Geometry and the Imagination


Author: David Hilbert,Stephan Cohn-Vossen
Publisher: University of Pennsylvania Press
ISBN: 9780821819982
Category: Mathematics
Page: 357
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This remarkable book has endured as a true masterpiece of mathematical exposition. There are few mathematics books that are still so widely read and continue to have so much to offer - even after more than half a century has passed! The book is overflowing with mathematical ideas, which are always explained clearly and elegantly, and above all, with penetrating insight. It is a joy to read, both for beginners and experienced mathematicians. 'Hilbert and Cohn-Vossen' is full of interesting facts, many of which you wish you had known before. It's also likely that you have heard those facts before, but surely wondered where they could be found. The book begins with examples of the simplest curves and surfaces, including thread constructions of certain quadrics and other surfaces.The chapter on regular systems of points leads to the crystallographic groups and the regular polyhedra in $\mathbb{R}^3$. In this chapter, they also discuss plane lattices. By considering unit lattices, and throwing in a small amount of number theory when necessary, they effortlessly derive Leibniz's series: $\pi/4 = 1 - 1/3 1/5 - 1/7 - \ldots$. In the section on lattices in three and more dimensions, the authors consider sphere-packing problems, including the famous Kepler problem.One of the most remarkable chapters is 'Projective Configurations'. In a short introductory section, Hilbert and Cohn-Vossen give perhaps the most concise and lucid description of why a general geometer would care about projective geometry and why such an ostensibly plain setup is truly rich in structure and ideas. Here, we see regular polyhedra again, from a different perspective. One of the high points of the chapter is the discussion of Schlafli's Double-Six, which leads to the description of the 27 lines on the general smooth cubic surface. As is true throughout the book, the magnificent drawings in this chapter immeasurably help the reader.A particularly intriguing section in the chapter on differential geometry is Eleven Properties of the Sphere. Which eleven properties of such a ubiquitous mathematical object caught their discerning eye and why? Many mathematicians are familiar with the plaster models of surfaces found in many mathematics departments. The book includes pictures of some of the models that are found in the Gottingen collection. Furthermore, the mysterious lines that mark these surfaces are finally explained!The chapter on kinematics includes a nice discussion of linkages and the geometry of configurations of points and rods that are connected and, perhaps, constrained in some way. This topic in geometry has become increasingly important in recent times, especially in applications to robotics. This is another example of a simple situation that leads to a rich geometry. It would be hard to overestimate the continuing influence Hilbert-Cohn-Vossen's book has had on mathematicians of this century. It surely belongs in the 'pantheon' of great mathematics books.

Anschauliche Geometrie


Author: David Hilbert,Stefan Cohn-Vossen
Publisher: Springer-Verlag
ISBN: 3662366851
Category: Mathematics
Page: 312
View: 4165

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Dieser 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.

An Introduction to Differentiable Manifolds and Riemannian Geometry


Author: William Munger Boothby
Publisher: Gulf Professional Publishing
ISBN: 9780121160517
Category: Mathematics
Page: 419
View: 5295

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The second edition of this text has sold over 6,000 copies since publication in 1986 and this revision will make it even more useful. This is the only book available that is approachable by "beginners" in this subject. It has become an essential introduction to the subject for mathematics students, engineers, physicists, and economists who need to learn how to apply these vital methods. It is also the only book that thoroughly reviews certain areas of advanced calculus that are necessary to understand the subject. Line and surface integrals Divergence and curl of vector fields

Elementary Geometry


Author: Ilka Agricola,Thomas Friedrich
Publisher: American Mathematical Soc.
ISBN: 0821843478
Category: Mathematics
Page: 243
View: 8880

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Elementary geometry provides the foundation of modern geometry. For the most part, the standard introductions end at the formal Euclidean geometry of high school. Agricola and Friedrich revisit geometry, but from the higher viewpoint of university mathematics. Plane geometry is developed from its basic objects and their properties and then moves to conics and basic solids, including the Platonic solids and a proof of Euler's polytope formula. Particular care is taken to explain symmetry groups, including the description of ornaments and the classification of isometries by their number of fixed points. Complex numbers are introduced to provide an alternative, very elegant approach to plane geometry. The authors then treat spherical and hyperbolic geometries, with special emphasis on their basic geometric properties. This largely self-contained book provides a much deeper understanding of familiar topics, as well as an introduction to new topics that complete the picture of two-dimensional geometries. For undergraduate mathematics students the book will be an excellent introduction to an advanced point of view on geometry. For mathematics teachers it will be a valuable reference and a source book for topics for projects. The book contains over 100 figures and scores of exercises. It is suitable for a one-semester course in geometry for undergraduates, particularly for mathematics majors and future secondary school teachers.

Modern Geometric Structures and Fields


Author: Сергей Петрович Новиков,Искандер Асанович Тайманов
Publisher: American Mathematical Soc.
ISBN: 0821839292
Category: Mathematics
Page: 633
View: 9679

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The book presents the basics of Riemannian geometry in its modern form as geometry of differentiable manifolds and the most important structures on them. The authors' approach is that the source of all constructions in Riemannian geometry is a manifold that allows one to compute scalar products of tangent vectors. With this approach, the authors show that Riemannian geometry has a great influence to several fundamental areas of modern mathematics and its applications. In particular, Geometry is a bridge between pure mathematics and natural sciences, first of all physics. Fundamental laws of nature are formulated as relations between geometric fields describing various physical quantities. The study of global properties of geometric objects leads to the far-reaching development of topology, including topology and geometry of fiber bundles. Geometric theory of Hamiltonian systems, which describe many physical phenomena, led to the development of symplectic and Poisson geometry. Field theory and the multidimensional calculus of variations, presented in the book, unify mathematics with theoretical physics. Geometry of complex and algebraic manifolds unifies Riemannian geometry with modern complex analysis, as well as with algebra and number theory. Prerequisites for using the book include several basic undergraduate courses, such as advanced calculus, linear algebra, ordinary differential equations, and elements of topology.

Smarandache Notions

(book Series).
Author: Jack Allen,Feng Liu,Drago Constantinescu
Publisher: N.A
ISBN: 9781931233569
Category: Number theory
Page: N.A
View: 6916

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Hilbert


Author: Constance Reid,Hermann Weyl
Publisher: Springer-Verlag
ISBN: 3662286157
Category: Mathematics
Page: 290
View: 7261

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Levels of Infinity

Selected Writings on Mathematics and Philosophy
Author: Hermann Weyl,Peter Pesic
Publisher: Courier Corporation
ISBN: 0486489035
Category: Mathematics
Page: 240
View: 4658

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This original anthology collects 10 of Weyl's less-technical writings that address the broader scope and implications of mathematics. Most have been long unavailable or not previously published in book form. Subjects include logic, topology, abstract algebra, relativity theory, and reflections on the work of Weyl's mentor, David Hilbert. 2012 edition.

Einfachste Grundbegriffe der Topologie


Author: Paul Alexandroff,David Hilbert
Publisher: Springer-Verlag
ISBN: 3642911854
Category: Mathematics
Page: 50
View: 3014

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Dieser 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.

Symmetries and Laplacians

Introduction to Harmonic Analysis, Group Representations and Applications
Author: David Gurarie
Publisher: Courier Corporation
ISBN: 0486462889
Category: Mathematics
Page: 453
View: 445

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Designed as an introduction to harmonic analysis and group representations, this book examines concepts, ideas, results, and techniques related to symmetry groups and Laplacians. Its exposition is based largely on examples and applications of general theory, covering a wide range of topics rather than delving deeply into any particular area. Author David Gurarie, a Professor of Mathematics at Case Western Reserve University, focuses on discrete or continuous geometrical objects and structures, such as regular graphs, lattices, and symmetric Riemannian manifolds. Starting with the basics of representation theory, Professor Gurarie discusses commutative harmonic analysis, representations of compact and finite groups, Lie groups, and the Heisenberg group and semidirect products. Among numerous applications included are integrable hamiltonian systems, geodesic flows on symmetric spaces, and the spectral theory of the Hydrogen atom (Schrodinger operator with Coulomb potential) explicated by its Runge-Lenz symmetry. Three helpful appendixes include supplemental information, and the text concludes with references, a list of frequently used notations, and an index.

Elementare Differentialgeometrie


Author: Christian Bär
Publisher: Walter de Gruyter
ISBN: 3110224593
Category: Mathematics
Page: 356
View: 5676

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This textbook presents an introduction to the differential geometry of curves and surfaces. This second, revised edition has been expanded to include solutions and applications in cartography. Topics include Euclidean geometry, curve theory, surface theory, curvature concepts, minimal surfaces, Riemann geometry and the Gauss-Bonnet theorem.

Convexity and related combinatorial geometry

proceedings of the second University of Oklahoma conference
Author: David C. Kay,Marilyn Breen,University of Oklahoma
Publisher: Marcel Dekker Inc
ISBN: N.A
Category: Mathematics
Page: 243
View: 4768

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Walther von Dyck (1856-1934)

Mathematik, Technik und Wissenschaftsorganisation an der TH München
Author: Ulf Hashagen
Publisher: Franz Steiner Verlag
ISBN: 9783515083591
Category: Science
Page: 802
View: 9503

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Der Mathematiker Walther von Dyck gehorte als mehrfach wiedergewahlter Rektor der TH Muenchen zu den einflussreichsten bayerischen Wissenschaftlern seiner Zeit. Den Hohepunkt seines Einflusses erreichte Dyck in der Weimarer Republik als stellvertretender Vorsitzender des Hochschulverbandes sowie der aNotgemeinschafto (Vorlaufer der DFG). Diese umfassende biographische Studie bettet die wissenschaftliche Biographie des Mathematikers Dyck in eine fundierte Darstellung der Wissenschaftspolitik des Kaiserreichs und der Weimarer Republik ein. Darueber hinaus werden der Forschungs- und Lehrbetrieb an einer Technischen Hochschule, die innerfachlichen Streitigkeiten und Kampfe in der deutschen Mathematik sowie nicht zuletzt die Rolle Dycks als Organisator der aFlamisierungo der Universitat Gent im Rahmen der deutschen Kriegszielpolitik in Belgien analysiert. "a Hashagen hat mit seiner enorm informativen Studie Leben und Werk Walther von Dycks ein Worte gefasst und zu Papier gebrachtae und die eProbleme mit einer Biographieae nicht versteckt, sondern hervorragend gemeistert." sehepunkte "a eine bedeutende Biographie und zugleich eine beachtliche institutionen- wie disziplingeschichtliche Darstellunga" Das Historisch-Politische Buch "a the definitive biography of Walther von Dyck." H-Net Review "a ein nuetzliches Nachschlagewerka" Gesnerus "ainformativ, interessant, anregend a" Sudhoffs Archiv ."