Book Description

This is a reissue of Professor Coxeter's classic text on non-Euclidean geometry. It begins with a historical introductory chapter, and then devotes three chapters to surveying real projective geometry, and three to elliptic geometry. After this the Euclidean and hyperbolic geometries are built up axiomatically as special cases of a more general 'descriptive geometry'. This is essential reading for anybody with an interest in geometry. ... Read more

Reviews (1)

The beauty of geometry is captured
Originally published in 1942, this book has lost none of its power in the last half century. It is a commentary on the recent demise of geometry in many curricula that 33 years elapsed between the publication of the fifth and sixth editions. Fortunately, like so many things in the world, trends in mathematics are cyclic, and one can hope that the geometric cycle is on the rise. We in mathematics owe so much to geometry. It is generally conceded that much of the origins of mathematics is due to the simple necessity of maintaining accurate plots in settlements. The only book from the ancient history of mathematics that all mathematicians have heard of is the Elements by Euclid. It is one of the most read books of all time, arguably the only book without a religious theme still in widespread use over 2000 years after the publication of the first edition. The geometry taught in high schools today is with only minor modifications found in the Euclidean classic.
There are other reasons why geometry should occupy a special place in our hearts. Most of the principles of the axiomatic method, the concept of the theorem and many of the techniques used in proofs were born and nurtured in the cradle of geometry. For many centuries, it was nearly an act of faith that all of geometry was Euclidean. That annoying fifth postulate seemed so out of place and yet it could not be made to go away. Many tried to remove it, but finally the Holmsean dictum of ,"once you have eliminated the impossible, what is left, not matter how improbable, must be true", had to be admitted. There were in fact three geometries, all of which are of equal validity. The other two, elliptic and hyperbolic, are the main topics of this wonderful book.
Coxeter is arguably the best geometer of this century but there can be no argument that he is the best explainer of geometry of this century. While fifty years is a mere spasm compared to the time since Euclid, it is certainly possible that students will be reading Coxeter far into the future with the same appreciation that we have when we read Euclid. His explanations of the non-Euclidean geometries is so clear that one cannot help but absorb the essentials. In so many ways, Euclidean geometry is but the middle way between the two other geometries. A point well made and in great detail by Coxeter.
Geometry is a jewel that was born on the banks of the Nile river and we should treasure and respect it as the seed from which so much of our basic reasoning processes sprouted. For this reason, you should buy this book and keep a copy on your shelf.

Published in Smarandache Notions Journal, reprinted with permission. ... Read more

Book Description

This remarkable book has endured as a true masterpiece of mathematical exposition. There are few mathematics books that are still so widely read and continue to have so much to offer--after more than half a century! The book is overflowing with mathematical ideas, which are always explained clearly and elegantly, and above all, with penetrating insight. It is a joy to read, both for beginners and experienced mathematicians.

"Hilbert and Cohn-Vossen" is full of interesting facts, many of which you wish you had known before, or had wondered where they could be found. The book begins with examples of the simplest curves and surfaces, including thread constructions of certain quadrics and other surfaces. The chapter on regular systems of points leads to the crystallographic groups and the regular polyhedra in $\mathbb{R}^3$. In this chapter, they also discuss plane lattices. By considering unit lattices, and throwing in a small amount of number theory when necessary, they effortlessly derive Leibniz's series: $\pi/4 = 1 - 1/3 + 1/5 - 1/7 + - \ldots$. In the section on lattices in three and more dimensions, the authors consider sphere-packing problems, including the famous Kepler problem.

One of the most remarkable chapters is "Projective Configurations". In a short introductory section, Hilbert and Cohn-Vossen give perhaps the most concise and lucid description of why a general geometer would care about projective geometry and why such an ostensibly plain setup is truly rich in structure and ideas. Here, we see regular polyhedra again, from a different perspective. One of the high points of the chapter is the discussion of Schlafli's Double-Six, which leads to the description of the 27 lines on the general smooth cubic surface. As is true throughout the book, the magnificent drawings in this chapter immeasurably help the reader.

A particularly intriguing section in the chapter on differential geometry is Eleven Properties of the Sphere. Which eleven properties of such a ubiquitous mathematical object caught their discerning eye and why? Many mathematicians are familiar with the plaster models of surfaces found in many mathematics departments. The book includes pictures of some of the models that are found in the Göttingen collection. Furthermore, the mysterious lines that mark these surfaces are finally explained!

The chapter on kinematics includes a nice discussion of linkages and the geometry of configurations of points and rods that are connected and, perhaps, constrained in some way. This topic in geometry has become increasingly important in recent times, especially in applications to robotics. This is another example of a simple situation that leads to a rich geometry.

It would be hard to overestimate the continuing influence Hilbert-Cohn-Vossen's book has had on mathematicians of this century. It surely belongs in the "pantheon" of great mathematics books. ... Read more

Reviews (4)

Beautiful, Rewarding, and Deep.
I have some 47 books in the geometry section of my shelves. If I had to discard 40 of these, Geometry and the Imagination would be among the 7 remaining.

Geometry is the study of relationships between shapes, and this book helps you see how shapes fit together. Ultimately, you must make the connections in your mind using your mind's eye. The illustrations and text help you make these connections. This is a book that requires effort and delivers rewards.

A glimpse of mathematics as Hilbert saw it
The leading mathematician of the 20th century, David Hilbert liked to quote "an old French mathematician" saying "A mathematical theory should not be considered complete until you have made it so clear that you can explain it to the first man you meet on the street". By that standard, this book by Hilbert was the first to complete several branches of geometry: for example, plane projective geometry and projective duality, regular polyhedra in 4 dimensions, elliptic and hyperbolic non-Euclidean geometries, topology of surfaces, curves in space, Gaussian curvature of surfaces (esp. that fact that you cannot bend a sphere without stretching some part of it, but you can if there is just one hole however small), and how lattices in the plane relate to number theory.

It is beautiful geometry, beautifully described. Besides the relatively recent topics he handles classics like conic sections, ruled surfaces, crystal groups, and 3 dimensional polyhedra. In line with Hilbert's thinking, the results and the descriptions are beautiful because they are so clear.

More than that, this book is an accessible look at how Hilbert saw mathematics. In the preface he denounces "the superstition that mathematics is but a continuation ... of juggling with numbers". Ironically, some people today will tell you Hilbert thought math was precisely juggling with formal symbols. That is a misunderstanding of Hilbert's logical strategy of "formalism" which he created to avoid various criticisms of set theory. This book is the only written work where Hilbert actually applied that strategy by dividing proofs up into intuitive and infinitary/set-theoretic parts. Alongside many thoroughly intuitive proofs, Hilbert gives several extensively intuitive proofs which also require detailed calculation with the infinite sets of real of complex numbers. In those cases Hilbert says "we would use analysis to show ..." and then he wraps up the proof without actually giving the analytic part.

If you find it terribly easy to absorb Hilbert's THEORY OF ALGEBRAIC NUMBER FIELDS and also Hilbert and Courant METHODS OF MATHEMATICAL PHYSICS, then of course you'll get a fuller idea of his math by reading them--but only if you find it very easy. Hilbert did. And that ease is a part of how he saw the subject. I do not mean he found the results easily but he easily grasped them once found. And you'll have to read both, and a lot more, to see the sweep of his view. For Hilbert the lectures in GEOMETRY AND THE IMAGINATION were among the crowns of his career. He showed the wide scope of geometry and finally completed the proofs of recent, advanced results from all around it. He made them so clear he could explain them to you or me.

A Book to Put under Your Pillow
There might be less than 10 mathematics books in the world that I am glad to put under my pillow when I go to sleep. And this book is one of the top three.

A masterpiece!
This is one of the best books on Mathematics ever written. The author is arguably the best mathematician of the century. Here he treats geometry, including topology, in an elementary, though profound, way, with no formalism. A work of art. Books like this shouldn't ever become "out-of-print". ... Read more

Book Description

The basic goals of the book are: (i) to introduce the subject to those interested in discovering it, (ii) to coherently present a number of basic techniques and results, currently used in the subject, to those working in it, and (iii) to present some of the results that are attractive in their own right, and which lend themselves to a presentation not overburdened with technical machinery. ... Read more

Book Description

This book is meant for a one year course in Riemannian Geometry. The approach the author has taken deviates in some ways from the standard path. Instead of discussing variational calculus, the author introduces a more elementary approach which simply uses standard calculus together with some techniques from differential equations. This book will appeal to readers with a knowledge of standard manifold theory, including such topics as tensors and Stoke's theorem. Scattered throughout the text is a variety of exercises which will help the readers to deepen their understanding of the subject. ... Read more

Book Description

This book is an exposition of the theoretical foundations of hyperbolic manifolds. It is intended to be used both as a textbook and as a reference. The reader is assumed to have a basic knowledge of algebra and topology at the first year graduate level of an American university. The book is divided into three parts. The first part, Chapters 1-7, is concerned with hyperbolic geometry and discrete groups. The second part, Chapters 8-12, is devoted to the theory of hyperbolic manifolds. The third part, Chapter 13, integrates the first two parts in a development of the theory of hyperbolic orbifolds. There are over 500 exercises in this book and more than 180 illustrations. ... Read more

Reviews (1)

An excellent overview for mathematicians and physicists
The advent of non-Euclidean geometry resulted in many different areas of mathematics, some being specifically related to geometry, others being more general, such as proof theory and model theory. This book is an excellent overview of a particular branch of non-Euclidean geometry called hyperbolic geometry. There are good exercises in the book, and the author gives a detailed history of the subjects after the end of each chapter. After a brief review of Euclidean geometry in chapter 1, emphasizing the metric properties of Euclidean space, orthogonal transformations, and isometries, the author discusses spherical geometry in chapter 2. Spherical and hyperbolic geometries are dual to each other, in the sense that in spherical geometry, a line through a point outside a given line is never parallel to the given line; but in hyperbolic geometry there are infinitely many such lines. Also, the sum of the angles of a spherical triangle is always greater than 180 degrees ; but in hyperbolic geometry less than 180 degrees. Hyperbolic geometry is of crucial importance in physics, particularly in the theory of relativity, and the author begins a discussion of this kind of geometry in chapter 3. Hyperbolic n-space is viewed more as dual to elliptic geometry in the sense that it is modeled as a unit sphere of imaginary radius with only the positive sheet of this (disconnected) set retained. The author outlines in detail the important properties of hyperbolic geometry along with its trigonometry. This is followed in the next chapters by a model of hyperbolic n-space as a conformal ball and an upper half-space, and a consideration of the isometries of hyperbolic space. The Mobius transformations are given detailed treatment. The famous classical discrete groups are introduced, along with the crystallographic groups. The discussion gets more abstract in some parts here, for the author introduces some algebraic notions such as valuation rings, in order to prove Selberg's lemma. The author finally lays the groundwork for a theory of hyperbolic manifolds in chapter 8, by first introducing geometric spaces. These are defined by four axioms, which are generalizations of Euclid's first four axioms, and two of these axioms imply that any geometric manifold is an n-manifold. The discussion is specialized in the next chapter to geometric surfaces, where the famous Gauss-Bonnet theorem, relating the area of a surface to its Euler characteristic, is proved for spherical, Euclidean, or hyperbolic surfaces. The author studies the collection of similarity equivalence classes of complete structures for a geometric surface, namely the moduli space of such structures. Physicists, particularly string theorists, will appreciate the resulting discussion on Teichmuller space and the Dehn-Nielsen theorem. Considerations of a nature more familiar to geometric topologists follows in the next chapter, where it is shown how to explicitly construct hyperbolic 3-manifolds. Dehn surgery is employed to study the complement of the figure 8 knot. The discussion is very interesting, for it employs explicit detailed constructions that would take many hours to dig out of the literature. The general case of n-dimensional hyperbolic manifolds is the subject of chapter 11, with the constructions in chapter 10 generalized to deal with high dimensions. The author considers also the two closed, orientable, hyperbolic manifolds of the same homotopy type have the same volume by using the Gromov invariant, a quantity defined in terms of the singular homology on the manifold. The reader will get a taste of the Haar measure in the proof of the result, and later an overview of measure homology. The later is very interesting, as it brings in techniques from differential topology and the de Rham complex, and it also defines a notion of a "straightening" and smearing of a singular complex. Mostow rigidity, which says that for any two closed, connected, orientable, hyperbolic n-manifolds, with n greater than 2, a homotopy between these will also be an isometry, is also proven here. The next chapter is more involved than the rest, and deals with the case of geometrically finite n-manifolds. Dealing with cusps and "sharp corners" from the actions of discrete groups is given detailed and rigorous discussion here. The discussion leads naturally to a treatment of orbifolds in the next chapter. These objects have been extremely important in string theories in high energy physics, and the author does an excellent job of detailing their properties. ... Read more

Book Description

This updated edition will continue to meet the needs for an authoritative comprehensive analysis of the rapidly growing field of basic hypergeometric series, or q-series.It includes deductive proofs, exercises, and useful appendices. Three new chapters have been added to this edition covering q-series in two and more variables; linear- and bilinear-generating functions for basic orthogonal polynomials; and summation and transformation formulas for elliptic hypergeometric series. In addition, the text and bibliography have been expanded to reflect recent developments. First Edition Hb (1990): 0-521-35049-2 ... Read more

Book Description

Anyone who gambles, plays cards, loves puzzles, or simply seeks an intellectual challenge will love this amusing and thought-provoking book. With wit and clarity, the authors deftly progress from simple arithmetic to calculus and non-Euclidean geometry. Their subjects: assorted geometries—plane and fancy; famous puzzles that made mathematical history; tantalizing paradoxes; and more. "Charming and exciting"—Saturday Review of Literature. 169 text figures.
... Read more

Reviews (6)

Great for high schoolers with interest
My only complaint is its lack of rigor and the fact that it is getting rather out-of-date; besides that, this is the sort of book that everyone interested in math should read while they're in high school.

Indulge your enjoyment of mathematics and expand your mind
My school teacher gave me this book to read when I was 13 years old, based on the interest I showed in Mathematics that went beyond the curriculum at school. In many ways it was way beyond my comprehension at the time, but little did I know that it would have such a lasting effect on me. Reading about concepts of infinity, that you could only describe to a fellow teenager as "different sizes of infinity", I realized that there really is a philosophy of mathematics that transcends all other subjects and that there is also an art to working with the subject. I can't recommend this book enough, and I never did give it back to my teacher!

Somewhat dated but still well worth reading
Originally published in 1940, the material in this book is beginning to show a little age. However, the quality of the writing renders those defects to near irrelevancy. Popular descriptions of mathematics are differentiated by the quality of the writing rather than the distinctiveness of the mathematics, and this one shines.
I like this book, starting with the title. It takes an enormous amount of imagination to do mathematics, something unappreciated by the public. It is easy to understand the use of linear segments to approximate the length of a curve. However, it requires an enormous leap of abstraction to believe that if they are made of zero length and then summed up, the result is the true length. Calculus students dutifully record and apply this, but in most cases don't appreciate the significance of the idea. In nearly all cases of major mathematical advancement, a fundamental change in thought processes was necessary. Those changes require imagination and the advances explained in this book are well documented and described.
Mathematicians are containers of some of the greatest concentrations of imagination that humans possess. Their leaps of abstraction often include descriptions of objects that cannot be visualized. Kasner and Newman capture this essential ingredient, serving it up in palatable portions.

Mathematics and the Imagination
This book came to me by chance.Instantly got my attention.It is written in such a way,that makes interesant to travel through different chapters.In each one you have the mathematical theme mixed with stories , mathematicians histories and puzzles.You learn about people with the greatest imagination.Their personality and a lot of other things,that make you enjoy the reading,no matter if you love mathematics or you have hated it all the time.I've enjoyed specially the chapter about Mathematical Analysis,with the story about rivalry between the egocentric Newton and the humble Leibnitz. This book is the opportunity to learn that mathematics are not the boring thing we have learned at school.

Oldie but Goodie
Having had this book around the house for ages, I picked it up and to my surprise within a few minutes really understood (not just enough to use, but actually understood) what logarithms really are, where they come from. The chapter on e, pi, and i is another great one to get the story behind the story, as it were. For me the book could better have continued in this vein of explaining concepts we've seen before but never really grasped intuitively, and perhaps because I'm not terribly interested in mathematical games I found that segment less fun. But in fairness, they've done a good job getting away from textbook math and into some interesting themes. I don't know if it's all still valid, as it is so old--references to Fermat's last theorem are at least outdated! ... Read more

Book Description

Janos Bolyai (1802-1860) was a mathematician who changed our fundamental ideas about space. As a teenager he started to explore a set of nettlesome geometrical problems, including Euclid's parallel postulate, and in 1832 he published a brilliant twenty-four-page paper that eventually shook the foundations of the 2000-year-old tradition of Euclidean geometry. Bolyai's "Appendix" (published as just that--an appendix to a much longer mathematical work by his father) set up a series of mathematical proposals whose implications would blossom into the new field of non-Euclidean geometry, providing essential intellectual background for ideas as varied as the theory of relativity and the work of Marcel Duchamp. In this short book, Jeremy Gray explains Bolyai's ideas and the historical context in which they emerged, were debated, and were eventually recognized as a central achievement in the Western intellectual tradition. Intended for nonspecialists, the book includes facsimiles of Bolyai's original paper and the 1898 English translation by G. B. Halstead, both reproduced from copies in the Burndy Library at MIT. ... Read more

Book Description

This book provides a geometrical experience that unifies Euclidean conceptsgenerally discussed in traditional high school geometry courses with variousNon-Euclidean views of the world.The book offers the readera "map" for a voyage through plane geometry and its various branches. It takesan informal tone while presenting the material in a reasonably rigorous manneras well as organizing it into a logical progression. Chapters are separatedinto independent units so readers can learn information in bites. Contains asummary at the conclusion of each chapter that includes a list of newdefinitions and theorems to aid in the organization of the material. PresentsEuclidean and non-Euclidean geometrics with a significant amount of backgroundinformation, that places much of the development of geometry in an historicalcontext.A valuable reference book on basic geometry for almostany reader seeking additional information on the subject. ... Read more

Reviews (4)

Not bad, not great
I am currently using this book for an advanced Geometry class. The book works well in conjunction with a well taught class, which I thankfully have. The text contains a lot of information, although not all of the mathematical subtleties are brought to the readers attention, and are either left as exercises or for the student to ponder further. Greater analysis of deep and complex concepts would have been appreciated, and some answers to the problems would have been helpful as well -- the book contains none, which is a shame because the exercises are pretty good and are a requirement to master the material. I can understand why my professor likes this book, but it can be difficult to learn from if you haven't been exposed to the material before, and especially if you need to rely on it as a primary source of information.

Roads to Geometry - Traffic Jam
I am currently enrolled in a college course that uses this text.
The book contains no study guide or solutions section and the frequent statement throughout the text is "proof of ... is left as an excercise." If you don't know the solutions before you read the text, then spend your money and time reading another book.

Confusing and overly-condensed
*Note: this refers to the second edition.

Author: CS/Math double major at Cameron University.

This is the book I unfortunately got stuck with for my College Geometry class at Cameron. Thankfully I had a great teacher and I enjoyed the class in addition to getting an 'A', but that was no thanks to this textbook. Our class and the textbook were proof-oriented and designed to introduce the student to the basic axioms, theorems, and developments of several geometries.The book doesn't go extremely in-depth into any geometry but it was only designed to introduce the basic logic and principles of the geometries anyway.The problem is the extreme conciseness and unclarity of the material.Chapter 1 is written well enough, giving the reader a good foundation in axiomatic development.From then on, each chapter starts with a list of axioms and explanations and then it's the traditional theorem-proof-theorem-proof format.Unfortunately, the proofs are condensed into as compact a format as possible.The way the proofs were written, it kind of made the book feel like 1000 page book crammed into 400.The drawings are somewhat helpful but even those are lacking.Also, a series of exercises are given at the end of each section yet no answers are given in the back.And, like most geometry books, there's no available study guide.On the plus side, the exercises have a nice range of difficulty, from trivial to virtually impossible and everything in between.

Avoid if you can or pray you get a good teacher like I did.

A basic book in need of more graphics
This book is conveniently sized for easy transportation.This book is lacking any answer key or study guide to help the reader fully understand the content of the questions asked.It has many typo's making the readerconfused when referring to text.The print for the exercise section isvery small for the adult reader of a mathematical study journal.As aneducator I would choose an additional study or reference list of books foruse. ... Read more

Book 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

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

Book Description

The geometry of the hyperbolic plane has been an active and fascinating field of mathematical inquiry for most of the past two centuries. This book provides a self-contained introduction to the subject, taking the approach that hyperbolic geometry consists of the study of those quantities invariant under the action of a natural group of transformations. Topics covered include the upper half-space model of the hyperbolic plane, Möbius transformations, the general Möbius group and the subgroup preserving path length in the upper half-space model, arc-length and distance, the Poincaré disc model, convex subsets of the hyperbolic plane, and the Gauss-Bonnet formula for the area of a hyperbolic polygon and its applications. The style and level of the book, which assumes few mathematical prerequisites, make it an ideal introduction to this subject, providing the reader with a firm grasp of the concepts and techniques of this beautiful area of mathematics. ... Read more

Reviews (1)

great book
this is a really great introduction to hyperbolic geometry.especially if you want to study gammas acting on the upper half plane.it starts at a much lower level then any other text. ... Read more

Book Description

In this engaging book, Donald Davis explains some of the most fascinating ideas in mathematics to the nonspecialist, highlighting their philosophical and historical interest, their often surprising applicability, and their beauty. The three main topics discussed are non- Euclidean geometry, with its application to the theory of relativity; number theory, with its application to cryptography; and fractals, which are an application in art, among other areas, of early mathematical work on iteration. Other topics include the influence of Greek mathematics on Kepler's laws of planetary motion, and the theoretical work that led to the development of computers. Assuming the reader has some background in basic algebra and geometry, Davis relies on exercises to develop some of the important concepts. These exercises are designed to improve the reader's ability in logic, and enable him or her actually to experience mathematics at increasingly advanced levels. ... Read more

Reviews (5)

Excellent for high school teachers and students
I use the ideas in this book in my mathematics teaching in high school. Students learn to think of the world as Euclidean through most of their instruction; Taxicab Geoemetry gives teachers a very straghtforward way to introduce non-Eucliean Geometry. Admittedly, this book is not thorough, and it is very open ended (which I consider to be positive). Nevertheless, for its intended audience it is outstanding.

Disappointing
Very simplistic treatment, with some results left for the reader to work through exercises. The chapters are almost non-existent, with all the book being mainly exercises.

Excellent for what it is
Before purchasing this book, realize what it is. This is a book about non-euclidean geometry. Specifically, a specialized form of non-euclidian geometry affectionately referred to as taxi-cab geometry. This is not a table top book, but is a book for mathemeticians and those interested in mathematics. Others need not apply (regardless of how interesting the topic is). This is an excellent introduction to non-euclidean geometry because it strips away common misconceptions about the nature of non-euclidean geometries. This text is excellent for grade school children and those who would like to branch into more advanced non-euclidean geometries like hyperbolic.

This is a book for a math student only.
I thought that this book would be about geometry of exotic (but real) places in outer space (such as a black hole, for example). Instead, it turned out to be a lethally boring book, full of math problems, that was LESS interesting than my geometry book in high school!

A simple, real-world example of non-Euclidean geometry
Some years ago, I was employed by a company that built mapping software. One of the projects I worked on was the design of features that allowed for the computation of the shortest path from one position to another following only roads. This form of travel is similar to the taxicab geometry in that movement is restricted to lines. The only difference is that roads can be placed at any location or angle whereas the lines in taxicab geometry are equidistant and perpendicular. Think of it as the geometry of graph paper. As I constructed the program, I was struck by how so much of classical Euclidean geometry does not apply. Yet, the geometry is generally easier to understand because it is almost always how we move from place to place.
The phrase non-Euclidean geometry generally conjures up thoughts of curved space and Riemannian geometry. However, in this delightfully simple book, a natural non-Euclidean geometry is developed in great detail. Very little mathematics is needed to understand the geometry, if you can mark and understand the points on a grid of graph paper, nearly all of the topics will make sense. A large number of problems are included at the end of each chapter and solutions to many appear in an appendix.
The problems cover topics such as finding the point(s) of minimum distance between two or more points and what the taxicab analogues of circles and ellipses are. Determining the point of minimum distance between three or more points is a hard problem in standard geometry, but fairly simple in the taxicab geometry.
Geometry is the godfather of abstract mathematics, being first used to codify the parceling of land and the construction of cities. In this book, you will learn how to minimize functions based on the restrictions of traveling through cities, a task with many applications in the world. ... Read more

Reviews (1)

A great introduction to hyperbolic geometry
This is the only book I've ever seen that makes the hyperbolic plane seemso natural and accessible.It takes ordinary Euclidean plane geometry asits basis, which allows the book to cover a lot of material in a verysatisfying manner, without requiring advanced background like group theoryor differential geometry.(In fact, the only background required isbasically exposure to proofs, and occasionally a little calculus or linearalgebra.)It's also extremely well-written, and the problems are wellthought out. A great text for college juniors or seniors (or even advancedsophomores), and also great for self-study. ... Read more

Book Description

Geometry and Its Applications combines traditional geometry with ideas of recent decades to present a new approach for the 21st century. It balances the deductive approach with discovery learning, and introduces axiomatic, Euclidian geometry, non-Euclidian geometry, and transformational geometry. The text integrates realistic applications throughout, includes historical notes in many chapters, and contains student and instructor's guides that support Geometer's Sketchpad. Includes a free instructor's manual to professors of adopting universities.

* A unique blend of modern applications and theory
* Excellent balance of mathematical rigor and informal style
* CD-ROM (included) offers courseware for use with The Geometer's Sketchpad
* Covers polyhedra and planar maps
* Offers balance between deductive geometry and coordinate geometry using vectors
* Contains over 700 exercises with complete solutions available
* Includes Student and Instructor Guides which support the software ... Read more

Reviews (1)

THE BOOK OF IDEAS
I got this book as a second hand and shortly its very very nice book.
The applications are very smart and clear ,
Its contexts and illustrations are adequate ,precise and really easy to read and understand.
I realy loved this book ,and i guess this is how the geometry Should be taught as rich ideas with apps not in abstract form.
You will find a nice proof for fermat's least time principle,
and lots lots more intersting ideas good for physics and computer
graphics programming.
This book really worth any price. ... Read more

Book Description

This book, which grew out of Steven Bleiler's lecture notes from a course given by Andrew Casson at the University of Texas, is designed to serve as an introduction to the applications of hyperbolic geometry to low dimensional topology. In particular it provides a concise exposition of the work of Neilsen and Thurston on the automorphisms of surfaces. The reader requires only an understanding of basic topology and linear algebra, while the early chapters on hyperbolic geometry and geometric structures on surfaces can profitably be read by anyone with a knowledge of standard Euclidean geometry desiring to learn more abour other 'geometric structures'. ... Read more

Book Description

This practical monograph is the only volume devoted exclusively to the design and analysis of structures in which nonlinear effects arecritical. Following a general overview focusing on the phenomena ofgeometric nonlinearities, the authors detail a hierarchy of discreteand continuous systems, from trusses to frames and from beams to amembrane finite element. They discuss topics including linearstructural analysis, exact analysis of trusses, nonlinear analysis ofmembranes, plane frames and space frames, cablenets and fabricstructures and three-dimensional beam-columns. Appendices provideinformation on determinants, the rotatin matrix, perturbation methodsapplied to plane beams, member stiffness when beam-column effects areincluded and graphics on a PC. An invaluable teaching and designtool, the text is accompanied by a FORTRAN disk accessible to PCsrunning DOS, and it presents computer programs as integral, both inthe classroom and in the workplace. Taking the first unified approachto geometric nonlinearities, this book allows readers to: * analyzeand design cable nets and fabric structures * incorporate exactcomputer analysis as a replacement for approximate buckling methods *approach nonlinear structural analysis as a simple application ofNewton's method by employing perturbation theory ... Read more

Book Description

Although Riemann surfaces are a time-honoured field, this book is novel in its broad perspective that systematically explores the connection with other fields of mathematics. It can serve as an introduction to contemporary mathematics as a whole as it develops background material from algebraic topology, differential geometry, the calculus of variations, elliptic PDE, and algebraic geometry. It is unique among textbooks on Riemann surfaces in including an introduction to Teichmüller theory. The analytic approach is likewise new as it is based on the theory of harmonic maps. For this 2nd edition the author has further improved aspects of presentation of various parts of the text. ... Read more