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Pseudo-Riemannian Homogeneous Structures
This book provides an up-to-date presentation of homogeneous pseudo-Riemannian structures, an essential tool in the study of pseudo-Riemannian homogeneous spaces. Benefiting from large symmetry groups, these spaces are of high interest in Geometry and Theoretical Physics. Since the seminal book by Tricerri and Vanhecke, the theory of homogeneous structures has been considerably developed and many applications have been found. The present work covers a gap in the literature of more than 35 years, presenting the latest contributions to the field in a modern geometric approach, with special focus on manifolds equipped with pseudo-Riemannian metrics. This unique reference on the topic will be of interest to researchers working in areas of mathematics where homogeneous spaces play an important role, such as Differential Geometry, Global Analysis, General Relativity, and Particle Physics.
Polynomial Automorphisms and the Jacobian Conjecture
- The Shestakov-Umirbaev Theory and Nagata's Conjecture. - Counterexamples to Hilbert's Fourteenth Problem. - Prime Characteristic Methods and the Cancellation Problem. - The Jacobian Conjecture: New Equivalences. - Mathieu-Zhao Spaces.
Elementary Overview of Mathematical Structures, An: Algebra, Topology and Categories
Since the last century, a large part of Mathematics is concerned with the study of mathematical structures, from groups to fields and vector spaces, from lattices to Boolean algebras, from metric spaces to topological spaces, from topological groups to Banach spaces.More recently, these structured sets and their transformations have been assembled in higher structures, called categories.We want to give a structural overview of these topics, where the basic facts of the different theories are unified through the 'universal properties' that they satisfy, and their particularities stand out, perhaps even more.This book can be used as a textbook for undergraduate studies and for self-study. It can provide students of Mathematics with a unified perspective of subjects which are often kept apart. It is also addressed to students and researchers of disciplines having strong interactions with Mathematics, like Physics and Chemistry, Statistics, Computer Science, Engineering.
Geometric Analysis
- Extremal Eigenvalue Problems and Free Boundary Minimal Surfaces in the Ball. - Applications of Min-Max Methods to Geometry. - Ricci Flow and Ricci Limit Spaces. - Pseudo-Hermitian Geometry in 3D.
General Topology
The first half of the book provides an introduction to general topology, with ample space given to exercises and carefully selected applications. The second half of the text includes topics in asymmetric topology, a field motivated by applications in computer science. Recurring themes include the interactions of topology with order theory and mathematics designed to model loss-of-resolution situations.
Real and Complex Singularities
This text offers a selection of papers on singularity theory presented at the Sixth Workshop on Real and Complex Singularities held at ICMC-USP, Brazil. It should help students and specialists to understand results that illustrate the connections between singularity theory and related fields.
The Joy of Geometry
A veteran math educator reveals the hidden fascinations of geometry and why this staple of math education is important.If you remember anything about high school geometry class, it's probably doing proofs. But geometry is more than axioms, postulates, theorems, and proofs. It's the science of beautiful and extraordinary geometric relationships--most of which is lost in high school classrooms where the focus is on the rigor of logically proving those relationships. This book will awaken readers to the appeal of geometry by placing the focus squarely on geometry's visually compelling features and intrinsic elegance. Who knew that straight lines, circles, and area could be so interesting? Not to mention optical illusions. So get out the rulers, compasses, or even a software program, and discover geometry for the first time.
The first six books of the Elements of Euclid, in which coloured diagrams and symbols are
This book has been considered by academicians and scholars of great significance and value to literature. This forms a part of the knowledge base for future generations. So that the book is never forgotten we have represented this book in a print format as the same form as it was originally first published. Hence any marks or annotations seen are left intentionally to preserve its true nature.
Von Euklids Elementen zur zehnten DimensionGeometrie und Topologie von Raum und Zeit
Wenn man in der Nacht zum Sternenhimmel aufschaut, so blickt man in eine unendliche Weite. Ist unser Universum tats瓣chlich unendlich gro?? Alles was wir aus unserer Erfahrung kennen ist endlich. Etwas, was eine unendliche Ausdehnung hat, k繹nnen wir nicht begreifen. Aber auch die Vorstellung eines Weltraums mit endlichem Volumen f羹hrt zu Problemen: Das Universum m羹sste dann einen Rand haben. Wie k繹nnte ein solcher Rand aber aussehen und was liegt hinter der Grenze? Ein weiteres Problem kommt hinzu: Unser Blick in den Himmel ist ein Blick in die Vergangenheit. Das Licht der Sterne hat Tausende von Jahren gebraucht, um uns zu erreichen. Wie sieht der Raum da drau?en heute aus? Macht es 羹berhaupt Sinn, von einem heutigen Ereignis auf einem fremden Stern zu sprechen? Vielleicht gibt es den Himmelsk繹rper und den umgebenden Raum gar nicht mehr. Fragen dieser Art besch瓣ftigen Philosophen, Physiker und Mathematiker seit Jahrtausenden. Und obwohl wir inzwischen eine F羹lle von Informationen gesammelt haben, gibt es noch keine endg羹ltigen Antworten. Das vorliegende Buch befasst sich mit der Natur von Raum und Raumzeit aus der Sicht eines Mathematikers. Es zeigt, dass die Mathematik Modelle sowohl von endlichen als auch unendlichen R瓣umen bereitstellt, ohne sich in Widerspr羹che zu verwickeln. Es beschreibt, dass man mit logischen Mitteln R羹ckschl羹sse sowohl auf die globale Gestalt als auch auf lokale Eigenschaften des Universums ziehen kann, wenn man bestimmte Axiome als wahr annimmt. Das Buch folgt damit der Tradition von Euklid, welcher als erster solche Raum-Axiome in systematischer Weise pr瓣zisierte. Ausgehend von der euklidischen Geometrie wird ein Bogen gespannt bis zu den Mannigfaltigkeiten, welche vielen physikalischen Theorien als Basis dienen. Dabei lassen wir uns nicht von der Fragestellung nach der wahren Natur des Raumes leiten, sondern fragen uns: Welche Modelle des Universums sind aus logischer Sicht m繹glich? So werden wir uns in die Gedankeng瓣nge eines zwei
MathematicsFrom Beginning to Infinity
Need to learn mathematics but don't know where to start? For a long time, there has been a need for a single-book series containing all of mathematics explained step-by-step. This is that series-starting from the basics of learning how to count and stretching all the way to the most brilliant mathematical discoveries in history. This book is concise, efficient, and straight to the point, without any unnecessary and irrelevant information, such as historical contexts, graphing devices, and application of mathematics. Nothing is developed without proof, and the reader finds the answer independently. This book is a guide that teaches the reader how to find the answer through simple reasoning, and the answer only comes at the very end of a thoroughly developed explanation. Learn and understand mathematics completely and you'll never have to memorize another equation!
Introduction to the Geometrical Analysis of Vector Fields, An: With Applications to Maximum Principles and Lie Groups
This book provides the reader with a gentle path through the multifaceted theory of vector fields, starting from the definitions and the basic properties of vector fields and flows, and ending with some of their countless applications, in the framework of what is nowadays called Geometrical Analysis. Once the background material is established, the applications mainly deal with the following meaningful settings: ODE theory; Maximum Principles (weak, strong and propagation principles); Lie groups (with an emphasis on the construction of Lie groups). This book also provides an introduction to the basic theory of Geometrical Analysis, with a new foundational presentation based on Ordinary Differential Equation techniques, in a unitary and self-contained way.
Quantum Gravity in a Nutshell1
Just like Richard P. Feynman, winner of the Nobel Prize in physics, who thrived on outrageous adventures. In this lively work that "can shatter the stereotype of the stuffy scientist" (SUSP Science Foundation), Balungi unifies Quantum mechanics with Albert Einstein General Relativity theory in a remarkable way. This thought experimental book is a good introduction to quantum gravity and has a lot of interesting history about the development of the theory since 1899. It's an informal introduction to a very difficult and doubtfully intelligible theory.- doubted even by its most ingenious contributors. The reader should expect that he/she will have to concentrate hard on what Balungi says but the rewards are significant. He is a talented physicist and a good writer. If you read it carefully and stop to think about the message as it unfolds then you will get a worthwhile if imperfect picture of what the theory is saying and how it was invented, It's buried treasure and you will have to do some digging. It is a really serious attempt to do all that can be done in an informal style. Balungi explains and re-defines Einstein's theory of general relativity, quantum mechanics, black holes, the complex architecture of the universe, elementary particles, gravity, and the nature of the mind. This wonderful and exciting book is optimal for physics graduate students and researchers. Not since Stephen W Hawking's celebrated best-seller Brief History of Time or Carol Rovelli's Reality is not what it seems ( The journey to quantum gravity) has physics been so vividly, intelligently and entertainingly revealed.
A Geometric Analysis of the Platonic Solids and Other Semi-Regular Polyhedra
Getting Inside the Mind of Nature Discover the simple but powerful mathematics of the underlying geometric figures that shape our worldCan geometry be exciting?It can for those who can appreciate the beauty of numbers and their relationships.This book contains a meticulous geometric investigation of the 5 Platonic Solids and 5 other important polyhedra, as well as reference charts for each solid.Poly (many) hedron (face) means "many faces." Polyhedra are 3 dimensional figures with 4 or more faces, or sides. These polyhedra are reflections of Nature herself, and a study of them provides insight into the way the world is structured. Nature is not only beautiful, but highly intelligent. As you explore the polyhedra in this book, this will become apparent over and over again.The book contains a geometric explanation of the Phi Ratio and Fibonacci series, and a detailed analysis of the pentagon, which forms the basis for many of these solids. The pentagon is composed entirely of Phi relationships, and is integral to a proper understanding of sacred geometry.With over 140 full-color illustrations, this book is perfect for teachers and students of geometry alike. It is a must for those who are serious about sacred geometry.This book is dedicated to those who can appreciate the logic of numbers and the beauty of nature, for they are both aspects of the same unifying principle.Prerequisites: knowledge of simple algebra and elementary trigonometry. No brainiac math skills required! Only the ability to appreciate nature's own logic.Book #1 in the Geometric Explorations SeriesWhat's New in the Second Edition: An analysis of the Truncated Icosahedron (soccer ball) Rewrites of several sections for greater clarity Correction of typos from the First Edition. From Loving Healing Press (www.LovingHealing.com)
A Geometric Analysis of the Platonic Solids and Other Semi-Regular Polyhedra
Getting Inside the Mind of Nature Discover the simple but powerful mathematics of the underlying geometric figures that shape our worldCan geometry be exciting?It can for those who can appreciate the beauty of numbers and their relationships.This book contains a meticulous geometric investigation of the 5 Platonic Solids and 5 other important polyhedra, as well as reference charts for each solid.Poly (many) hedron (face) means "many faces." Polyhedra are 3 dimensional figures with 4 or more faces, or sides. These polyhedra are reflections of Nature herself, and a study of them provides insight into the way the world is structured. Nature is not only beautiful, but highly intelligent. As you explore the polyhedra in this book, this will become apparent over and over again.The book contains a geometric explanation of the Phi Ratio and Fibonacci series, and a detailed analysis of the pentagon, which forms the basis for many of these solids. The pentagon is composed entirely of Phi relationships, and is integral to a proper understanding of sacred geometry.With over 140 full-color illustrations, this book is perfect for teachers and students of geometry alike. It is a must for those who are serious about sacred geometry.This book is dedicated to those who can appreciate the logic of numbers and the beauty of nature, for they are both aspects of the same unifying principle.Prerequisites: knowledge of simple algebra and elementary trigonometry. No brainiac math skills required! Only the ability to appreciate nature's own logic.Book #1 in the Geometric Explorations SeriesWhat's New in the Second Edition: An analysis of the Truncated Icosahedron (soccer ball) Rewrites of several sections for greater clarity Correction of typos from the First Edition. From Marvelous Spirit Press
Symmetry
Why do cars have the same symmetry as dragonflies? Is there really a beautiful swirling pattern lurking in every dripping tap? What do insect eggs have in common with planets, and why? In this exquisite book, the smallest and most concise ever produced, designer David Wade introduces the main principles of symmetry, and shows how, despite opinions over exactly what it is, symmetry can be found in almost every corner of science, nature, and human culture.
Classical Geometry
Sacred geometry is at the heart of a thousand years of art and architecture as represented in mosques, temples and churches around the world. Stunning in their search of perfection amid countless symmetries the art achieves a beauty inspired by its divine motivations. On a more practical side, sacred art represents an excellent application of the principles of geometry as illustrated by Euclid in the Elements. Countless constructions and theorems first discovered by the ancient Greek mathematicians are carefully merged with craftsmanship to produce murals, paintings and mosaics of infinite variety. This book grew out of a set of workshops done primarily through the Monterey Bay Area Math Project over the last several years. The book begins with a discussion of compass-straightedge constructions of polygons and the variety of regular and semi-regular tilings. The Polygon-in-Contact method as initially documented in the Topkapi Scrolls and further developed by contemporary scholars and artists is introduced as a method of generating traditional Islamic Geometric patterns. Many examples are illustrated with varying degrees of complexity suitable for all age groups. In addition to developing traditional patterns, the methods shown illustrate areas of generalization constrained only by students imagination.
Mechanical Drawing
"Mechanical Drawing" aims to provide a fundamental course on all theory, principles, and methods needed to create a practical working drawing. It outlines a systematic method for mechanical drawing and offers useful information on convention and design, making it ideal for students and novices. Contents include: "Instruments and Their Uses", "Drafting Room Conventions", "Freehand Lettering and Geometric Drawing", "Orthographic Projection", "Representation of Points and Lines", "Representation of Planes", "Orthographic Projection Applied", "Isometrical Projection", "Working Drawings", "Lettering, continued", "Geometrical Drawing, continued", et cetera. Many vintage books such as this are becoming increasingly scarce and expensive. We are republishing this book now in an affordable, high-quality, modern edition complete with a specially commissioned new introduction on technical drawing and drafting. First published in 1919.
The American Text-Book of Art Education - Teachers' Manual for The Primary Course of Instruction in Drawing
This book contains a teacher's manual for an introductory course on technical drawing and drafting. Written in clear, plain language and profusely illustrated, this volume will be of considerable utility to teachers and instructors of technical drawing. Contents include: "General Directions for Teachers", "Preliminary Study", "Use of the Manual-Methods", "Pan of the Course", "Geometric Name or Term", "Reduction and Enlargement", "Use of the Blackboard", "Memory Exercises", "Drawing from Dictation", "Drawing from Objects", "Busy Work", "Design", "Particular Methods", "Drill Work", et cetera. Many vintage books such as this are becoming increasingly scarce and expensive. We are republishing this book now in an affordable, high-quality, modern edition complete with a specially commissioned new introduction on technical drawing and drafting.
Geometry Essentials for Dummies
Geometry Essentials For Dummies (9781119590446) was previously published as Geometry Essentials For Dummies (9781118068755). While this version features a new Dummies cover and design, the content is the same as the prior release and should not be considered a new or updated product. Just the critical concepts you need to score high in geometry This practical, friendly guide focuses on critical concepts taught in a typical geometry course, from the properties of triangles, parallelograms, circles, and cylinders, to the skills and strategies you need to write geometry proofs. Geometry Essentials For Dummies is perfect for cramming or doing homework, or as a reference for parents helping kids study for exams. Get down to the basics -- get a handle on the basics of geometry, from lines, segments, and angles, to vertices, altitudes, and diagonals Conquer proofs with confidence -- follow easy-to-grasp instructions for understanding the components of a formal geometry proof Take triangles in strides -- learn how to take in a triangle's sides, analyze its angles, work through an SAS proof, and apply the Pythagorean Theorem Polish up on polygons -- get the lowdown on quadrilaterals and other polygons: their angles, areas, properties, perimeters, and much more
Joseph Fourier 250th Birthday
For the 250th birthday of Joseph Fourier, born in 1768 in Auxerre, France, this MDPI Special Issue will explore modern topics related to Fourier Analysis and Heat Equation.Modern developments of Fourier analysis during the 20th century have explored generalizations of Fourier and Fourier-Plancherel formula for non-commutative harmonic analysis, applied to locally-compact, non-Abelian groups. In parallel, the theory of coherent states and wavelets has been generalized over Lie groups. One should add the developments, over the last 30 years, of the applications of harmonic analysis to the description of the fascinating world of aperiodic structures in condensed matter physics. The notions of model sets, introduced by Y. Meyer, and of almost periodic functions, have revealed themselves to be extremely fruitful in this domain of natural sciences.The name of Joseph Fourier is also inseparable from the study of the mathematics of heat. Modern research on heat equations explores the extension of the classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. In parallel, in geometric mechanics, Jean-Marie Souriau interpreted the temperature vector of Planck as a space-time vector, obtaining, in this way, a phenomenological model of continuous media, which presents some interesting properties.One last comment concerns the fundamental contributions of Fourier analysis to quantum physics: Quantum mechanics and quantum field theory.The content of this Special Issue will highlight papers exploring non-commutative Fourier harmonic analysis, spectral properties of aperiodic order, the hypoelliptic heat equation, and the relativistic heat equation in the context of Information Theory and Geometric Science of Information.
Introduction to Algebraic Geometry
Author Serge Lang defines algebraic geometry as the study of systems of algebraic equations in several variables and of the structure that one can give to the solutions of such equations. The study can be carried out in four ways: analytical, topological, algebraico-geometric, and arithmetic. This volume offers a rapid, concise, and self-contained introductory approach to the algebraic aspects of the third method, the algebraico-geometric. The treatment assumes only familiarity with elementary algebra up to the level of Galois theory.Starting with an opening chapter on the general theory of places, the author advances to examinations of algebraic varieties, the absolute theory of varieties, and products, projections, and correspondences. Subsequent chapters explore normal varieties, divisors and linear systems, differential forms, the theory of simple points, and algebraic groups, concluding with a focus on the Riemann-Roch theorem. All the theorems of a general nature related to the foundations of the theory of algebraic groups are featured.
Mcgraw-hill Education Geometry Review and Workbook
This engaging review guide and workbook is the ideal tool for sharpening your Geometry skills!This review guide and workbook will help you strengthen your Geometry knowledge, and it will enable you to develop new math skills to excel in your high school classwork and on standardized tests. Clear and concise explanations will walk you step by step through each essential math concept. 500 practical review questions, in turn, provide extensive opportunities for you to practice your new skills. If you are looking for material based on national or state standards, this book is your ideal study tool!Features: - Aligned to national standards, including the Common Core State Standards, as well as the standards of non-Common Core states and Canada- Designed to help you excel in the classroom and on standardized tests- Concise, clear explanations offer step-by-step instruction so you can easily grasp key concepts- You will learn how to apply Geometry to practical situations- 500 review questions provide extensive opportunities for you to practice what you've learned
When Opposites Attract
What happens when opposites attract? When the North Pole meets the South Pole? When a boring activity is paired up with an exciting one? Let's find out through this coloring book! Coloring gives you the opportunity to create your own world. You can say that it's a creative journey of sort and the destination is your imagination. Grab a copy and start coloring today!
Algebraic Geometry and Number Theory
This lecture notes volume presents significant contributions from the "Algebraic Geometry and Number Theory" Summer School, held at Galatasaray University, Istanbul, June 2-13, 2014.It addresses subjects ranging from Arakelov geometry and Iwasawa theory to classical projective geometry, birational geometry and equivariant cohomology. Its main aim is to introduce these contemporary research topics to graduate students who plan to specialize in the area of algebraic geometry and/or number theory. All contributions combine main concepts and techniques with motivating examples and illustrative problems for the covered subjects. Naturally, the book will also be of interest to researchers working in algebraic geometry, number theory and related fields.
The Changing Shape of Geometry
Celebrating a century of geometry and geometry teaching, this volume includes popular articles on Pythagoras, the golden ratio and recreational geometry. Thirty "Desert Island Theorems" from distinguished mathematicians and educators disclose surprising results. (Contributors include a Nobel Laureate and a Pulitzer Prize winner.) Co-published with The Mathematical Association of America.
Schaum's Outline of Geometry
Tough Test Questions? Missed Lectures? Not Enough Time? Textbook too Pricey?Fortunately, there's Schaum's. This all-in-one-package includes more than 650 fully-solved problems, examples, and practice exercises to sharpen your problem-solving skills. Plus, you will have access to 25 detailed videos featuring math instructors who explain how to solve the most commonly tested problems--it's just like having your own virtual tutor! You'll find everything you need to build confidence, skills, and knowledge for the highest score possible.More than 40 million students have trusted Schaum's to help them succeed in the classroom and on exams. Schaum's is the key to faster learning and higher grades in every subject. Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. Helpful tables and illustrations increase your understanding of the subject at hand.Schaum's Outline of Geometry, Sixth Edition features: - Updated content to matches the latest curriculum- Over 650 problems, solved step by step- An accessible format for quick and easy review- Clear explanations for all geometry concepts- Access to revised Schaums.com website with access to 25 problem-solving videos, and more
Geometry for Dummies
Hit the geometry wall? Get up and running with this no-nonsense guide! Does the thought of geometry make you jittery? You're not alone. Fortunately, this down-to-earth guide helps you approach it from a new angle, making it easier than ever to conquer your fears and score your highest in geometry. From getting started with geometry basics to making friends with lines and angles, you'll be proving triangles congruent, calculating circumference, using formulas, and serving up pi in no time. Geometry is a subject full of mathematical richness and beauty. But it's a subject that bewilders many students because it's so unlike the math they've done before--it requires the use of deductive logic in formal proofs. If you're having a hard time wrapping your mind around what that even means, you've come to the right place! Inside, you'll find out how a proof's chain of logic works and even discover some secrets for getting past rough spots along the way. You don't have to be a math genius to grasp geometry, and this book helps you get un-stumped in a hurry! Find out how to decode complex geometry proofs Learn to reason deductively and inductively Make sense of angles, arcs, area, and more Improve your chances of scoring higher in your geometry class There's no reason to let your nerves get jangled over geometry--your understanding will take new shape with the help of Geometry For Dummies.
Fractals: A Very Short Introduction
From the contours of coastlines to the outlines of clouds, and the branching of trees, fractal shapes can be found everywhere in nature. In this Very Short Introduction, Kenneth Falconer explains the basic concepts of fractal geometry, which produced a revolution in our mathematical understanding of patterns in the twentieth century, and explores the wide range of applications in science, and in aspects of economics. About the Series: Oxford's Very Short Introductions series offers concise and original introductions to a wide range of subjects--from Islam to Sociology, Politics to Classics, Literary Theory to History, and Archaeology to the Bible. Not simply a textbook of definitions, each volume in this series provides trenchant and provocative--yet always balanced and complete--discussions of the central issues in a given discipline or field. Every Very Short Introduction gives a readable evolution of the subject in question, demonstrating how the subject has developed and how it has influenced society. Eventually, the series will encompass every major academic discipline, offering all students an accessible and abundant reference library. Whatever the area of study that one deems important or appealing, whatever the topic that fascinates the general reader, the Very Short Introductions series has a handy and affordable guide that will likely prove indispensable.
Shaping Space
Molecules, galaxies, art galleries, sculptures, viruses, crystals, architecture, and more: Shaping Space--Exploring Polyhedra in Nature, Art, and the Geometrical Imagination is an exuberant survey of polyhedra and at the same time a hands-on, mind-boggling introduction to one of the oldest and most fascinating branches of mathematics. Some of the world's leading geometers present a treasury of ideas, history, and culture to make the beauty of polyhedra accessible to students, teachers, polyhedra hobbyists, and professionals such as architects and designers, painters and sculptors, biologists and chemists, crystallographers, physicists and earth scientists, engineers and model builders, mathematicians and computer scientists. The creative chapters by more than 25 authors explore almost every imaginable side of polyhedra. From the beauty of natural forms to the monumental constructions made by man, there is something to fascinate every reader. The book is dedicated to the memory of the legendary geometer H. S. M. Coxeter and the multifaceted design scientist Arthur L. Loeb.
Hidden Geometry of Flowers
Can we imagine a world without flowers? Flowers are beautiful, offering us delight in their color, fragrance and form, as well as their medicinal benefits. Flowers also speak to us in the language of the plant form itself, as cultural symbols in different societies, and at the highest levels of inspiration. In this beautiful and original book, renowned thinker and geometrist Keith Critchlow has chosen to focus on an aspect of flowers that has received perhaps the least attention. This is the flower as teacher of symmetry and geometry (the 'eternal verities', as Plato called them). In this sense, he says, flowers can be treated as sources of remembering -- a way of recalling our own wholeness, as well as awakening our inner power of recognition and consciousness. What is evident in the geometry of the face of a flower can remind us of the geometry that underlies all existence. Working from his own flower photographs and with every geometric pattern hand-drawn, the author reviews the role of flowers within the perspective of our relationship with the natural world. His illuminating study is an attempt to re-engage the human spirit in its intimate relation with all nature.
Introduction to Non-Euclidean Geometry
One of the first college-level texts for elementary courses in non-Euclidean geometry, this concise, readable volume is geared toward students familiar with calculus. A full treatment of the historical background explores the centuries-long efforts to prove Euclid's parallel postulate and their triumphant conclusion. Numerous original exercises form an integral part of the book.Topics include hyperbolic plane geometry and hyperbolic plane trigonometry, applications of calculus to the solutions of some problems in hyperbolic geometry, elliptic plane geometry and trigonometry, and the consistency of the non-Euclidean geometries. Extensive appendixes offer background information on the foundation of Euclidean geometry, circular and hyperbolic functions, the theory of orthogonal circles and allied topics, and the elements of inversion.
Curvature in Mathematics and Physics
This original text for courses in differential geometry is geared toward advanced undergraduate and graduate majors in math and physics. Based on an advanced class taught by a world-renowned mathematician for more than fifty years, the treatment introduces semi-Riemannian geometry and its principal physical application, Einstein's theory of general relativity, using the Cartan exterior calculus as a principal tool. Starting with an introduction to the various curvatures associated to a hypersurface embedded in Euclidean space, the text advances to a brief review of the differential and integral calculus on manifolds. A discussion of the fundamental notions of linear connections and their curvatures follows, along with considerations of Levi-Civita's theorem, bi-invariant metrics on a Lie group, Cartan calculations, Gauss's lemma, and variational formulas. Additional topics include the Hopf-Rinow, Myer's, and Frobenius theorems; special and general relativity; connections on principal and associated bundles; the star operator; superconnections; semi-Riemannian submersions; and Petrov types. Prerequisites include linear algebra and advanced calculus, preferably in the language of differential forms.
Secrets of Triangles
Everyone knows what a triangle is, yet very few people appreciate that the common three-sided figure holds many intriguing "secrets." For example, if a circle is inscribed in any random triangle and then three lines are drawn from the three points of tangency to the opposite vertices of the triangle, these lines will always meet at a common point - no matter what the shape of the triangle. This and many more interesting geometrical properties are revealed in this entertaining and illuminating book about geometry. Flying in the face of the common impression that mathematics is usually dry and intimidating, this book proves that this sometimes-daunting, abstract discipline can be both fun and intellectually stimulating. The authors, two veteran math educators, explore the multitude of surprising relationships connected with triangles and show some clever approaches to constructing triangles using a straightedge and a compass. Readers will learn how they can improve their problem-solving skills by performing these triangle constructions. The lines, points, and circles related to triangles harbor countless surprising relationships that are presented here in a very engaging fashion.Requiring no more than a knowledge of high school mathematics and written in clear and accessible language, this book will give all readers a new insight into some of the most enjoyable and fascinating aspects of geometry.
Polyhedra
Polyhedra have cropped up in many different guises throughout recorded history. Recently, polyhedra and their symmetries have been cast in a new light by combinatorics and group theory. This unique text comprehensively documents the many and varied ways that polyhedra have come to the fore throughout the development of mathematics. The author strikes a balance between covering the historical development of the theory surrounding polyhedra and rigorous treatment of the mathematics involved. Attractively illustrated--including 16 color plates--Polyhedra elucidates ideas that have proven difficult to grasp. Mathematicians, as well as historians of mathematics, will find this book fascinating.
Flexagons Inside Out
Flexagons, paper models that can be bent in different ways to change their shape, are easy to make and work in surprising ways. This book contains numerous diagrams that the reader can photocopy and use to construct a variety of fascinating flexagons. The author also explains the mathematics behind these amazing creations. Although knowledge of the technical details requires a mathematical background, the models can be made and used by anyone. Flexagons appeals to all readers interested in puzzles and recreational mathematics.
The Pythagorean Theorem
The Pythagorean theorem may be the best-known equation in mathematics. Its origins reach back to the beginnings of civilization, and today every student continues to study it. What most nonmathematicians don't understand or appreciate is why this simply stated theorem has fascinated countless generations. In this entertaining and informative book, a veteran math educator makes the importance of the Pythagorean theorem delightfully clear.He begins with a brief history of Pythagoras and the early use of his theorem by the ancient Egyptians, Babylonians, Indians, and Chinese, who used it intuitively long before Pythagoras's name was attached to it. He then shows the many ingenious ways in which the theorem has been proved visually using highly imaginative diagrams. Some of these go back to ancient mathematicians; others are comparatively recent proofs, including one by the twentieth president of the United States, James A. Garfield. After demonstrating some curious applications of the theorem, the author then explores the Pythagorean triples, pointing out the many hidden surprises of the three numbers that can represent the sides of the right triangle (e.g, 3, 4, 5 and 5, 12, 13). And many will truly amaze the reader. He then turns to the "Pythagorean means" (the arithmetic, geometric, and harmonic means). By comparing their magnitudes in a variety of ways, he gives the reader a true appreciation for these mathematical concepts. The final two chapters view the Pythagorean theorem from an artistic point of view - namely, how Pythagoras's work manifests itself in music and how the Pythagorean theorem can influence fractals. The author's lucid presentation and gift for conveying the significance of this key equation to those with little math background will inform, entertain, and inspire the reader, once again demonstrating the power and beauty of mathematics!
Geometry
A no-nonsense practical guide to geometry, providing concise summaries, clear model examples, and plenty of practice, making this workbook the ideal complement to class study or self-study, preparation for exams or a brush-up on rusty skills. About the BookEstablished as a successful practical workbook series with more than 20 titles in the language learning category, Practice Makes Perfect now provides the same clear, concise approach and extensive exercises to key fields within mathematics.The key to the Practice Makes Perfect series is the extensive exercises that provide learners with all the practice they need for mastery. Not focused on any particular test or exam, but complementary to most geometry curriculaDeliberately all-encompassing approach: international perspective and balance between traditional and newer approaches.Large trim allows clear presentation of worked problems, exercises, and explained answers.FeaturesNo-nonsense approach: provides clear presentation of content.Over 500 exercises and answers covering all aspects of geometrySuccessful series: "Practice Makes Perfect" has sales of 1,000,000 copies in the language category - now applied to mathematicsWorkbook is not exam specific, yet it provides thorough coverage of the geometry skills required in most math tests.
Ruler and Compass
From the practical fundamentals to the more demanding, this pocket-sized book introduces the origins and basic principles of geometric constructions using ruler and compass. Since the earliest times mankind has employed the simple geometric forms of straight line and circle. Originally marked out by eye and later using a stretched cord, in time these came to be made with the simple tools of ruler and compass. This small book introduces the origins and basic principles of geometric constructions using ruler and compass, before going on to cover dozens of geometric constructions, from the practical fundamentals to the more demanding. - Constructions for more than thirty regular polygons - Introduces geometric construction using compasses alone and using rulers alone - Explains why trisecting an angle is impossible using classical rules and how to trisect an angle anyway - Highlights the relationship of geometric construction with many fundamental developments throughout the history of mathematics
Foundations of Geometry
Geared toward students preparing to teach high school mathematics, this text is also of value to professionals, as well as to students seeking further background in geometry. It explores the principles of Euclidean and non-Euclidean geometry, and it instructs readers in both generalities and specifics of the axiomatic method.Well-written and accessible, the text begins by acquainting students with the axiomatic method as well as a general pattern of thought. Subsequent chapters present in-depth coverage of Euclidean geometry, including the geometry of four dimensions, plane hyperbolic geometry, and a Euclidean model of the hyperbolic plane. Detailed definitions, corollaries, theorems, and postulates are explained incrementally and illustrated by numerous figures. Each chapter concludes with multiple exercises that test and reinforce students' understanding of the material.
Space and Counterspace
'Einstein did not attempt to explain the constancy of the velocity of light: he assumed it and derived his theories accordingly. But we have explained it.'from Chapter 8 of Space and Counterspace Many people feel alienated by modern science and its impersonal view of our world, based on the concept of the 'detached observer'. Our human intuitions suggest that we need a broader-based science which can encompass phenomena currently excluded, such as human consciousness, qualities and values. In this groundbreaking book, Nick Thomas presents a wider view of science using the theory of 'counterspace'. Counterspace exists alongside space as we know it, and was first proposed by Rudolf Steiner, and developed by the Cambridge mathematician George Adams. Through its startling lens, key aspects of our world -- such as gravity, time, light and color, as well as the stars, the solar system, and the classical elements -- can be viewed and understood in dynamically new ways. Thomas's work and ideas are on the cusp of a true revolution in the way modern scientific method can penetrate even deeper into the mysteries of our natural world.
Introduction to Projective Geometry
This lucid introductory text offers both an analytic and an axiomatic approach to plane projective geometry. The analytic treatment builds and expands upon students' familiarity with elementary plane analytic geometry and provides a well-motivated approach to projective geometry. Subsequent chapters explore Euclidean and non-Euclidean geometry as specializations of the projective plane, revealing the existence of an infinite number of geometries, each Euclidean in nature but characterized by a different set of distance- and angle-measurement formulas. Outstanding pedagogical features include worked-through examples, introductions and summaries for each topic, and numerous theorems, proofs, and exercises that reinforce each chapter's precepts. Two helpful indexes conclude the text, along with answers to all odd-numbered exercises. In addition to its value to undergraduate students of mathematics, computer science, and secondary mathematics education, this volume provides an excellent reference for computer science professionals.
Fifth Postulate
The great discovery that no one wanted to makeIt's the dawn of the Industrial Revolution, and Euclidean geometry has been profoundly influential for centuries. One mystery remains, however: Euclid's fifth postulate has eluded for two thousand years all attempts to prove it. What happens when three nineteenth-century mathematicians realize that there is no way to prove the fifth postulate and that it ought to be discarded-along with everything they'd come to know about geometry? Jason Socrates Bardi shares the dramatic story of the moment when the tangible and easily understood world we live in gave way to the strange, mind-blowing world of relativity, curved space-time, and more."Jason Socrates Bardi tells the story of the discovery of non-Euclidian geometry--one of the greatest intellectual advances of all time--with tremendous clarity and verve. I loved this book." --John Horgan, author, The End of Science and Rational Mysticism"An accessible and engrossing blend of micro-biography, history and mathematics, woven together to reveal a blockbuster discovery." --David Wolman, author of Righting the Mother Tongue and A Left-Hand Turn Around the World
Invitation to Geometry
Intended for students of many different backgrounds with only a modest knowledge of mathematics, this text features self-contained chapters that can be adapted to several types of geometry courses. Only a slight acquaintance with mathematics beyond the high-school level is necessary, including some familiarity with calculus and linear algebra. This text's introductions to several branches of geometry feature topics and treatments based on memorability and relevance. The author emphasizes connections with calculus and simple mechanics, focusing on developing students' grasp of spatial relationships. Subjects include classical Euclidean material, polygonal and circle isoperimetry, conics and Pascal's theorem, geometrical optimization, geometry and trigonometry on a sphere, graphs, convexity, and elements of differential geometry of curves. Additional material may be conveniently introduced in several places, and each chapter concludes with exercises of varying degrees of difficulty.
