Proof in Geometry
This single-volume compilation of two books explores the construction of geometric proofs. In addition to offering useful criteria for determining correctness, it presents examples of faulty proofs that illustrate common errors. High-school geometry is the sole prerequisite.Proof in Geometry, the first in this two-part compilation, discusses the construction of geometric proofs and presents criteria useful for determining whether a proof is logically correct and whether it actually constitutes proof. It features sample invalid proofs, in which the errors are explained and corrected.Mistakes in Geometric Proofs, the second book in this compilation, consists chiefly of examples of faulty proofs. Some illustrate mistakes in reasoning students might be likely to make, and others are classic sophisms. Chapters 1 and 3 present the faulty proofs, and chapters 2 and 4 offer comprehensive analyses of the errors.
Geometry Workbook for Dummies
Make gerometry paractice easy and dun!Geometry is one of the oldest mathematical subjects in history. Unfortunately, few geometry study guides offer clear explanations, causing many people to get tripped up or lost when trying to solve a proof--even when they know the terms and concepts like the back of their hand. However, this problem can be fixed with practice and some strategies for slicing through all the mumbo-jumbo and getting right to the heart of the proof. Geometry Workbook For Dummies ensures that practice makes perfect, especially when problems are presented without the stiff, formal style that you'd find in your math textbook. Written with a commonsense, street-smart approach, this guide gives you the step-by-step process to solve each proof, along with tips, shortcuts, and mnemonic devices to make sure the solutions stick. It also gives you plenty of room to work out your solutions, providing you with space to breathe and a clear head. This book provides you with the tools you need to solve all types of geometry problems, including: Congruent triangles Finding the area, angle, and size of quadrilaterals Angle-arc theorems and formulas Touching radii and tangents Connecting radii and chords Parallel, perpendicular, and intersecting lines and planes Slope, distance, and midpoint formulas Line and circle equations Handling rotations, reflections, and other transformations Packed with tons of strategies for solving proofs and a review of key concepts, Geometry Workbook For Dummies is the ultimate study aid for students, parents, and anyone with an interest in the field.
Analytical Geometry of Three Dimensions
Brief but rigorous, this text is geared toward advanced undergraduates and graduate students. It covers the coordinate system, planes and lines, spheres, homogeneous coordinates, general equations of the second degree, quadric in Cartesian coordinates, and intersection of quadrics. Mathematician, physicist, and astronomer, William H. McCrea conducted research in many areas and is best known for his work on relativity and cosmology. McCrea studied and taught at universities around the world, and this book is based on a series of his lectures.
Projective Geometry
This text explores the methods of the projective geometry of the plane. Some knowledge of the elements of metrical and analytical geometry is assumed; a rigorous first chapter serves to prepare readers. Following an introduction to the methods of the symbolic notation, the text advances to a consideration of the theory of one-to-one correspondence. It derives the projective properties of the conic and discusses the representation of these properties by the general equation of the second degree. A study of the relationship between Euclidean and projective geometry concludes the presentation. Numerous illustrative examples appear throughout the text.
Algebraic Geometry
This text for advanced undergraduate students is both an introduction to algebraic geometry and a bridge between its two parts -- the analytical-topological and the algebraic. Because of its extensive use of formal power series (power series without convergency), the treatment will appeal to readers conversant with analysis but less familiar with the formidable techniques of modern algebra. The book opens with an overview of the results required from algebra and proceeds to the fundamental concepts of the general theory of algebraic varieties: general point, dimension, function field, rational transformations, and correspondences. A concentrated chapter on formal power series with applications to algebraic varieties follows. An extensive survey of algebraic curves includes places, linear series, abelian differentials, and algebraic correspondences. The text concludes with an examination of systems of curves on a surface.
Gauge Theory And Variational Principles
This text provides a framework for describing and organizing the basic forces of nature and the interactions of subatomic particles. A detailed and self-contained mathematical account of gauge theory, it is geared toward beginning graduate students and advanced undergraduates in mathematics and physics. This well-organized treatment supplements its rigor with intuitive ideas.Starting with an examination of principal fiber bundles and connections, the text explores curvature; particle fields, Lagrangians, and gauge invariance; Lagrange's equation for particle fields; and the inhomogeneous field equation. Additional topics include free Dirac electron fields; interactions; calculus on frame bundle; and unification of gauge fields and gravitation. The text concludes with references, a selected bibliography, an index of notation, and a general index.
General Investigations of Curved Surfaces
Gauss's theory of surfaces is among the purely mathematical achievements inspired by ideas that arose in connection with surveys of the surface of the earth. Long regarded as a masterpiece in content and form, this work features one of the author's most original contributions to mathematics--the discovery that Gauss termed the "Theorema Egregium." It consists of his penetrating definition of the concept of surface curvature and the theorem that the "Gauss curvature" is invariant under arbitrary isometric deformation of a curved surface. The profound effects of these concepts were soon generalized by Bernhard Riemann, and subsequent development included the important role of the Gauss-Riemann concept of curvature in modern relativity theory.This edition of Gauss's classic work features a new introduction, bibliography, and notes by science historian Peter Pesic. In addition, an informative appendix offers historical background.
The Fourth Dimension Simply Explained
Twenty-two essays examine the fourth dimension: how it may be studied, its relationship to non-Euclidean geometry, analogues to three-dimensional space, its absurdities and curiosities, and its simpler properties. 1910 edition.
History Of Analytic Geometry
Specifically designed as an integrated survey of the development of analytic geometry, this classic study takes a unique approach to the history of ideas. The author, a distinguished historian of mathematics, presents a detailed view of not only the concepts themselves, but also the ways in which they extended the work of each generation, from before the Alexandrian Age through the eras of the great French mathematicians Fermat and Descartes, and on through Newton and Euler to the "Golden Age," from 1789 to 1850. Appropriate as an undergraduate text, this history is accessible to any mathematically inclined reader. 1956 edition. Analytical bibliography. Index.
The Geometry of Meaning
From Pythagoras's harmonic sequence to Einstein's theory of relativity, geometric models of position, proximity, ratio, and the underlying properties of physical space have provided us with powerful ideas and accurate scientific tools. Currently, similar geometric models are being applied to another type of space--the conceptual space of information and meaning, where the contributions of Pythagoras and Einstein are a part of the landscape itself. The rich geometry of conceptual space can be glimpsed, for instance, in internet documents: while the documents themselves define a structure of visual layouts and point-to-point links, search engines create an additional structure by matching keywords to nearby documents in a spatial arrangement of content. What the Geometry of Meaning provides is a much-needed exploration of computational techniques to represent meaning and of the conceptual spaces on which these representations are founded.
Extremal Graph Theory
The ever-expanding field of extremal graph theory encompasses a diverse array of problem-solving methods, including applications to economics, computer science, and optimization theory. This volume, based on a series of lectures delivered to graduate students at the University of Cambridge, presents a concise yet comprehensive treatment of extremal graph theory.Unlike most graph theory treatises, this text features complete proofs for almost all of its results. Further insights into theory are provided by the numerous exercises of varying degrees of difficulty that accompany each chapter. Although geared toward mathematicians and research students, much of Extremal Graph Theory is accessible even to undergraduate students of mathematics. Pure mathematicians will find this text a valuable resource in terms of its unusually large collection of results and proofs, and professionals in other fields with an interest in the applications of graph theory will also appreciate its precision and scope.
Euclidean Geometry and Transformations
"A good textbook." ― Mathematical Gazette. This introduction to Euclidean geometry emphasizes both the theory and the practical application of isometries and similarities to geometric transformations. Each chapter begins with an optional commentary on the history of geometry. Contents include modern elementary geometry, isometries and similarities in the plane, vectors and complex numbers in geometry, inversion, and isometries in space. Numerous exercises appear throughout the text, many of which have corresponding answers and hints at the back of the book. Prerequisites for this text, which is suitable for undergraduate courses, include high school algebra, geometry, and elementary trigonometry. 1972 edition.
Prime Obsession
The definitive story of the Riemann Hypothesis, a fascinating and epic mathematical mystery that continues to challege the world. In 1859, Bernhard Riemann, a little-known thirty-two year old mathematician, made a hypothesis while presenting a paper to the Berlin Academy titled "On the Number of Prime Numbers Less Than a Given Quantity." Today, after 150 years of careful research and exhaustive study, the Riemann Hyphothesis remains unsolved, with a one-million-dollar prize earmarked for the first person to conquer it.Alternating passages of extraordinarily lucid mathematical exposition with chapters of elegantly composed biography and history, Prime Obsession is a fascinating and fluent account of an epic mathematical mystery that continues to challenge and excite the world.
Pythagorean Triangles
The Pythagorean Theorem is one of the fundamental theorems of elementary geometry, and Pythagorean triangles -- right triangles whose sides are natural numbers -- have been studied by mathematicians since antiquity. In this classic text, a brilliant Polish mathematician explores the intriguing mathematical relationships in such triangles.Starting with "primitive" Pythagorean triangles, the text examines triangles with sides less than 100, triangles with two sides that are successive numbers, divisibility of one of the sides by 3 or by 5, the values of the sides of triangles, triangles with the same arm or the same hypotenuse, triangles with the same perimeter, and triangles with the same area. Additional topics include the radii of circles inscribed in Pythagorean triangles, triangles in which one or more sides are squares, triangles with natural sides and natural areas, triangles in which the hypotenuse and the sum of the arms are squares, representation of triangles with the help of the points of a plane, right triangles whose sides are reciprocals of natural numbers, and cuboids with edges and diagonals expressed by natural numbers.
Practical Conic Sections
Illustrated with interesting examples from everyday life, this text shows how to create ellipses, parabolas, and hyperbolas and presents fascinating historical background on their ancient origins. The text starts with a discussion of techniques for generating the conic curves, showing how to create accurate depictions of large or small conic curves and describing their reflective properties, from light in telescopes to sound in microphones and amplifiers. It further defines the role of curves in the construction of auditoriums, antennas, lamps, and numerous other design applications. Only a basic knowledge of plane geometry needed; suitable for undergraduate courses. 1993 edition. 98 figures.
Ruler and the Round
An intriguing look at the ""impossible"" geometric constructions (those that defy completion with just a ruler and a compass), this book covers angle trisection and circle division. 1970 edition.
Euclids Elements
The classic Heath translation, in a completely new layout with plenty of space and generous margins. An affordable but sturdy student and teacher sewn softcover edition in one volume, with minimal notes and a new index/glossary.
The Works of Archimedes
The complete works of antiquity's great geometer appear here in a highly accessible English translation by a distinguished scholar. Remarkable for his range of thought and his mastery of treatment, Archimedes addressed such topics as the famous problems of the ratio of the areas of a cylinder and an inscribed sphere; the measurement of a circle; the properties of conoids, spheroids, and spirals; and the quadrature of the parabola. This edition offers an informative introduction with many valuable insights into the ancient mathematician's life and thought as well as the views of his contemporaries. Modern mathematicians, physicists, science historians, and logicians will find this volume a source of timeless fascination.
Euclid's Window
Through Euclid's Window Leonard Mlodinow brilliantly and delightfully leads us on a journey through five revolutions in geometry, from the Greek concept of parallel lines to the latest notions of hyperspace. Here is an altogether new, refreshing, alternative history of math revealing how simple questions anyone might ask about space -- in the living room or in some other galaxy -- have been the hidden engine of the highest achievements in science and technology. Based on Mlodinow's extensive historical research; his studies alongside colleagues such as Richard Feynman and Kip Thorne; and interviews with leading physicists and mathematicians such as Murray Gell-Mann, Edward Witten, and Brian Greene, Euclid's Window is an extraordinary blend of rigorous, authoritative investigation and accessible, good-humored storytelling that makes a stunningly original argument asserting the primacy of geometry. For those who have looked through Euclid's Window, no space, no thing, and no time will ever be quite the same.
Spherical Models
Well known through the work of Buckminster Fuller, models of spherical polyhedral excite the imaginations not only of mathematicians, architects and engineers but also of artists and designers. This book, concerned with polyhedrons, sphere tilings, and dome structures, offers a well-illustrated, practical approach to creating a host of beautiful and interesting models, including starfaced spherical forms that can serve as basic structures for geodesic domes.Employing only the most elementary mathematical principles, the text initially provides complete instructions for making spherical models from five regular solids, using only circular bands of paper, a ruler and a compass. Also discussed is tessellation, or tiling, on a sphere and how to make spherical models of all the semiregular solids. The volume concludes with a discussion of the relationship of polyhedral to geodesic domes and directions for building models of the domes.Abundantly illustrated with photographs, drawings, and computer graphics of attractive geometrical models, this volume will appeal to a wide range of readers--from students and teachers of mathematics, art, design, architecture and engineering, to recreational mathematics enthusiasts and builders of geodesic domes.
Curvature and Homology
This systematic and self-contained treatment examines the topology of differentiable manifolds, curvature and homology of Riemannian manifolds, compact Lie groups, complex manifolds, and curvature and homology of Kaehler manifolds. It generalizes the theory of Riemann surfaces to that of Riemannian manifolds. Includes four helpful appendixes. "A valuable survey." -- Nature. 1962 edition.
Topological Spaces
Set topology, the subject of the present volume, studies sets in topological spaces and topological vector spaces; whenever these sets are collections of n-tuples or classes of functions, the book recovers well-known results of classical analysis.In order to provide a topological tool of equal interest to students of both pure and applied mathematics, this work includes a systematic development of the properties of multi-valued functions. Topics include families of sets, mappings of one set into another, ordered sets, topological spaces, topological properties of metric spaces, mappings from one topological space into another, mappings of one vector space into another, convex sets and convex functions in the space R" and topological vector spaces. To render the expositions more concrete, examples have been included from very different domains. 1963 edition.
The Green Book of Mathematical Problems
This convenient volume offers a popular selection of 100 practice problems -- with hints and solutions -- for students preparing for undergraduate-level math competitions. The challenging brainteasers will also appeal to anyone interested in problems concerning real numbers, differential equations, integrals, polynomials, sets, and other mathematical topics. The hints are very helpful and the solutions are easy to follow. Questions drawn from geometry, group theory and linear algebra involve subjects ranging from multivariate integration to finite series to infinite sums and classical analysis. The only prerequisite is a high school-level background in mathematics. Problem-solvers at varying degrees of proficiency will find this treasury of top-notch numerical puzzles an invitation to hone their mathematical skills as well as a source of stimulating entertainment.
Theory of Linear Operators in Hilbert Space/Two Volumes Bound As One
Shapes, Space, and Symmetry
"Altogether a most instructive, entertaining, and esthetically pleasing book." -- Science.Since the ancient Greeks, the visualization of space has been a challenge that has intrigued men of learning. Through centuries of thoughtful looking, a number of three-dimensional figures or polyhedral, as the Greeks called them, have been discovered, admired, and wondered at for their mathematical elegance and beauty. And they have been put to use in remarkably diverse ways by engineers and builders, chemists and crystallographers, architects and sculptors.This book describes very clearly and simply, and illustrates with beautiful photographs of models, a great number of three-dimensional figures, all but a few consisting of plane faces bounded by straight lines. It examines the nine regular solids -- the five commonly called Platonic, described by Theaetetus in the fourth century B.C., and the four called Kepler-Poinsot, two each of which were discovered by Kepler and Poinsot many centuries later. And it examines many variations obtained by truncation, stellation, dualization, and compounding.Writing for the layman as well as the student or professional in mathematics, Alan Holden explains the structure of the figures and demonstrates how they can be used to explain mathematics visually rather than by symbol systems, an effort hailed by Scientific American magazine as "a victory of clear, connected thinking over the theorematic method." At the end of the book the author includes a section containing instructions for constructing cardboard models.
Differential Geometry
This outstanding textbook by a distinguished mathematical scholar introduces the differential geometry of curves and surfaces in three-dimensional Euclidean space. The subject is presented in its simplest, most essential form, but with many explanatory details, figures and examples, and in a manner that conveys the geometric significance and theoretical and practical importance of the different concepts, methods and results involved.The first chapters of the book focus on the basic concepts and facts of analytic geometry, the theory of space curves, and the foundations of the theory of surfaces, including problems closely related to the first and second fundamental forms. The treatment of the theory of surfaces makes full use of the tensor calculus.The later chapters address geodesics, mappings of surfaces, special surfaces, and the absolute differential calculus and the displacement of Levi-Civit?. Problems at the end of each section (with solutions at the end of the book) will help students meaningfully review the material presented, and familiarize themselves with the manner of reasoning in differential geometry.
Sacred Geometry
The thinkers of ancient Egypt, Greece and India recognized that numbers governed much of what they saw in their world and hence provided an approach to its divine creator. Robert Lawlor sets out the system that determines the dimension and the form of both man-made and natural structures, from Gothic cathedrals to flowers, from music to the human body. By also involving the reader in practical experiments, he leads with ease from simple principles to a grasp of the logarithmic spiral, the Golden Proportion, the squaring of the circle and other ubiquitous ratios and proportions.Art and Imagination: These large-format, gloriously-illustrated paperbacks cover Eastern and Western religion and philosophy, including myth and magic, alchemy and astrology. The distinguished authors bring a wealth of knowledge, visionary thinking and accessible writing to each intriguing subject.
Geometry
"A lucid and masterly survey." -- Mathematics GazetteProfessor Pedoe is widely known as a fine teacher and a fine geometer. His abilities in both areas are clearly evident in this self-contained, well-written, and lucid introduction to the scope and methods of elementary geometry. It covers the geometry usually included in undergraduate courses in mathematics, except for the theory of convex sets. Based on a course given by the author for several years at the University of Minnesota, the main purpose of the book is to increase geometrical, and therefore mathematical, understanding and to help students enjoy geometry.Among the topics discussed: the use of vectors and their products in work on Desargues' and Pappus' theorem and the nine-point circle; circles and coaxal systems; the representation of circles by points in three dimensions; mappings of the Euclidean plane, similitudes, isometries, mappings of the inversive plane, and Moebius transformations; projective geometry of the plane, space, and n dimensions; the projective generation of conics and quadrics; Moebius tetrahedra; the tetrahedral complex; the twisted cubic curve; the cubic surface; oriented circles; and introduction to algebraic geometry.In addition, three appendices deal with Euclidean definitions, postulates, and propositions; the Grassmann-Pluecker coordinates of lines in S3, and the group of circular transformations. Among the outstanding features of this book are its many worked examples and over 500 exercises to test geometrical understanding.
Lectures on Classical Differential Geometry
Elementary, yet authoritative and scholarly, this book offers an excellent brief introduction to the classical theory of differential geometry. It is aimed at advanced undergraduate and graduate students who will find it not only highly readable but replete with illustrations carefully selected to help stimulate the student's visual understanding of geometry. The text features an abundance of problems, most of which are simple enough for class use, and often convey an interesting geometrical fact. A selection of more difficult problems has been included to challenge the ambitious student.Written by a noted mathematician and historian of mathematics, this volume presents the fundamental conceptions of the theory of curves and surfaces and applies them to a number of examples. Dr. Struik has enhanced the treatment with copious historical, biographical, and bibliographical references that place the theory in context and encourage the student to consult original sources and discover additional important ideas there.For this second edition, Professor Struik made some corrections and added an appendix with a sketch of the application of Cartan's method of Pfaffians to curve and surface theory. The result was to further increase the merit of this stimulating, thought-provoking text -- ideal for classroom use, but also perfectly suited for self-study. In this attractive, inexpensive paperback edition, it belongs in the library of any mathematician or student of mathematics interested in differential geometry.
Taxicab Geometry
This entertaining, stimulating textbook offers anyone familiar with Euclidean geometry -- undergraduate math students, advanced high school students, and puzzle fans of any age -- an opportunity to explore taxicab geometry, a simple, non-Euclidean system that helps put Euclidean geometry in sharper perspective.In taxicab geometry, the shortest distance between two points is not a straight line. Distance is not measured as the crow flies, but as a taxicab travels the "grid" of the city street, from block to block, vertically and horizontally, until the destination is reached. Because of this non-Euclidean method of measuring distance, some familiar geometric figures are transmitted: for example, circles become squares.However, taxicab geometry has important practical applications. As Professor Krause points out, "While Euclidean geometry appears to be a good model of the 'natural' world, taxicab geometry is a better model of the artificial urban world that man has built."As a result, the book is replete with practical applications of this non-Euclidean system to urban geometry and urban planning -- from deciding the optimum location for a factory or a phone booth, to determining the most efficient routes for a mass transit system.The underlying emphasis throughout this unique, challenging textbook is on how mathematicians think, and how they apply an apparently theoretical system to the solution of real-world problems.
Fundamental Concepts of Geometry
Fundamental Concepts of Geometry demonstrates in a clear and lucid manner the relationships of several types of geometry to one another. This highly regarded work is a superior teaching text, especially valuable in teacher preparation, as well as providing an excellent overview of the foundations and historical evolution of geometrical concepts. Professor Meserve (University of Vermont) offers students and prospective teachers the broad mathematical perspective gained from an elementary treatment of the fundamental concepts of mathematics. The clearly presented text is written on an undergraduate (or advanced secondary-school) level and includes numerous exercises and a brief bibliography. An indispensable taddition to any math library, this helpful guide will enable the reader to discover the relationships among Euclidean plane geometry and other geometries; obtain a practical understanding of "proof"; view geometry as a logical system based on postulates and undefined elements; and appreciate the historical evolution of geometric concepts.
Geometry of Complex Numbers
"This book should be in every library, and every expert in classical function theory should be familiar with this material. The author has performed a distinct service by making this material so conveniently accessible in a single book." -- Mathematical ReviewSince its initial publication in 1962, Professor Schwerdtfeger's illuminating book has been widely praised for generating a deeper understanding of the geometrical theory of analytic functions as well as of the connections between different branches of geometry. Its focus lies in the intersection of geometry, analysis, and algebra, with the exposition generally taking place on a moderately advanced level. Much emphasis, however, has been given to the careful exposition of details and to the development of an adequate algebraic technique.In three broad chapters, the author clearly and elegantly approaches his subject. The first chapter, Analytic Geometry of Circles, treats such topics as representation of circles by Hermitian matrices, inversion, stereographic projection, and the cross ratio. The second chapter considers in depth the Moebius transformation: its elementary properties, real one-dimensional projectivities, similarity and classification of various kinds, anti-homographies, iteration, and geometrical characterization. The final chapter, Two-Dimensional Non-Euclidean Geometries, discusses subgroups of Moebius transformations, the geometry of a transformation group, hyperbolic geometry, and spherical and elliptic geometry. For this Dover edition, Professor Schwerdtfeger has added four new appendices and a supplementary bibliography.Advanced undergraduates who possess a working knowledge of the algebra of complex numbers and of the elements of analytical geometry and linear algebra will greatly profit from reading this book. It will also prove a stimulating and thought-provoking book to mathematics professors and teachers.
Lie Algebras
Lie group theory, developed by M. Sophus Lie in the nineteenth century, ranks among the more important developments in modern mathematics. Lie algebras comprise a significant part of Lie group theory and are being actively studied today. This book, by Professor Nathan Jacobson of Yale, is the definitive treatment of the subject and can be used as a text for graduate courses.Chapter 1 introduces basic concepts that are necessary for an understanding of structure theory, while the following three chapters present the theory itself: solvable and nilpotent Lie algebras, Cartan's criterion and its consequences, and split semi-simple Lie algebras. Chapter 5, on universal enveloping algebras, provides the abstract concepts underlying representation theory. The basic results on representation theory are given in three succeeding chapters: the theorem of Ado-Iwasawa, classification of irreducible modules, and characters of the irreducible modules. In Chapter 9 the automorphisms of semi-simple Lie algebras over an algebraically closed field of characteristic zero are determined. These results are applied in Chapter 10 to the problems of sorting out the simple Lie algebras over an arbitrary field. The reader, to fully benefit from this tenth chapter, should have some knowledge about the notions of Galois theory and some of the results of the Wedderburn structure theory of associative algebras.Nathan Jacobson, presently Henry Ford II Professor of Mathematics at Yale University, is a well-known authority in the field of abstract algebra. His book, Lie Algebras, is a classic handbook both for researchers and students. Though it presupposes knowledge of linear algebra, it is not overly theoretical and can be readily used for self-study.
