Product Description

Product Description
Many secondary mathematics teachers and teacher educators feel ill-equipped to teach a course in geometry. Most feel fortunate to stay one chapter ahead of the very students they are teaching! Most college-level geometry texts don't address their specific needs.Advanced Euclidian Geometry was written to fill this void in many ways:• Provides both in-service and pre-service teachers with a solid review of the essentials of the high school geometry course - AND MORE!• Offers further elaboration of those concepts (advanced Euclidian geometry, the 19th and 20th century expansions of Euclid's work) to give teachers more confidence in guiding student explorations and answering questions.• Contains hundreds of illustrations created in The Geometer's Sketchpad(r) Dynamic Geometry(tm) software.• Is packaged with a dual (Windows(r)/Macintosh(r)) platform CD-ROM containing over 100 interactive sketches using The Geometer's Sketchpad (assumes that users has access to the Sketchpad program) ... Read more

Customer Reviews (3)

Book Good but CD Has no Program - Just Prog Help
I purchased this book that came with "Geometry Sketchpad" CD except CD contained no program file, only help files.

Just a scam to purchase program?

Instructions and program purchase made unclear on website and within book itself.

This book just made teaching the course so much easier!
What an invaluable supplemental resource to have at my side while teaching a course in College Geometry! Although the class text is excellent (Geometry for College Students, by Isaacs), having this as a back-up made all the difference. The explanations are detailed and pitched at a level that students can easily follow, and the (Geometer's Sketchpad) diagrams show all the required information. Having the CD saved me from recreating the diagrams myself for class use. I highly recommend this book to anyone teaching high school or college geometry, or anyone who just wants to advance their knowledge of Euclidean and synthetic geometry.

A must-have book for every geometry teacher
As a high-school geometry teacher, I have often wished for a book like this. Sadly, the typical one-year geometry course comes to an end just as students are within reach of some truly beautiful and intriguing theorems. Dr. Posamentier's book begins where most high-school geometry textbooks end, and presents many wonderful results that lie just beyond their purview: the nine-
point circle; the golden rectangle; the theorems of Ceva, Menelaus, Ptolemy, Pascal, Desargues, and Brianchon; excircles and incircles; cyclic quadrilaterals; and much more. This book provides a rich geometric feast.

There are several books that cover much of the same ground that Dr. Posamentier surveys: GEOMETRY REVISITED, by Coxeter and Greitzer, and EPISODES IN NINETEENTH AND TWENTIETH CENTURY EUCLIDEAN GEOMETRY, by Honsberger, are two of the best. What makes Dr. Posamentier's book stand out is its usefulness as a textbook. Theorems are fully proved, and arranged in a logically coherent sequence. (The book is Euclidean in format as well as in subject matter.)

The book is thoughtfully designed. The pages are large, the type is easy to read, the diagrams are clear, and the book lies flat when opened.

EVERY HIGH-SCHOOL GEOMETRY TEACHER SHOULD HAVE THIS BOOK. It's a rich source of supplementary material for regular sections, and an ideal textbook for the second semester of an honors-level class, or for a student who wants to pursue the study of geometry on an independent-study basis.

I know that I'll be turning to this book again and again next year, and for as long as I teach geometry. In fact, I plan to buy another copy so that I'll have one at home and one in my classroom.

Bravo, Dr. Posamentier, and thank you! ... Read more

Product Description
The Foundations of Geometry and the Non-Euclidean Plane is a self-contained text for junior, senior, and first-year graduate courses. Historical material is interwoven with a rigorous ruler- and protractor axiomatic development of the Euclidean and hyperbolic planes. Additional topics include the classical axiomatic systems of Euclid and Hilbert, axiom systems for three and four dimensional absolute geometry, and Pieri's system based on rigid motions. Models, such as Taxicab Geometry, are used extensively to illustrate theory. ... Read more

Customer Reviews (1)

Wonderfully clear and complete
Though perfectly clear to the mathematician, Non-Euclidean geometry is surronded by an aura of mystery and mistrust among the general public, and even a good many mathematicians would be hard pressed to explain exactly how the negation of the parallel postulate leads to all those strange formulas teeming with hyperbolic functions and other exotica. G.E. Martin explains everything beautifully, with exemplary clarity and just the right amount of detail. The reader also gets a complete construction of Euclidean geometry starting with the Birkhoff-Halsted axiom system, as well as a wealth of historical information into the bargain. Every serious math major or amateur ought to read this book, and many a professional could well benefit from it. ... Read more

Product Description

This is a book on Euclidean geometry that covers the standard material in a completely new way, while also introducing a number of new topics that would be suitable as a junior-senior level undergraduate textbook. The author does not begin in the traditional manner with abstract geometric axioms. Instead, he assumes the real numbers, and begins his treatment by introducing such modern concepts as a metric space, vector space notation, and groups, and thus lays a rigorous basis for geometry while at the same time giving the student tools that will be useful in other courses.

Product Description

"From nothing I have created a new different world," wrote János Bolyai to his father, Wolgang Bolyai, on November 3, 1823, to let him know his discovery of non-Euclidean geometry, as we call it today. The results of Bolyai and the co-discoverer, the Russian Lobachevskii, changed the course of mathematics, opened the way for modern physical theories of the twentieth century, and had an impact on the history of human culture.

The papers in this volume, which commemorates the 200th anniversary of the birth of János Bolyai, were written by leading scientists of non-Euclidean geometry, its history, and its applications. Some of the papers present new discoveries about the life and works of János Bolyai and the history of non-Euclidean geometry, others deal with geometrical axiomatics; polyhedra; fractals; hyperbolic, Riemannian and discrete geometry; tilings; visualization; and applications in physics.

Product Description
1. The origins of geometry.
Geometry, according to ~erodo.!lJ.s, and the Greek derivation
of the vord, had its origin in Egypt in the mensuration
of land, and the fixing of boundaries necessitated by the
repeated inundations of the Nile. It consisted at first of'
isolated facts of observation and rude rules for calculation,
until it came under the influence of Greek thought. The
honour of having introduced the study of geometry from
Egypt falls to THALES of ~Iiletus (640-546 B.C.), one of the
seven" vise men" of Greece. This marks the first step in
the raising of geometry from its lowly level; geometric elements
were abstracted from their material clothing, and
the geometry of lines emerged. 'Vith P.¥THAGORAS (about
580-500 B.C.) geometry really began to be a metrical science,
and in the hands of his followers and the succeeding
Platonists the advance in geometrical knowledge was fairly
rapid. Already, also, attempts were made, by HIPPOCRATES
of.Chios (about 43

Table of Contents

CHAPTER 1; HISTORICAL; 1 The origins of geometry; 2 Euclid's Elements: axioms and postulates; 3 The Parallel-postu1ate; attempts to prove it -; 4 Playfair;s Axiom; 5 Thibaut's rotation proof; the direction faJlaey; 6 Bertrand's proof by infinite areas -; 7 Equidistant Hnes -; 8 First glimpses of non-euclidean geometry; Saccheri; 9 Lambert; PAOP; I; 2; 3; " 6; 7; 9; ll,; 13; 10 Gauss 14; 11 SchweikaTt~ Taurinus and Vachtcr 14; 12 Logendre 1 fJ; 13 The theory of parallels in Britain - H); 14 The discovery of non-euclidean geometry: Lobachevsky 20; 15 Bolyai 21; 16 Tho later development 24; Examples I 25; CHAPTER II; ELE)lENTARY HYPERBOLIC GEOMETRY; I Fundamental assumptions or a;xjoms; 2 ... Read more

Product Description

Curves and surfaces are objects that everyone can see, and many of the questions that can be asked about them are natural and easily understood. Differential geometry is concerned with the precise mathematical formulation of some of these questions. It is a subject that contains some of the most beautiful and profound results in mathematics yet many of these are accessible to higher-level undergraduates.

Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces suitable for a first course on the subject. Prerequisites are kept to an absolute minimum – nothing beyond first courses in linear algebra and multivariable calculus – and the most direct and straightforward approach is used throughout.

New features of this revised and expanded second edition include:

• a chapter on non-Euclidean geometry, a subject that is of great importance in the history of mathematics and crucial in many modern developments. The main results can be reached easily and quickly by making use of the results and techniques developed earlier in the book.
• Coverage of topics such as: parallel transport and its applications; map colouring; holonomy and Gaussian curvature.
• Around 200 additional exercises, and a full solutions manual for instructors, available via www.springer.com

Praise for the first edition:

"The text is nicely illustrated, the definitions are well-motivated and the proofs are particularly well-written and student-friendly…this book would make an excellent text for an undergraduate course, but could also well be used for a reading course, or simply read for pleasure."

Australian Mathematical Society Gazette

"Excellent figures supplement a good account, sprinkled with illustrative examples."

Times Higher Education Supplement

Customer Reviews (7)

Definately a good begineers book
If you want a very general introduction of Differential Geometry, this is the book to start. Very nicely written text. Understandable examples. Broad coverage of materials . Explains space curves and surface properties with amazing quality. Recommended as a beginners level introduction

Very appropriate for self-study
It's a very good book overall, especially if you like to spend more time reading on your own than in a classroom.

Written to teach rather than to impress
I have purchased hundreds of technical books and really treasure the ones that seem to have been written in order to really convey the material rather than impress the reader with how smart the author is. This is such a book. The material is remarkably clear and the author's style strikes me as a notable example of the mathematical writing styles put forth in the articles comprising the text "How to Write Mathematics." For example, the material proceeds in a logical chain such that the reader is never confronted with a term or concept before it has been explained. The notation is defined meticulously and repeatedly so the reader is not forced to continually refer backwards through the text to remember the meaning of the symbols. This also is a boon for "grasshopper readers" who will use the text as a reference, as opposed to a linear reader. Symbols don't change meaning, are not overloaded, and seem to have been chosen for intuitive appeal. For example, a lower-case gamma denotes a parametric function for a curve and, to me, the shape of the gamma suggests the sorts of curves being discussed. In my experience, this book is best in class.

An enjoyable text on the subject!
I've been looking for a decent book on differential geometry for years now.Most of the good ones are fairly pricey, or require the reader to have a deep knowledge of mathematics.This fits in neither category.You only need multi variable calculus, linear algebra, and some experience with reading/writing proofs.This book will also appeal to those who want to learn on their own, as every problem has a hint/solution in the back.

Dissapointing
The book starts ok, but very quickly deteriorates into the classical boring math style of theorem-proof. There are a million books on the subject matter, and I don't see the need of another one which is pretty much identical. It is not a bad book, but has absolutely no added value - just pick any of the differential geometry books out there, and they will be the exact same thing. Why do they bother writing the same book over and over?? ... Read more

Product Description

Customer Reviews (2)

Non-Euclidean Geometry
I am an engineer by education, and it has been many years since my college days. I bought this book to refresh my mind about mathematics and the specifics of non-Euclidean geometry.It is not an easy read, but it does help if one is interested in the subject.Introductory Non-Euclidean Geometry

What does this mean?
I quote from page 10.

"12. Theorem. If the two angles C and D are equal, the perpendiculars are equal, and if the angles are unequal, the perpendiculars are unequal, and the larger perpendicular makes the smaller angle."

There is no accomapanying diagram and no proof provided. I choose this particular theorem because it is used in distinguishing Euclidean, hyperbolic and elliptic geometries, where it is needed for the theorem to hold when comparing a perpendicular to either an acute or an obtuse angle.

After reading this gobbledygook, I gave up in despair.
... Read more

Product Description
Ideal for users who may have little previous experiencewith abstraction and proof, this book provides a rigorous and unified—yetstraightforward and accessible—exposition of the foundationsof Euclidean, hyperbolic, and spherical geometry. Unique in approach,it combines an extended theme—the study of a generalized absoluteplane from axioms through classification into the three fundamentalclassical planes—with a leisurely development that allows ampletime for mathematical growth. It is purposefully structuredto facilitate the development of analytic and reasoning skills andto promote an awareness of the depth, power, and subtlety of the axiomaticmethod in general, and of Euclidean and non-Euclidean plane geometryin particular.Focus on one main topic—The axiomatic developmentof the absolute plane—which is pursued through a classificationinto Euclidean, hyperbolic, and spherical planes. Presents specific models such as the sphere, the Klein-Betrami hyperbolic model, and the “gap” plane. Gradually presents axioms for absolute plane geometry. ... Read more

Product Description
This text is designed for a one-semester undergraduate course in projective geometry. In incorporates a synthetic approach starting with axioms from which the general theory is deduced, together with an analytic approach using the real projective plane as a model. The first is refined as the second is generalized until the two coincide via the introduction of coordinates in an abstract projective plane. Special attention is paid to the role of Desargues' and Pappus' axioms in the theory. At the end of the book is a list of problems that can be used as exercises while reading. The emphasis on the various groups of transformations that arise in projective geometry introduces the reader to group theory in a practical context. While the book does not assume any previous knowledge of abstract algebra, some familiarity with group theory would be useful. First published in 1967 and long out of print, this book is now reissued with a new preface, an appendix on the simple group of order 168, which appears as the group of automorphisms of a projective plane of seven points, and a list of errata. ... Read more

Customer Reviews (2)

Good for math majors; perhaps annoying to graphics experts
As the preface says, this book approaches the subject from two different directions: analytic and synthetic. The synthetic approach seems to contain both (1) the parts of the subject best suited to students who know only high-school math, and (2) the parts ill-suited to those who are not "pure" mathematicians. An example of the former is a high-school-geometry-style proof that any three points on any line may be transformed to any three points on any other line via a composition of two perspectivities. An example of the latter is the projective plane that is freely generated by a configuration. The inclusion of things like this latter example is the reason why the word "foundations" is in the title; "foundations" does not mean that which should be learned before everything else; perhaps "foundations" should be defined as "that which will annoy those who want to apply projective geometry to computer graphics and which will delight `pure' mathematicians who want to know how the subject interacts with abstract algebra." The latter group will like this book and will find it very readable.

This book is short and omits some topics found in many books. Conics and polarities are not dealt with.

axiom-theorem-example very good introduction in the area
This is one of thoseboth popular and scientific books. It dealsrigorouslyand reader-friendly with projective (and, subsequently, affine) planes. Desargues and Pappus axioms/theorems interactions are described;projective collineations are studied. Best for freshmen and their advisors,as well. ... Read more

Product Description
Engaging, accessible, and extensively illustrated, this brief, but solid introduction to modern geometry describes geometry as it is understood and used by contemporary mathematicians and theoretical scientists. Basically non-Euclidean in approach, it relates geometry to familiar ideas from analytic geometry, staying firmly in the Cartesian plane. It uses the principle geometric concept of congruence or geometric transformation--introducing and using the Erlanger Program explicitly throughout. It features significant modern applications of geometry--e.g., the geometry of relativity, symmetry, art and crystallography, finite geometry and computation.Covers a full range of topics from plane geometry, projective geometry, solid geometry, discrete geometry, and axiom systems.For anyone interested in an introduction to geometry used by contemporary mathematicians and theoretical scientists. ... Read more

Customer Reviews (1)

This book is not great.
First of all, there are numerous minor errors in the printing; they get to be annoying at best, and extremely confusing at their worst.

The book also is too much of an overview--it makes a good introduction but a poor reference text.It is also very poorly indexed, which can make it hard to find things.The exercises are also poor--many new concepts are introduced in the exercises at the end of the chapters.

The writing is actually pretty good, for the most part.I think that the stuff that is explained in the book is explained well in most places, and the author does a very good job of tieing things together and bringing in historical background and significance of the topics being discussed.

I lastly might add that the name is very misleading--the geometries described in this book were mostly discovered over 100 years ago--there is nothing drastically "modern" about them.

Overall, this book was not prepared for being published--it needs a new edition to correct errors and tie up loose ends. ... Read more

Product Description
Although extensively revised, this new edition continues in the fine tradition of its predecessor. Major changes include: a notation that formalizes the distinction between equality and congruence and between line, ray and line segment; a completely rewritten chapter on mathematical logic with inclusion of truth tables and the logical basis for the discovery of non-Euclidean geometries; expanded coverage of analytic geometry with more theorems discussed and proved with coordinate geometry; two distinct chapters on parallel lines and parallelograms; a condensed chapter on numerical trigonometry; more problems; expansion of the section on surface areas and volume; and additional review exercises at the end of each chapter. Concise and logical, it will serve as an excellent review of high school geometry. ... Read more

Customer Reviews (1)

Great text for high school or college
I had used this text in a geometry class I took in college as a math major (class was not required, just needed hours to fill). After using this book and finishing the course, I found it to be much better than the geometry book I used in high school.

This book, like many high school texts, breaks geometry down into constructions using only a straight-edge and compass and then introduces definitions and theorems (some proved, some left to prove as exercises). The part that hooked me in was the fill-in-the-blank-step proofs that were partially completed but let the reader finish so that he or she will become familiar with "Statement/Reason" geometric proofs.

Finally, after ten chapters which deal with such topics as quadrilaterals, parallel lines, polygon areas, similar and congruent triangles, and circles, the authors introduce logic by methods of truth tables, Euler circles, and tautologies.

Last but not least, a chapter connecting algebra to 2-Dimensional (Cartesian) geometry offer the bridge between the "abstract" world of algebra and the "concrete" world of geometry. ... Read more

Product Description
This lavishly illustrated book provides an unusually accessible approach to geometry by placing it in historical context.With concise discussions and carefully chosen illustrations the author brings the material to life by showing what problems motivated early geometers throughout the world. Geometry Civilized covers classical plane geometry, emphasizing the methods of Euclid but also drawing on advances made in China and India.It includes a wide range of problems, solutions, and illustrations, as well as a chapter on trigonometry, and prepares its readers for the study of solid geometry and conic sections. ... Read more

Customer Reviews (5)

Beautiful book, enlightening geometry
I'll be brief since others have written a bit more. The book is beautifully illustrated and argued. Perhaps 25% of the book is about applications of geometry and 75%, especially the problems, is solving "famous" or intriguing geometry problems.It does have a significant of Chinese geometry, but the focus is Euclidian and its descendants, though it does have lot of intriguing methods used by medieval masons/architects.If you are interested in an analogous book, consider Squaring the Circle, but its focus is less on classic geometry and more on finding applications of geometry in art, so you get far more on the golden ratio, etc. I thought Geometry Civilized could add a lot of intriguing applications to a sophisticated geometry class.

Opaque on geometry, vague and thin on civilization
Heilbron's greatest accomplishment in this work is the very thorough cutting and pasting that brings us many pretty pictures, especially from the worlds of art and architecture and old textbooks. Other than that there is little of value. The bulk of the book is the same old terse Euclidean geometry that you can find in just about any geometry book. You might as well read Euclid because Heilbron adds basically nothing in terms of insight and readability when it comes to the geometry itself. In fact, he repeatedly manages to create technical obstacles even in clear terrain; see for example what must surely be the most incomprehensible introduction ever of radian angle measure on page 278. Still, the book also discusses many diverse applications which perhaps makes it worthwhile? Unfortunately, no. First of all there are some horrendously formulated statements, such as the claim that pi "cannot be expressed as a number, even an irrational one" (p. 241) and the implicit claim that three points need not lie in a plane: "Assuming, what is more or less true, that Rhodes and Alexandria lie on the same noon circle or meridian (that is, that Rhodes, Alexandria, and the centre of the earth lie in the same plane), ..." (p. 66). One wonders how such things survived into the "corrected" paperback edition. More seriously, Heilbron frequently breaks the rule that in science and mathematics everything should be explained and nothing should be pulled out of a hat. He is forced to do so because he doesn't have very many interesting applications of Euclidean geometry to offer and so has to discuss applications that are thoroughly incompatible with the mathematics covered. This is completely unnecessary since Euclidean geometry has many wonderful applications, but Heilbron simply ignores them: remarkably, conic sections, for example, are never mentioned even though there is a section on burning mirrors (!?), where we are told in a parenthesis that "a slightly different surface, whose intersection with the plane of the paper makes a parabola, gives a more intense focus" than a spherical mirror (p. 282, this is the only occurrence of the word parabola in the book). Instead, for example, we learn that "a Dutch geometer named Willebrord Snel" simply "proposed" the law of refraction (p. 107), apparently on a whim, and there is no indication of why nature would choose to obey this curious law. Later this law is fundamental when we study the rainbow, yet another topic that our methods are completely incapable of handling. Lacking calculus, we resort to the use of a table of values and read off that "evidently" the properties of refraction in raindrops are such-and-such (p. 197). If this is a valid method then why did we bother toiling with proofs of the Pythagorean theorem, for example? We could have just thrown up a bunch of numerical calculations and said that "evidently" the theorem is true. Geometry is not "civilized" by betraying the very soul of rational inquiry.

Geometry and history intertwined
The rise of geometry was simultaneous with the rise of civilization. When people are aggregated into permanent population centers, it becomes necessary to precisely measure areas so that the proper taxes can be collected and also measure in the third dimension so that magnificent buildings can be constructed. While both require knowledge of geometry, they are experiential in nature.
The true rise of Western civilization as we know it took place in Greece, and the most permanent feature of that culture was the development of abstract mathematics. From this point one, mathematics was an endeavor that involved objects whose true nature was only in the mind. Any geometric diagram could only at best be a crude approximation of the true situation. It is hard to underestimate how much of a change this was from earlier forms of reasoning. I for one, firmly believe that all of the other ideas of democratic government, ethics, logic and philosophy that arose at the same time and place were a consequence of the new, more abstract and theoretical thinking that was taking place in geometry.
This book is a combination of history and geometry, showing how intertwined the two are. It is also one of the more extensive collections of solved geometry problems that exists today. Heilbron poses many problems, solving nearly all of them immediately after they are posed. As you step through the solutions, it is with a sense of wonderment as the steps are so direct, sequential and easy to understand, which is the hallmark of good geometric proofs.
Packed with figures, this book is suitable as a text for courses in geometry that the students will love, and I will point it out to anyone who claims that geometry is dull. With supplements, it could also be used as a text for courses in the history of mathematics.

Geometry and history intertwined
The rise of geometry was simultaneous with the rise of civilization. When people are aggregated into permanent population centers, it becomes necessary to precisely measure areas so that the proper taxes can be collected and also measure in the third dimension so that magnificent buildings can be constructed. While both require knowledge of geometry, they are experiential in nature.
The true rise of Western civilization as we know it took place in Greece, and the most permanent feature of that culture was the development of abstract mathematics. From this point one, mathematics was an endeavor that involved objects whose true nature was only in the mind. Any geometric diagram could only at best be a crude approximation of the true situation. It is hard to underestimate how much of a change this was from earlier forms of reasoning. I for one, firmly believe that all of the other ideas of democratic government, ethics, logic and philosophy that arose at the same time and place were a consequence of the new, more abstract and theoretical thinking that was taking place in geometry.
This book is a combination of history and geometry, showing how intertwined the two are. It is also one of the more extensive collections of solved geometry problems that exists today. Heilbron poses many problems, solving nearly all of them immediately after they are posed. As you step through the solutions, it is with a sense of wonderment as the steps are so direct, sequential and easy to understand, which is the hallmark of good geometric proofs.
Packed with figures, this book is suitable as a text for courses in geometry that the students will love, and I will point it out to anyone who claims that geometry is dull. With supplements, it could also be used as a text for courses in the history of mathematics.

A truly unique book
All books are unique, as George Orwell might have said, but some are more unique than others. And Heilbron's "Geometry Civilized" may be the most unique of all. It is, on the one hand, a coffee table book, insize and presentation, with beautiful illustations. On the other hand, itis a serious geometry text with full proofs of many theorems in Euclideangeometry, and plenty of interesting exercises for the reader. But perhapsmost of all, it is a fascinating ramble through a wide range of topics,written by a leading historian of science with a strong esthetic sense andequally strong views on math and science education. He is, in the words ofW.S. Gilbert,"Teeming with a lot o' news", including "Manycheerful facts about the square of the hypotenuse" -- the title of hischapter on the Pythagorean Theorem. Another chapter, "From Polygons toPi," includes the exact geometry of a Gothic arch and much of theaccompanying ornamentation, as well as other topics ranging from Stonehengeto the Pentagon building, and from the idea behind burning mirrorsattributed to Archimedes and actually constructed by Lavoisier and others,to the octagonal room designed by Thomas Jefferson. Anybody who enjoyedgeometry in high school should love this book, and many people who fearedor hated high school geometry may discover what they missed by not having aJohn Heilbron to show them the wonderful richness and flavor of what,presented badly, can appear a dry and useless subject. ... Read more

Product Description
The central theme of this book is the interaction between the curvature of a complete Riemannian manifold and its topology and global geometry. The first five chapters are preparatory in nature. They begin with a very concise introduction to Riemannian geometry, followed by an exposition of Toponogov's theorem--the first such treatment in a book in English. Next comes a detailed presentation of homogeneous spaces in which the main goal is to find formulas for their curvature. A quick chapter of Morse theory is followed by one on the injectivity radius. Chapters 6-9 deal with many of the most relevant contributions to the subject in the years 1959 to 1974. These include the pinching (or sphere) theorem, Berger's theorem for symmetric spaces, the differentiable sphere theorem, the structure of complete manifolds of non-negative curvature, and finally, results about the structure of complete manifolds of non-positive curvature. Emphasis is given to the phenomenon of rigidity, namely, the fact that although the conclusions which hold under the assumption of some strict inequality on curvature can fail when the strict inequality on curvature can fail when the strict inequality is relaxed to a weak one, the failure can happen only in a restricted way, which can usually be classified up to isometry. Much of the material, particularly the last four chapters, was essentially state-of-the-art when the book first appeared in 1975. Since then, the subject has exploded, but the material covered in the book still represents an essential prerequisite for anyone who wants to work in the field. ... Read more

Customer Reviews (2)

The basic reference
I am of course partial to the book ,I was graduate student at SUNY Stony Brook.
Still,I think this is the basic reference in the subject , a book that I like
to go back to review the most important theorems.
Clear, consise ,well written.
It is quite impissible to work in Riemanian Geometry without "Cheeger Ebin"

all you ever wanted to know about differential geometry
But thought the book was out of print. This classic book was published by North Holland in the late seventies, was photocopied by every grad student in geometry, and finally has been reprinted by the American Math Society. The book's goal (achieved) is to get you up to speed and working as quickly as possible. Riemannian geometry is covered from scratch (a la Milnor's Morse Theory (Annals of Mathematic Studies AM-51) but they don't stop there, and prove all of the basic comparison results. ... Read more

Product Description

How unique and definitive is Euclidean geometry in describing the "real" space in which we live?

Richard Trudeau confronts the fundamental question of truth and its representation through mathematical models in The Non-Euclidean Revolution. First, the author analyzes geometry in its historical and philosophical setting; second, he examines a revolution every bit as significant as the Copernican revolution in astronomy and the Darwinian revolution in biology; third, on the most speculative level, he questions the possibility of absolute knowledge of the world.

Trudeau writes in a lively, entertaining, and highly accessible style. His book provides one of the most stimulating and personal presentations of a struggle with the nature of truth in mathematics and the physical world. A portion of the book won the Pólya Prize, a distinguished award from the Mathematical Association of America.

"Trudeau meets the challenge of reaching a broad audience in clever ways...(The book) is a good addition to our literature on non-Euclidean geometry and it is recommended for the undergraduate library."--Choice (review of 1st edition)

"...the author, in this remarkable book, describes in an incomparable way the fascinating path taken by the geometry of the plane in its historical evolution from antiquity up to the discovery of non-Euclidean geometry. This 'non-Euclidean revolution', in all its aspects, is described very strikingly here...Many illustrations and some amusing sketches complement the very vividly written text."--Mathematical Reviews

Customer Reviews (2)

Propaganda history
This is on the whole a standard semi-historical text on non-Euclidan geometry. What sets this text apart from the others in this genre is Trudeau's candidness regarding conclusions and interpretation of this story. Trudeau's conclusions and interpretations are precisely those of the blue-eyed research mathematician of today, not realising that he is distorting history for the purpose of propaganda. His propaganda goal is simple: intuition is bad, formalism is good. The history of non-Euclidean geometry does not support this conclusion, so Trudeau has to fake it. He does this in the form of fictitious dialogues with a student, whose stupid "intuitions" are always set straight. For example, this alleged student has the alleged "intuition" that Playfair's axiom is necessarily true. His feeble struggle with Trudeau's iron intellect even needs to be condensed by this statement:

"Time goes by. More proofs are suggested. Trudeau, with obvious pleasure, demonstrates how each proposal turns on some assertion logically equivalent to Postulate 5. Finally the other lapses into exasperated silence." (p. 161).

The conclusion sanctioned by Trudeau is that intuition should be given up in favour of formalism. But this is complete bogus-history. The whole point of the debate concerning Postulate 5 was precisely that it was *not* intuitively obvious. Which is why Trudeau is reduced to fighting, "with obvious pleasure," a fictitious opponent. While the result was not intuitive, the nature of the formal system suggested that it could be proved. Thus *it was the formal system that fooled us,* not our intuitions about geometry. Therefore, if any lessons are to be learned from the history of non-Euclidean geometry it is the exact opposite of that sanctioned by Trudeau.

Among the many other things which Trudeau gets backwards because of his doctrinal blindness are the relation of logic to mathematics (p. 14) and why Kant was wrong (p. 250). But I do not have the patience to detail these things.

A nicely designed book, with accepted views of the subject
As indicated in my other reviews, my views of the subject differ from accepted ones, and I will try to explain them further in relation to this book.

The book assumes a somewhat condescending attitude, with imaginary dialogues between the author and presumably a student, possibly from the author's experience. The student asks supposedly "common sense" questions, and the author answers with lengthy explanations, sounding to me like excuses that make the teacher come out the loser.

Thus in a section about "points" (pp.23-30) the author defends the current concept of lines as composed of points, although a point is taken to have zero dimension, and zeros plus zeros are understood to add up to zero. The concept of points as the constituents of lines is indeed recent and questionable. The author like others overlooks Euclid's definition 3, "The extremities of a line are points", and points are in fact used to delimit lines, as do "breadthless" lines delimit areas, either usage not adding to dimension.

Another dialogue discusses "line" (p.170), which has caused similar confusion. Today the word is used for "straight line", while contrariwise the term is also applied to curves like great circles on a sphere. But the sticking point to me is the way it is justified to leave "primitive terms" undefined and then "interpret" them as desired (e.g. p.169).

The justification is roughly as follows. Basic logical principles are so general that one needn't specify what they are about, and then can apply them to particular cases. This is indeed true if a principle holds for anything whatsoever. But, for instance, Euclid's 5th postulate applies specifically to straight lines in a plane, which is why to reinterpret those terms as curvatures, and say the postulate then does not apply and is hence unprovable, commits the fallacy of equivocation.

What is disappointing is that undoubtedly good heads so carelessly perpetuate illogicalities while laying claim to increased rigor.
... Read more

Product Description
Professor Honsberger has succeeded in 'finding' and 'extricating' unexpected and little known properties of such fundamental figures as triangles, results that deserve to be better known. He has laid the foundations for his proofs with almost entirely synthetic methods easily accessible to students of Euclidean geometry early on. While in most of his other books Honsberger presents each of his gems, morsels, and plums, as self contained tidbits, in this volume he connects chapters with some deductive treads. He includes exercises and gives their solutions at the end of the book. In addition to appealing to lovers of synthetic geometry, this book will stimulate also those who, in this era of revitalizing geometry, will want to try their hands at deriving the results by analytic methods. Many of the incidence properties call to mind the duality principle; other results tempt the reader to prove them by vector methods, or by projective transformations, or complex numbers. ... Read more

Customer Reviews (2)

A honeypot of sweet problems in geometry
Geometry is such a sweet science and this book is a honeypot. Ross Honsberger is one of the best in creating crafty, delightful problems. In reading this book, you will be amazed at how hard the problems appear whenstated and how simple and elegant the proofs are.
Some topics arerelatively obscure. I am sure that Honsberger is quite correct in statingthat few modern mathematicians have heard of the symmedian point of atriangle. I certainly had not. However, the level of the material is suchthat even teachers of high school geometry will be able to find somesuitable exercises. College instructors will find it essential. Problemsare given at the end of each chapter and detailed solutions are included inan appendix.
It is hard to read this book and not understand why theGreeks were so captivated by geometry. Properly presented, as is done here,it is addictive.

Published in Smarandache Notions Journal,reprinted with permission.

A great geometry book for Olympiad training!
Ross Honsberger presents the articles of modern euclidean geometric properties in a way that is easy to follow and understand.The method of proofs and development of ideas illustrate different solving strategieswhich is valuable for self-studying young students.With the help ofsimple diagrams, I found no difficulties in understanding the articles. Problems are well-graded with solutions are well-presented at the back.AMUST-READ book for students participating maths contest! ... Read more

Product Description
Focussing on the geometry of hyperbolic manifolds, the aim here is to provide an exposition of some fundamental results, while being as self-contained, complete, detailed and unified as possible. Following some classical material on the hyperbolic space and the Teichmüller space, the book centers on the two fundamental results: Mostow's rigidity theorem (including a complete proof, following Gromov and Thurston) and Margulis' lemma. These then form the basis for studying Chabauty and geometric topology; a unified exposition is given of Wang's theorem and the Jorgensen-Thurston theory; and much space is devoted to the 3D case: a complete and elementary proof of the hyperbolic surgery theorem, based on the representation of three manifolds as glued ideal tetrahedra. ... Read more

Customer Reviews (2)

Quite a nice approach
I found this book to be quite helpful. It is a nice compliment to either Thurston's or Ratcliffe's book. Results are generally proved in a somewhat more restricted setting than in Thurston (making results like Margulislemma easier to understand on first meeting). The proofs in Ratcliffe seemvery dry in comparison with the present work (the authors here tend to bemore geometric in their arguments). As a matter of fact, I would put thisbook somewhere about the midpoint of the geometric intuition spectrumbetween Ratcliffe and Thurston.

Good advice for the student.
More drawings should have been included, the rest of the book was clearly worked out. It is alsosuitable for new competitors in Noneuclidean Geometry. Only minus the few drawings. ... Read more

Product Description
Quasicrystals and Geometry brings together for the first time the many strands of contemporary research in quasicrystal geometry and weaves them into a coherent whole. The author describes the historical and scientific context of this work, and carefully explains what has been proved and what is conjectured. This, together with a bibliography of over 250 references, provides a solid background for further study. The discovery in 1984 of crystals with 'forbidden' symmetry posed fascinating and challenging problems in many fields of mathematics, as well as in the solid state sciences. Increasingly, mathematicians and physicists are becoming intrigued by the quasicrystal phenomenon, and the result has been an exponential growth in the literature on the geometry of diffraction patterns, the behaviour of the Fibonacci and other nonperiodic sequences, and the fascinating properties of the Penrose tilings and their many relatives. ... Read more

Customer Reviews (1)

Worthwhile with lots of cool pictures
I bought the book on the strength of the pretty pictures scattered throughout the book (much much more interesting than the front cover).The text of the book is as impressive.

"Quasicrystals and Geometry" is a bit between an overview and a cookbook.Sections of the book are historical, parts are practical.Explicit and easy to follow instructions for generation of penrose tilings (and many other really cool tilings) are included.

Despite what the introduction says, this book is not written for a lay audience.A quarter or two of college level math seems to be assumed. ... Read more

Product Description
Geometry, this very ancient field of study of mathematics, frequently remains too little familiar to students. Michèle Audin, professor at the university of Strasbourg has written a book allowing them to remedy this situation and, and starting from linear algebra extend their knowledge of affine, euclidian and projective geometry, conic and quadric sections, 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 each exercise 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. ... Read more

Customer Reviews (1)

Math
Un libro con un contenido propicio para lo que buscaba, el lenguaje se entiende claramente y no deja espacio a dudas ... Read more

Product Description
This charming book introduces us to topics in hyperbolic geometry in a delightfully informal style.

Journey into Geometrics can be read at two levels. It can be studied as an informal introduction to post-Euclidean geometry, or it can serve as background material for university students. The material presented in the text is extended by carefully selected problems. The background required is minimal, standard high school geometry, yet the serious student, aided by problems attached to each chapter, should acquire a deeper understanding of the subject. ... Read more

Customer Reviews (1)

Another magical journey through the wonderland of geometry
One cannot say that Lewis Carroll would be pleased, since he helps with the narration of this account of some additional travels of his darling Alice. Many of the other original Wonderland characters also help Dr. WhatIf and him explain and illustrate a wide variety of geometric problems.
Although the author does an excellent job in presenting difficultconcepts, this is not a textbook, Thinkof it as an excursion into thewonderland of advanced geometry. Experts will frolic, amateurs with grinand eventually understand and beginners will smile right up to the point(pun intended), where they get lost.
Problems are given at the end ofeach chapter and detailed solutions appear in the back of the book. Apartial list of the topics covered includes inversions, reflections, ahyperbolic T-party, hyperbolic geometry, and projections. Verse with adecidedly Carrollean flavor appears throughout the text.
A joy toread and even better the second time through, it is a fitting sequel to theoriginal adventures.

Published in Journal of RecreationalMathematics, reprinted with permission. ... Read more