Euclidean-Geometry Books

A pearl! Anyone interested in Geometry shouldn't miss the lucid presentation of the great Hilbert.

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.

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.

This is a marvellous book. I will illustrate by one sample from each chapter (except chapter 1 on "the simplest curves and surfaces" which is the least exciting chapter). Chapter 2 on "regular system of points" contains a beautiful derivation of Leibnitz' series pi/4=1-1/3+1/5-1/7+... If we draw a large circle centred at the origin then of course a good measure of its area is the number of integer points it contains. Now, for any such point, x^2+y^2 is an integer less than r^2. So the number of such points can be obtained by going through all integers less than r^2 and counting how many times it can be written as the sum of two squares. But this is a classical problem in number theory and the solution is known. So this number theoretic result essentially tells us the area of a large circle, so it implies an expression for pi, namely Leibnitz' series. Chapter 3 is on projective geometry. We go through many projective configurations that are not seen very often today, but still the classics are the best, such as Desargues' theorem. If we have a triangular pyramid and cut it with two planes to get two triangles then the three points of intersection of the extensions of corresponding sides will or course be on a line (the intersection of the two planes), which is the three-dimensional Desargues' theorem. But by projecting the triangles onto one of the walls of the pyramid we get two projectively related plane triangles and the theorem holds for them also. All we have to do to prove the plane Desargues' theorem is to prove that all such configurations can be obtained in his way (i.e. that one can always erect an appropriate pyramid based on two projectively related plane triangles) which is practically obvious. Chapter 4 is on differential geometry. The fundamental concept of differential geometry is curvature, which is a number that indicates how curved a surface is at a given point. It may be defined as follows. We draw a little circle around the point on the surface and consider all the normals to the surface at these points. Take these normals and put them with their origin at the center of a sphere; then they will sweep out a section of the surface of the sphere. The curvature is the ratio of the area enclosed on the surface and that on the sphere as the circle is taken infinitesimally small. This quantity is seen to be invariant under bending by triangulating the surface; then the the circles are polygons with fixed angles and the theorem follows from the fact that the area of a spherical triangle is determined by its angles (proof omitted here; see any Stillwell geometry book for Harriot's beautiful proof (a.k.a. "Euler's proof")). Now, there are two fundamentally different types of points. Either the surface bends in the same direction in every direction, as on a sphere, or it bends in different directions like a saddle. In the first case the boundary on the sphere traced out by the normals has the same orientation as the boundary on the surface; in the second case the orientation is reversed. So, using signed area, the second type of points have negative curvature. A typical surface will have areas of positive curvature and areas of negative curvature and in between there will be lines of zero curvature. An absolutely wonderful, although perhaps not entirely successful, application of this concept is Klein's Apollo Belvidere hypothesis that the curves of zero curvature on a human face determine beauty. Chapter 5 on kinematics contains a determination of the curve that "we may observe ... every day in cups and tin cans when the light shines on them", i.e. the coffee cup caustic. With the sun at x=-infinity, the radius that makes an angle theta with the x-axis will point to a point where the angle of reflection is also theta. Consider a concentric circle of half the radius, and another circle with the other half of the radius as its diameter. The arc cut out of the inner circle by the radius and the x-axis is equal to the arc cut out of the outer circle by the radius and the reflected ray (arc with central angle theta in the big circle = arc with central angle 2*theta in the small cirlce). The shape of the caustic follows by rolling the outer circle on the inner. The reflected light rays are tangent to this curve since they are perpendicular to the line connecting the generating point with the center of motion (intersection of the two circles). From chapter 6 on topology one nice result is that any continuous mapping of a disc onto itself has a fixed point. For suppose it did not. Then any point in the circle can be connected with its image by an arrow. Now consider the point on the boundary. The arrow direction varies continuously as we walk once around the circle, and it end up where it started so it must have made an integer number of revolutions. But there is also a tangent at each point, and the tangent of course make one revolution as we walk once around. The arrows always point to some point in the disc so they could never point in a direction parallel to the tangent so the arrows in fact have to make one revolution also (they would have to be parallel to the tangent for a moment to overtake it, and if they stood still they would be parallel to the tangent "at six o'clock" so to speak). But if we consider the same situation for a concentric circle inside the disc then it too must have arrows making one revolution because the number of revolutions can not make jumps since the new circle is obtained by continuous shrinking of the circumference circle. But as we shrink this circle to infinitesimal radius then all its arrows point in the same direction, so they don't make one revolution, so we have a contradiction. One sees similarly that a continuous mapping of the sphere onto itself also has a fixed point. Since the projective plane is the sphere with diametrically opposite points identified this proves that any projective transformation has a fixed point.

I agree that this book, co-authored by the co-greatest mathematician of the first quarter of the twentieth century, is a masterpiece to be treasured and kept in print, as other reviewers have stated.

However: The Preface states: "This book was written to bring about a greater enjoyment of mathematics, by making it easier for the reader to penetrate to the essence of mathematics without having to weight himself down under a laborious course of studies."

All I can say is that if you read this and find it "easy," then you have terrific mathematical talent! Yes, the drawings and the intuitive descriptions are helpful, but much of the book is so obscure that I have been told that one of the world's leading geometers is working on an annotated edition explaining what the authors were talking about. On topics which I had already studied elsewhere, I found the presentation illuminating.

I still recommend this book.

One of the problems with modern mathematics is its obsession with rigor which has been attended, over the last few decades, by a mushrooming of symbols and jargon. Much of it is not clearly related to the ideas they serve to label, as evidenced by such terms as the topological use of "filter" whose etymology is obscure (ascribed by some to H. Cartan). Moreover, the particular subject of Lebesgue integration and its generalizations is made even more confusing by a wide variety of approaches depending on an author's penchants--many of whom are enamored with a purely axiomatic approach and who make little or no appeal to intuition or--God forbid!--pictures. The author of the present work is obviously someone who has actually taught mathematics and taught it lovingly. This book is an excellent read with lots of interesting topics well explained from a student's point of view. There seems to be a nice ramping from the truly elementary to the sophisticated, which means the book will interest experienced mathematicians, scientists and engineers. There are lots of "doable" problems that the reader can solve along the way. For the experienced mathematician these little problems help alot as a refresher (Oh!, now I remember, that's how you do it.). I like the emphasis on Euclidean space. Somehow, I always feel more comfortable there! It gives me things I can actually construct and doodle on paper. And, it allows the author to use a few figures in a meaningful way. Which is another of the book's strong points and if I could recommend a future improvement, it would be to bring on more of those pictures! Tristram Needham has done a nice job along these lines with his book "Visual Complex Analysis." (I ordered several copies as Christmas gifts--just kidding!). Anyone who has taught mathematics and genuinely wished to be understood by his students has, at various times, drawn them pictures. Inside the cover sheets are lists of integration formulae, a fourier transform table, and a table of "assorted facts" on things like the Gamma function; which show that this is not only a book on Lebesgue integration but a calculus book with the Lebesgue integral occupying center stage. Everyone who has been enamored by the notion of the integral--as I was as a freshman calculus student and have been ever since--will want to have this book on their shelf.

As someone who wasn't a math major but who has been trying to get up to speed on lebesgue measure and integration, I found this book to be truly accessible. Unlike other "introductory" texts (such as Kopp's "Measure, Integral and Probability") I could follow the reasoning in this book without much difficulty.

The only criticism I have of the book has to do with the first chapter. Its purpose is to provide background mathematical material and given the author's clear ability to explain difficult concepts, I wish that it covered that material in greater detail.

For others who may be looking to build a foundational understanding of this material but who may not be mathematicians, I'd also recommend Pitt's "Measure and Integration for Use" (1985) or his "Integration, Measure and Probability" (1963) (both out of print but fairly easy to find). Those books, along with Jones', are well-used items in my library.

This book is truly marvelous. Unlike many other books written by mathematicians, this one does not focus entirely on the overlycomplicated symbolism that has become so frighteningly common in modern textbooks on pure mathematics. The author, for the most part, demonstrates the concept before moving on to nonconstructive methods. This is accomplished via simple pictures, depicting simple figures such as rectangles. By using these to demonstrate the concepts of set theory, and lebesgue measure, the reader almost certainly will develop some awareness of lebesgue integration. For someone not already familiar with set theory (like myself) I suggest writing down the key terms of set theory (presented in the first 23 pages of the book) and keeping them with you while reading. Mathematical rigorousness should come secondary to understanding the key concepts, and therefore the reader might want to skip over nonessential ideas (usually some kind of proof related to an extension of a key concept) and go right to what is needed to understand what is going on.

"Lebesgue Integration on Euclidean Space" is a nearly ideal introduction to Lebesgue measure, integration, and differentiation. Though he omits some crucial theory, such as Egorov's Theorem, Jones strengthens his book by offereing as examples subjects that others leave as exercises. The best example of this is his section on L^p spaces for 0 < p < 1.

The book's greatest strength, however, is its readability. Whereas Royden gives no hint as to how much work is needed between steps, Jones highlights important steps in proofs, not just the important proofs. It is this motivated style that makes his book useful.

Jones is so careful in his construction of the theory that differentiation does not appear until Chapter 15, and specific results for R^1 come only in Chapter 16. But the wait is worth it.

While Jones has written a great introduction, the book cannot be used for more advanced courses. As the title suggests, the discussion is restricted to Euclidean spaces. In addition, his direct jump to measure on R^n and the use of "special rectangles" therein make the development incongruous with other books. But what is sacrificed in depth is made up for in breadth, with Jones hinting at how the theory is used in other branches of math. There's even an entire chapter devoted to the Gamma function!

As a student, I have found Jones's book more instructive on basic theory than Royden, Rudin, and Wheeden & Zygmund. I highly recommend it as a first-semester introduction to Lebesgue theory or as a source of clean, fundamental presentations of proofs.

This is a terrific text for a first course in graduate-level real analysis, and is suitable for self-study. It develops Lebesgue integration theory slowly, in a very clear manner. In addition, the latter part of the book covers the basics of Fourier Analysis and important topics in differentiation. I frequently refer to this book, as the results are easy to find.

".....Historians of mathematics as well as those more interested in the mathematics itself, WILL FIND THIS UNIQUE BOOK RICHLY REWARDING."
[from the book of the back cover]

This is the best book about polyhedra! But it's not always easy to read. He has chosen to take a chronological approach. That means that sometimes you have to look around a bit.

I picked up the book wanting to understand two things.

1. What are the exact definition of the Platonic and Archimedian solids, i.e., how to destinguish the Platonic from the the Deltahedra and the 13 Archimedian from their isomeric forms and the pyramids.

3. What's the reason behind the names for the Kepler-Poinsot solids. Why is the great stellated dodecahedron called the great stellated dodecahedron?

Cromwell answers the first question beautifully in Chapter 2. The second question is first discussed in Chapter 4, but I was still confused. It was only in Chapter 7 that it started to make sense.

I believe the book will answer most of your questions, but you may have to look around for it.

I've read many books on polyhedra, and this is the best I have seen. It covers the history and mathematics of many different polyhedra; the Platonic and Archimedean solids are just the beginning. Kepler's rhombic polyhedra, stellated polyhedra, Miller's solid, etc. -- it's all here. The diagrams are exceptional. I teach high school geometry, and have found this book to be an essential resource in class. The level of detail is quite high, making the book useful as a straight-through read (for someone who is really into math) or a book to flip around in (for those who find heavy math intimidating, but still like polyhedra). Includes helpful tips for model-making. Buy it!

I really like this book. It would be easy to say it's the best book on the subject that I've seen, but that doesn't say too much, because it's just about the only book I've ever seen devoted exclusively to this subject. So let me say instead that if you are at all interested in the geometric objects known as polyhedra, you will probably find something interesting in this book.

The author deals with the classical geometry of polyhedra, but not exclusively with that aspect. He covers the symmetry properties, best explained in terms of group theory concepts, and introduces and explains the notation of Schoenflies for describing symmetry groups (one of the two most common notations, and the one most used by people interested in things like molecular structure). This makes the book useful as well for those who want to learn about symmetry, and in fact this book is in many ways better for this purpose than many books I have seen with "symmetry" in their titles.

There is one thing with which I find fault: the index is inadequate. I had looked to see whether the book had a section describing the polyhedra known as Johnson solids, and found no reference to either "Norman Johnson" (after whom they are named) or "Johnson solids" in the index. But later, on scanning through the book, I found a very good treatment, explaining Johnson's terminology and with good illustrations of the Johnson solids and related polyhedra. The index made the book appear to be less adequate than it is. If this book ever goes into a second edition, it needs someone to make a new index.

It's a wonderful book for learning history of polyhedra, but I think it has too little 'mathematics' in. All in all, it's a masterpiece in my mathbook collection.

I'd taken Calculus ten years ago and needed to retake it. After working through 'Forgotten Algebra', this book was a godsend. It's B.S.-free and highly focused on what you need BEFORE Calculus. If you buy the first edition used (as I did), then it's cheap.

These guys are geniuses for presetnting complex information in a digestable, direct, witty, and accurate manner. Great for remembering forgotten trig and algebra. Awesome calc primer.

I salute a rare erudite math professor who can actually teach. Where can we find more? My school is full of passive aggressive math professors who know their stuff but cant teach.

thank you for the help

This book is totally great. I've seen very few books that explain mathematical concepts as well as this. The authors actually made math look easy by providing a lot of examples and writing in a more conversational, informal style (like the style in the Idiot's Guides series). It's a lot clearer than Barron's and Cliffs. Often there is a lot of humor added in.

As the title of this book indicates, this book is intended for calculus students in college who are struggling in their courses. However, for high school students who are using this book to teach themselves new concepts or using this as a tutorial, review, etc. I would recommend this as a supplement to another text. The book is somewhat brief and does not go through the advanced and nitty-gritry areas of algebra and trigonometry. This is intended to be a remedial "catch-up" book and is not targeted specifically towards the hig school student. But high school students, teachers, and curious readers will all find this book very helpful for making hard math look simple.

If you have previously studied and largely understood algebra and trigonometry and just need a quick review, then this book is right for you. However, if you haven't studied these topics or didn't understand them the first time, then it will be of little value. The coverage is thorough in breadth, starting with the basics of adding and subtracting all types of numbers through logarithms, exponential functions and trigonometry. Only a few pages are devoted to each topic with exercises at the end of each section. Solutions to some of the exercises are given at the end, although in my opinion there should have been more.
While the explanations are short, to the extent that it is possible in a small number of pages, they are through in depth. In a section that I found interesting, the dy/dx notation for a derivative is used. However, knowledge of calculus is not required, the reader is simply being asked to algebraically solve for dy/dx rather than use any knowledge of what it represents. Used in the manner for which it was intended this book is an effective tool in the study of precalculus material.
Note: This book is nearly identical to the companion book, "Just-In-Time: Algebra & Trigonometry for Early Transcendental Calculus." It would be a waste of money to buy both.

This book contains a detailed description of how to make (paper) models of each of the 75 uniform polyhedra, as well as some stellated polyhedra. Wenninger's descriptive and precise writing style is invaluable--his advice on construction methods and techniques are right on the mark. In nearly all cases, he provides sufficient data to allow the reader to draw his/her own templates, but especially in the most complex polyhedra the facial plane data is lacking, most likely because it is too lengthy to include. I would like to have seen more math in this text, and larger photographs. Otherwise, this is a must-have book for anyone interested in polyhedra.

My 8th grade math teacher let me borrow this book for the summer. I enjoyed making the modles, some of the more complex ones I havent been able to do but most are farly simple and easy to compleat with pacence

Since I discovered this book, I've easily spent hundreds of hours building these wonderful polyhedra. With a good general guide to model building and clear instructions, patterns and pictures of the completed model -- the book provides the raw material for a great hobby. The completed models are a continuing fascination, the relations between polyhedra and their symmetry can really only be appreciated when you have the models in your hands. While I loved building some of the over 120 models, it requires patience, a steady hand and practice.

I have been fascinated with these structures since my disciovery of this book in 1980. Magnus is a fine builder of models and a competent teacher. The explanations for building each model are concise. I also compliment the photographer. As one begins to understand the underlying principles of these solids, a vast array of options present themselves as topics of further study.

I have been using a number of the Sangakus from Sacred Geometry in my High School Pre-Calculus classes to get things rolling at the start of the class. The kids are loving them! Watching the kids last class get the problem and go to work on it reminded me of watching piranha's go after a hapless animal-- maybe a bit less graphic. The problems are just great-- they really hook the kids, really get them trying stuff, and they do a fantastic job of building up and connecting their skills. Of course I am having a great time with them too!

Further, the book is just a pleasure to read. Everything about it-- prose, graphics, mathematics, quality of production-- is just top notch.

I am always interested in what Tony Rothman has to say. He is the real deal, teaches physics at Princeton, Harvard, etc., who comes up with revolutionary insights you just can't find anywhere else. SACRED MATHEMATICS is a revelation and a tremendous challenge, another brilliant one in this writer's repertoire.

I began my Rothman studies after reading INSTANT PHYSICS, which pretty much brought me up to speed in what had always intrigued yet baffled me. Then I was amazed with his majestic DOUBT AND CERTAINTY followed by the jaw-dropping, myth-busting EVERYTHING'S RELATIVE. I couldn't get enough so I started backtracking and discovered the Pulitzer Prize nominated A PHYSICIST ON MADISON AVENUE and SCIENCE A LA MODE, where he maybe first established his continual theme of treating science with the skeptical irreverence it often deserves. In between, I discovered articles in SCIENTIFIC AMERICAN, DISCOVER, ISAAC ASIMOV'S SCIENCE FICTION MAGAZINE and THE NEW REPUBLIC, not to mention some weighty scientific papers and reports. Finally, I found his science fiction novel, THE WORLD IS ROUND, with which the movie industry might finally have the tools to do justice.

Tony Rothman is a great and gifted writer and SACRED MATHEMATICS is a beautifully illustrated book of art, religion, history and geometry. I see from his web site that a novel about The Great Seige of Malta is next. I anxiously anticipate that and hope that both APOCHRYPHA and the plays there mentioned will soon be published.

I strongly recommend SACRED MATHEMATICS and, in fact, everything written by Tony Rothman to anyone, who in a world too often full of nonsense and lies, cherishes instead reality and truth. Rothman's voice is beautiful and unique.

The last (for the moment) title of Fukagawa&Rothman is really excellent. Not only the printing is superb, but the mathematical content is also outstanding. Strongly recomended to every lover of geometry...

For anyone who truly loves mathematics, this book is a must have.
Simply put, the book tells the story of sangaku, geometry problems which were painted in color on wooden tablets and displayed at Buddhist temples and Shinto shrines throughout Japan. Most of the sangaku were composed by people from all walks of life-priests, farmers, children women, samurai, etc.-between 1600 and 1900. Approximately 900 of the old tablets have survived and even today one is occasionally found at an abandoned temple/shrine. Tony Rothman has assisted Mr. Fukagawa Hidetoshi, a retired Japanese high school teacher, who is one of the world's foremost experts in sangaku, in producing a beautiful book. Various chapters discuss Japan and temple geometry, the Chinese foundation of mathematics, Japanese mathematics and mathematicians of the Edo period. In addition, the book contains over 200 sangaku problems ranging from very elementary to extremely difficult. The book also contains extensive excerpts from the diary of Yamaguchi Kanzan, a Japanese mathematician, who treked through Japan during the 1800s collecting sangaku problems. Finally, there are chapters on East and West, Japanese attempts to handle differentiation and integration, and inversion. The book contains numerous diagrams which accompany the problems and there are 16 color plates. In summary, this book captures a beautiful form of vanished mathematics which was artistic/religious in nature. Mr. Fukagawa Hidetoshi and Mr. Rothman are to be congratulated for producing a superb book which tells the story of this vanished mathematical/religious art form. Buy your copy today. This book contains enough history, mathematics, art, and religion to keep one's intellect perplexed for years.

It provides wide range of practical applications, with plain English, colorful pages, step by step from basic to advanced approach. It has got answers at the back. I recommend it`s Instructor`s solutions manual as well...

one of the best trig books in the world. the same goes for algebra and trig (right triangle)by beecher and bittinger:1993 edition.

The strength of this book is its organization. The reader is first introduced to the relation of arc length, radius, and angle. Then degrees and radians are learned. Then the unit circle is introduced. This is one of the best ways to learn trig.

The textbook presents the theory in a clear way that is easy to follow. If you were to read the chapter, you know enough to answer any of the problems. And if you were decided between texts, the layout of the problems of this text would be the reason to choose it. That is because of the science and real world applications of the problems. This is not "plug and chug." It is applying what was learned.

For me this book and the class in which it was used formed the foundation of all my latter math courses. This book has some pre-calculus problems, but that isn't its focus. Calculus has its advantages, but I always found trig to be more visual than most things in calculus. It is easier to picture what is actually going on in the math problem. But if you can relate your newly learned problem solving skills when approaching calculus problems, you will have no trouble.

One of my favorite problems in this book, which was included in the sixth edition on page 281, problem 71, is about an arched doorway. I don't know if the current versions have this problem. However it is worth researching. On my website (see my profile), I discuss this problem. And the excellent problems is what make this the best trig book I've seen.

This is a wonderful book on both hyperbolic geometry **and** spherical geometry--non-Euclidean geometry in general. It's more comprehensive than all of the others. The prerequisites for this book vary greatly from chapter to chapter. If you want to read, and understand, all of the material right away, the prerequisites are somewhat steep. I would study smooth and riemannian manifolds first (I heavily recommend John Lee's two books). I would also get some basic algebraic topology (Hatcher's is a classic). If you have these, it's smooth sailing ahead.

The book is excellent as a reference book and approaches hyperbolic geometry from the Lorentzian viewpoint (which seems to be different from other authors). A great book to have for graduate students studying hyperbolic geometry.

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.

This book conclusively reveals that fourth dimension theories and related spiritual quests constitute, at the very least, a key pillar in the development of modern art, if not its foundation. The book gives the lie to strictly formalist interpretations of the historic impetus for modern art. Perhaps more importantly, it highlights that vast amounts of art history omits the intended metaphysical content of early modern painting. If the record were set straight sooner, one wonders whether the trajectory of art would have led to contemporary art that is more substantive and enriching than it so often is. Artists such as Brice Marden, Agnes Martin, Astrid Colomar, Anne Truitt and others whose work is infused with spiritual depth would be seen as those whose work is most linked to, rather than most divorced from, the artistic heritage of modernism. With even out-of-print paperback copies of this ground-breaking book selling for well over $100, a new printing would be as welcomed as it is deserved.

One should not practice so-called fine arts without reading this book!!! One should be able to fill the b l a n k in his/her understanding of the contemporary relationship between arts & science! Worthy of your valuable time!

i highly recommend this book to anyone interested in turn of the century art and/or concepts of higher space. very insightful on a subject once widespread and now obscure, i.e. the fourth dimension.

I used this text along with Tristan Needham's "Visual Complex Analysis" to get a full dose of the geometric beauty inherent in studying complex variables. I found it to be a nice complement to the second year course in geometry at Cambridge University. Anderson does a wonderful job of working out in detail lots of examples so that you can get the algorithmic practice of solving problems. However this is not merely a cookbook. Rather, core elements of the theory are presented from the ground up, with plenty of time spent on understanding the group structure of Mobius transformations in various settings. Disc and upper-half plane models are treated as well as more general models. I recommend you buy both this book and Needham's if you want to appreciate the world of complex numbers.

This is an excellent introduction to hyperbolic geometry. It assumes knowledge of euclidean geometry, trigonometry, basic complex analysis, basic abstract algebra, and basic point set topology. That material is very well presented, and the exercises shed more light on what is being discussed. Plus, solutions to all the exercises are at the end of the 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.