Beautiful Geometry by
If you've ever thought that mathematics and art don't mix, this stunning visual history of geometry will change your mind. As much a work of art as a book about mathematics, Beautiful Geometry presents more than sixty exquisite color plates illustrating a wide range of geometric patterns and theorems, accompanied by brief accounts of the fascinating history and people behind each. With artwork by Swiss artist Eugen Jost and text by math historian Eli Maor, this unique celebration of geometry covers numerous subjects, from straightedge-and-compass constructions to intriguing configurations involving infinity. The result is a delightful and informative illustrated tour through the 2,500-year-old history of one of the most important branches of mathematics.
The Cult of Pythagoras: math and myths by
In this follow-up to his popular Science Secrets, Alberto A. Martínez discusses various popular myths from the history of mathematics: that Pythagoras proved the hypotenuse theorem, that Archimedes figured out how to test the purity of a gold crown while he was in a bathtub, that the Golden Ratio is in nature and ancient architecture, that the young Galois created group theory the night before the pistol duel that killed him, and more. Some stories are partly true, others are entirely false, but all show the power of invention in history. Pythagoras emerges as a symbol of the urge to conjecture and "fill in the gaps" of history. He has been credited with fundamental discoveries in mathematics and the sciences, yet there is nearly no evidence that he really contributed anything to such fields at all. This book asks: how does history change when we subtract the many small exaggerations and interpolations that writers have added for over two thousand years? The Cult of Pythagoras is also about invention in a positive sense. Most people view mathematical breakthroughs as "discoveries" rather than invention or creativity, believing that mathematics describes a realm of eternal ideas. But mathematicians have disagreed about what is possible and impossible, about what counts as a proof, and even about the results of certain operations. Was there ever invention in the history of concepts such as zero, negative numbers, imaginary numbers, quaternions, infinity, and infinitesimals? Martínez inspects a wealth of primary sources, in several languages, over a span of many centuries. By exploring disagreements and ambiguities in the history of the elements of mathematics, The Cult of Pythagoras dispels myths that obscure the actual origins of mathematical concepts. Martínez argues that an accurate history that analyzes myths reveals neglected aspects of mathematics that can encourage creativity in students and mathematicians.
Euclid's Window: the story of geometry from parallel lines to hyperspace by
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. Mlodinow reveals how geometry's first revolution began with a "little" scheme hatched by Pythagoras: the invention of a system of abstract rules that could model the universe. That modest idea was the basis of scientific civilization. But further advance was halted when the Western mind nodded off into the Dark Ages. Finally in the fourteenth century an obscure bishop in France invented the graph and heralded the next revolution: the marriage of geometry and number. Then, while intrepid mariners were sailing back and forth acrossthe Atlantic to the New World, a fifteen-year-old genius realized that, like the earth's surface, space could be curved. Could parallel lines re
Geometry: our cultural heritage by
Mathematics is more important than ever, but phrases like "math avoidance" and "math anxiety" are very much in the public vocabulary. In addition to providing an invitation to mathematics in general, this book emphasizes the dynamic character of geometry and its role as part of the foundation for our cultural heritage. Aimed at an informed public and future teachers of mathematics, it seeks to heal the ills of math phobia in society.
Geometry Through History: Euclidean, hyperbolic, and projective geometries by
Presented as an engaging discourse, this textbook invites readers to delve into the historical origins and uses of geometry. The narrative traces the influence of Euclid's system of geometry, as developed in his classic text The Elements, through the Arabic period, the modern era in the West, and up to twentieth century mathematics. Axioms and proof methods used by mathematicians from those periods are explored alongside the problems in Euclidean geometry that lead to their work. Students cultivate skills applicable to much of modern mathematics through sections that integrate concepts like projective and hyperbolic geometry with representative proof-based exercises. For its sophisticated account of ancient to modern geometries, this text assumes only a year of college mathematics as it builds towards its conclusion with algebraic curves and quaternions. Euclid's work has affected geometry for thousands of years, so this text has something to offer to anyone who wants to broaden their appreciation for the field.
An Algebraic Approach to Geometry by
This is a unified treatment of the various algebraic approaches to geometric spaces. The study of algebraic curves in the complex projective plane is the natural link between linear geometry at an undergraduate level and algebraic geometry at a graduate level, and it is also an important topic in geometric applications, such as cryptography. 380 years ago, the work of Fermat and Descartes led us to study geometric problems using coordinates and equations. Today, this is the most popular way of handling geometrical problems. Linear algebra provides an efficient tool for studying all the first degree (lines, planes) and second degree (ellipses, hyperboloids) geometric figures, in the affine, the Euclidean, the Hermitian and the projective contexts. But recent applications of mathematics, like cryptography, need these notions not only in real or complex cases, but also in more general settings, like in spaces constructed on finite fields. And of course, why not also turn our attention to geometric figures of higher degrees? Besides all the linear aspects of geometry in their most general setting, this book also describes useful algebraic tools for studying curves of arbitrary degree and investigates results as advanced as the Bezout theorem, the Cramer paradox, topological group of a cubic, rational curves etc. Hence the book is of interest for all those who have to teach or study linear geometry: affine, Euclidean, Hermitian, projective; it is also of great interest to those who do not want to restrict themselves to the undergraduate level of geometric figures of degree one or two.
Euclid's Elements: all thirteen books complete in one volume by
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 Four Pillars of Geometry by
This book is unique in that it looks at geometry from 4 different viewpoints - Euclid-style axioms, linear algebra, projective geometry, and groups and their invariants Approach makes the subject accessible to readers of all mathematical tastes, from the visual to the algebraic Abundantly supplemented with figures and exercises
Geometry by
Geometry, this very ancient field of study of mathematics, frequently remains too little familiar to students. Michle Audin, professor at the University of Strasbourg, has written a book allowing them to remedy this situation and, starting from linear algebra, extend their knowledge of affine, Euclidean and projective geometry, conic sections and quadrics, curves and surfaces. It includes many nice theorems like the nine-point circle, Feuerbach's theorem, and so on. Everything is presented clearly and rigourously. Each property is proved, examples and exercises illustrate the course content perfectly. Precise hints for most of the exercises are provided at the end of the book. This very comprehensive text is addressed to students at upper undergraduate and Master's level to discover geometry and deepen their knowledge and understanding.
Geometry: Euclid and Beyond by
This book offers a unique opportunity to understand the essence of one of the great thinkers of western civilization. A guided reading of Euclid's Elements leads to a critical discussion and rigorous modern treatment of Euclid's geometry and its more recent descendants, with complete proofs. Topics include the introduction of coordinates, the theory of area, history of the parallel postulate, the various non-Euclidean geometries, and the regular and semi-regular polyhedra.
Essential Topology by
This book brings the most important aspects of modern topology within reach of a second-year undergraduate student. It successfully unites the most exciting aspects of modern topology with those that are most useful for research, leaving readers prepared and motivated for further study. Written from a thoroughly modern perspective, every topic is introduced with an explanation of why it is being studied, and a huge number of examples provide further motivation. The book is ideal for self-study and assumes only a familiarity with the notion of continuity and basic algebra.
Geometry and Topology by
Geometry provides a whole range of views on the universe, serving as the inspiration, technical toolkit and ultimate goal for many branches of mathematics and physics. This book introduces the ideas of geometry, and includes a generous supply of simple explanations and examples. The treatment emphasises coordinate systems and the coordinate changes that generate symmetries. The discussion moves from Euclidean to non-Euclidean geometries, including spherical and hyperbolic geometry, and then on to affine and projective linear geometries. Group theory is introduced to treat geometric symmetries, leading to the unification of geometry and group theory in the Erlangen program. An introduction to basic topology follows, with the Möbius strip, the Klein bottle and the surface with g handles exemplifying quotient topologies and the homeomorphism problem. Topology combines with group theory to yield the geometry of transformation groups,having applications to relativity theory and quantum mechanics. A final chapter features historical discussions and indications for further reading. With minimal prerequisites, the book provides a first glimpse of many research topics in modern algebra, geometry and theoretical physics. The book is based on many years' teaching experience, and is thoroughly class-tested. There are copious illustrations, and each chapter ends with a wide supply of exercises. Further teaching material is available for teachers via the web, including assignable problem sheets with solutions.
Introduction to Topology and Geometry by
A sweeping yet uniquely accessible introduction to a variety of central geometrical topics Covering over two centuries of innovations in many of the central geometrical disciplines, Introduction to Topology and Geometry is the most comprehensive introductory-level presentation of modern geometry currently available. Unique in both style and scope, the book covers an unparalleled range of topics, yet strikes a welcome balance between academic rigor and accessibility. By including subject matter previously relegated to higher-level graduate courses in mathematics and making it both interesting and accessible, the author presents a complete and cohesive picture of the science for students just entering the field. Historical notes throughout provide readers with a feel for how mathematical disciplines and theorems come into being. Students and teachers will benefit from a uniquely unified treatment of such topics as: Homeomorphism Graph theory Surface topology Knot theory Differential geometry Riemannian geometry Hyperbolic geometry Algebraic topology General topology Using a variety of theorems to tie these seemingly disparate topics together, the author demonstrates the essential unity of mathematics. A logical yet flexible organization makes the text useful for courses in basic geometry as well as those with a more topological focus, while exercises ranging from the routine to the challenging make the material accessible at varying levels of study.
Topology by
This is an introductory textbook on general and algebraic topology, aimed at anyone with a basic knowledge of calculus and linear algebra. It provides full proofs and includes many examples and exercises. The covered topics include: set theory and cardinal arithmetic; axiom of choice and Zorn's lemma; topological spaces and continuous functions; connectedness and compactness; Alexandrov compactification; quotient topologies; countability and separation axioms; prebasis and Alexander's theorem; the Tychonoff theorem and paracompactness; complete metric spaces and function spaces; Baire spaces; homotopy of maps; the fundamental group; the van Kampen theorem; covering spaces; Brouwer and Borsuk's theorems; free groups and free product of groups; and basic category theory. While it is very concrete at the beginning, abstract concepts are gradually introduced. It is suitable for anyone needing a basic, comprehensive introduction to general and algebraic topology and its applications.
Topology Now! by
Topology is a branch of mathematics packed with intriguing concepts, fascinating geometrical objects, and ingenious methods for studying them. The authors have written this textbook to make the material accessible to undergraduate students without requiring extensive prerequisites in upper-level mathematics. The approach is to cultivate the intuitive ideas of continuity, convergence, and connectedness so students can quickly delve into knot theory, the topology of surfaces and three-dimensional manifolds, fixed points and elementary homotopy theory. The fundamental concepts of point-set topology appear at the end of the book when students can see how this level of abstraction provides a sound logical basis for the geometrical ideas that have come before. This organization exposes students to the exciting world of topology now(!) rather than later. Students using this textbook should have some exposure to the geometry of objects in higher-dimensional Euclidean spaces together with an appreciation of precise mathematical definitions and proofs.
