volume II.- References.

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Book SynopsisComputational synthetic geometry deals with methods for realizing abstract geometric objects in concrete vector spaces. This research monograph considers a large class of problems from convexity and discrete geometry including constructing convex polytopes from simplicial complexes, vector geometries from incidence structures and hyperplane arrangements from oriented matroids. It turns out that algorithms for these constructions exist if and only if arbitrary polynomial equations are decidable with respect to the underlying field. Besides such complexity theorems a variety of symbolic algorithms are discussed, and the methods are applied to obtain new mathematical results on convex polytopes, projective configurations and the combinatorics of Grassmann varieties. Finally algebraic varieties characterizing matroids and oriented matroids are introduced providing a new basis for applying computer algebra methods in this field. The necessary background knowledge is reviewed briefly. The text is accessible to students with graduate level background in mathematics, and will serve professional geometers and computer scientists as an introduction and motivation for further research.Table of ContentsPreliminaries.- On the existence of algorithms.- Combinatorial and algebraic methods.- Algebraic criteria for geometric realizability.- Geometric methods.- Recent topological results.- Preprocessing methods.- On the finding of polyheadral manifolds.- Matroids and chirotopes as algebraic varieties.

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Book SynopsisDieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

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Book SynopsisDie Kreisgeometrie eignet sich in idealer Weise, den Reichtum der Geometrie zu erschließen. Ausgehend von den klassischen, über 2000 Jahre alten Sätzen der Kreisgeometrie spannt der Autor den Bogen bis in die Neuzeit, in der neue, vor allem von Jacob Steiner entwickelte Werkzeuge der Kreisgeometrie einen enormem Schub brachten. Damit gelingt es ihm, ein breites Themenspektrum anzusprechen, das nicht nur viele berühmte Sätze, sondern auch zahlreiche kaum bekannte Resultate umfasst.Um die Beweisideen und deren geometrischen Kern transparent zu machen, steht bei allen Beweisen die geometrische Argumentation im Vordergrund.Über 250 Abbildungen und ein lockerer, aber präziser Schreibstil begleiten den Leser bei dieser faszinierenden Reise durch die Kreisgeometrie.Table of ContentsOuvertüre: Kreise in gotischem Maßwerk.- Grundlagen.- Potenzgerade und Kreisbüschel.- Krummes soll gerade werden - die Inversion am Kreis.- Berühmte Kreise.- Vielecke in und um Kreisen.- Auch Geraden sind Kreise - die konforme Ebene.- Das Appolonische Berührproblem.- Kreisketten.- (K)eine runde Sache - Kurven konstanter Breite.- Konstruktionen - ohne Kreis(e) geht es nicht.- Literaturverzeichnis.- Abbildungsverzeichnis.- Index.

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Book SynopsisDieses Buch zeichnet ein – im wahrsten Sinne des Wortes – farbenfrohes Bild von Mathematik: Es stellt eine Auswahl von mathematischen Themen vor, die mithilfe durchgehend farbig gestalteter Abbildungen und unterstützt durch möglichst einfache Erläuterungen erschlossen werden. Auf Formeln und Rechnungen wurde soweit wie möglich verzichtet.Das Buch eignet sich insbesondere für Kinder ab etwa 8 Jahren und Jugendliche. Lehrenden und Eltern bietet es entsprechende Anregungen für gemeinsame mathematische Erkundungen. Die einzelnen Kapitel sind im Wesentlichen unabhängig voneinander lesbar, sodass sie möglichst vielseitig einsetzbar sind. Durch regelmäßig eingestreute Anregungen zum Nachdenken und Anstöße für eigene Untersuchungen wird mathematisches Entdecken erlebbar. Lösungshinweise zu diesen Anregungen sowie Kopiervorlagen werden online zur Verfügung gestellt.Die Mehrzahl der Themen wird Leserinnen und Lesern der beliebten Bücher Mathematik ist schön, Mathematik ist wunderschön und Mathematik ist wunderwunderschön bekannt vorkommen – es handelt sich um eine Auswahl von Inhalten aus diesen Büchern, die hier größtenteils völlig neu aufbereitet wurden: Die Darstellungen sind im vorliegenden „kunterbunten“ Buch nochmal sehr viel anschaulicher, ausführlicher und kindgerechter. Außerdem sind weitere Themen hinzugekommen, die sich besonders für jüngere Mathematikinteressierte eignen.Table of ContentsEinfache Bausteine zum Parkettieren.- Flächenteilungen.- Muster aus bunten Steinen.- Keltische Knoten.- Muster legen mit quadratischen Fliesen und Karten.- Wortschlangen und Wege im Rechteck-Raster.- Regelmäßige Vielecke und Sterne.- Das verschwundene Quadrat.- Parkettieren mit goldenen Dreiecken.- Magische Quadrate.- Würfel und Würfelspiele.- Entdeckungen mit Quadratzahlen und Potenzen.

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Book SynopsisDieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

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Book SynopsisThe origins of the word problem are in group theory, decidability and complexity. But through the vision of M. Gromov and the language of filling functions, the topic now impacts the world of large-scale geometry. This book contains accounts of many recent developments in Geometric Group Theory and shows the interaction between the word problem and geometry continues to be a central theme. It contains many figures, numerous exercises and open questions.Table of ContentsDehn Functions and Non-Positive Curvature.- The Isoperimetric Spectrum.- Dehn Functions of Subgroups of CAT(0) Groups.- Filling Functions.- Filling Functions.- Relationships Between Filling Functions.- Example: Nilpotent Groups.- Asymptotic Cones.- Diagrams and Groups.- Dehn’s Problems and Cayley Graphs.- Van Kampen Diagrams and Pictures.- Small Cancellation Conditions.- Isoperimetric Inequalities and Quasi-Isometries.- Free Nilpotent Groups.- Hyperbolic-by-free groups.

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Book SynopsisPer la prima volta sono riuniti e didatticamente rielaborati a fondo in un manuale e in un CD i testi di base della geometria della visione. L'opera raccoglie i testi originali di Euclide (l'Ottica), Menelao (La Sferica), Alberti (De Pictura), Piero della Francesca (De Prospectiva Pingendi), che hanno dato origine al disegno prospettico rinascimentale e alla moderna geometria proiettiva. L'opera offre la possibilità di unire intuizione e ragionamento, costruendo un'immagine chiara dello sviluppo della matematica legata alla visione, dalle origini classiche, al rinascimento, alla moderna geometria proiettiva. Numerose schede e animazioni interattive facilitano l'intuizione degli argomenti, che sono comunque trattati con il massimo rigore e chiarezza, in una esposizione didatticamente molto efficace.Table of ContentsIntroduzione; Cap.I, Euclide: Dagli Elementi all'Ottica; Cap. II, La visione di oggetti uguali; Cap. III, La visione della profondità; Cap. IV, La prospettiva; Cap. V, Corrispondenze conformi e omotetie; Cap. VI, Piero della Francesca; Cap. VII, La geometria proiettiva; Appendice.

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Book SynopsisThe book provides an introduction to Differential Geometry of Curves and Surfaces. The theory of curves starts with a discussion of possible definitions of the concept of curve, proving in particular the classification of 1-dimensional manifolds. We then present the classical local theory of parametrized plane and space curves (curves in n-dimensional space are discussed in the complementary material): curvature, torsion, Frenet’s formulas and the fundamental theorem of the local theory of curves. Then, after a self-contained presentation of degree theory for continuous self-maps of the circumference, we study the global theory of plane curves, introducing winding and rotation numbers, and proving the Jordan curve theorem for curves of class C2, and Hopf theorem on the rotation number of closed simple curves. The local theory of surfaces begins with a comparison of the concept of parametrized (i.e., immersed) surface with the concept of regular (i.e., embedded) surface. We then develop the basic differential geometry of surfaces in R3: definitions, examples, differentiable maps and functions, tangent vectors (presented both as vectors tangent to curves in the surface and as derivations on germs of differentiable functions; we shall consistently use both approaches in the whole book) and orientation. Next we study the several notions of curvature on a surface, stressing both the geometrical meaning of the objects introduced and the algebraic/analytical methods needed to study them via the Gauss map, up to the proof of Gauss’ Teorema Egregium. Then we introduce vector fields on a surface (flow, first integrals, integral curves) and geodesics (definition, basic properties, geodesic curvature, and, in the complementary material, a full proof of minimizing properties of geodesics and of the Hopf-Rinow theorem for surfaces). Then we shall present a proof of the celebrated Gauss-Bonnet theorem, both in its local and in its global form, using basic properties (fully proved in the complementary material) of triangulations of surfaces. As an application, we shall prove the Poincaré-Hopf theorem on zeroes of vector fields. Finally, the last chapter will be devoted to several important results on the global theory of surfaces, like for instance the characterization of surfaces with constant Gaussian curvature, and the orientability of compact surfaces in R3.Trade ReviewFrom the reviews:“The authors’ goal in writing this book is to present the theory of curves and surfaces from the viewpoint of contemporary mathematics. … New concepts and new definitions are fully motivated, and illustrated by numerous examples. … the book is beautifully written, very well organized, and most of all it may serve as both a less advanced text and a more advanced text for readers interested in the theory of curves and surfaces.” (Andrew Bucki, Mathematical Reviews, June, 2013)“It is dedicated to the study of curves and surfaces both from a local and global viewpoint. It is written and organised in such a way that it can be used by a large scope of students, not only for beginning, intermediate or advanced undergraduate courses in mathematics or physics, but also for engineering or computer science students. … the book can be useful for post-graduate students, too. The book is well written and includes many examples and figures.” (Raúl Oset Sinha, Zentralblatt MATH, Vol. 1238, 2012)

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