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.
Customer Reviews:
Many beautiful things.......2007-01-12
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.
Don't expect to find it "easy.".......2006-12-24
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.
Beautiful, Rewarding, and Deep........2003-07-21
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.......2001-11-09
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.......2000-10-20
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.
Customer Reviews:
Thanks!.......2007-09-23
Thank you for the book. It came on time and the condition of the book was very good, which the sender had promised.
A wealth of knowledge of geometry.......2007-08-09
This text provides a wealth of knowledge about geometry. For me, with only a minimum of college level geometry previously studied, it was my first meeting with a rigorous development of any type of geometry, euclidean or noneuclidean. It was very exciting to see how this subject can be so carefully developed. Even though I was exposed to a meticulous construction of real analysis and algebra ,there is quite a difference in the techniques used to develop geometry, which you might anticipate.
Each time I have reviewed Dr.Greenberg's text, I am not only able to retain the material easier, but also to achieve a new level of understanding, which is kind of surprising.
This text is a treasure of knowledge of geometry, but the reader, if not much better prepared than me, needs to understand that digesting this text requires a bit of a committment , but it is well worth the effort. If you are a prior football player, like me, you will probably remember the coach mentioning it will take a 110% effort to win. This is a different way of indicating how tenacious, I feel, you will need to be.
I am really looking forward to reading Dr. Greenberg's most recent edition of this text, which is now available.
Hard to get into without a math professor on hand.......2007-04-05
First of all, I must point out that i am reviewing the second edition of this book. I'm sure the third edition is different, but i think the main points of my review will still hold.
I bought this book because i needed to brush up on my geometry for the California Subject Examination for Teachers (CSET) in mathematics. While it is certainly a well written book (I found the historical aspects of it particularly interesting), its major flaw is that there are no answers to the end-of-chapter excercises! This makes the book virtually useless to anyone not in school wanting to learn geometry in their own time (i.e. not for a class). Whilst i managed to do most of the exercises at the end of the first chapter (at least i think i did), it seemed pointless to attempt subsequent problems as they were quite in depth and there would be no way for me to know whether they were right or not! A big improvement would be if the number of problems were cut down (seriously, it would take years for someone to finish all of the end-of-chapter problems!) and something resembling answers was in the back of the book.
Excellent Condition.......2006-02-21
I received the textbook within just a few days after placing my order online. The book came new, as was promised. The promptness of the delivery and the quality of the product would definitely persuade me to buy from this seller again.
Excellent Book.......2005-12-24
This book is written like a mystery, and I thoroughly enjoyed the way it led me into an understanding of non-Euclidean geometry. It builds the foundation - neutral geometry, while keeping you into suspense as to whether the parallel postulate can be proved. It includes just enough history of the mathematicians who spent their lives trying to prove the parallel postulate, with excellent referencing for further study. I hate to give away the high point of the mystery, but it has to do with the parallel postulate being independent of neutral geometry! (Read the book if you don't realize the significance of that!) The book then goes into detail on hyperbolic geometric models, such as those of Poincare and Klein. The referencing is complete and thorough. It is just a well written book, as fun to read as a math book can ever be. A classic. I highly recommend it for students and anyone interested in geometry.
Book Description
Exposition of 4th dimension, concepts of relativity as Flatland characters continue adventures. Popular, easily followed yet accurate, profound. Topics include curved space time as a higher dimension, special relativity, and shape of space-time. Accessible to lay readers but also of interest to specialists. Includes 141 illustrations.
Customer Reviews:
With few exceptions, it is a readable, stepwise explanation of how the universe is structured.......2007-06-29
To understand relativity, it is necessary to understand geometry, specifically how a straight line can be curved. For nearly everyone, any attempt to understand four-dimensional space begins with understanding how a three-dimensional creature would appear to a two-dimensional one. One of the earliest and still the greatest of all introductions to going up a dimension is "Flatland" by Edwin A. Abbott. Quite naturally and sensibly, Rucker starts with Abbott's rendition of the properties of Flatland.
Rucker then moves on to the idea of curved space, where the shortest distance between two points is a "straight line", which is curved by the properties of the space. The space that we occupy is curved by the presence of matter, as Einstein claimed in his relativity theories. Furthermore, movement causes shrinkage in the direction of the movement and the slowing of time, which causes time to become just another dimension of space. As counterintuitive as this may appear, Einstein's relativity theory has been verified over and over again to a large number of significant figures.
One of the best things about this book is that Rucker has included problems at the end of each chapter. These problems reinforce the concepts of the chapter; it is unfortunate that no solutions were included.
In this book, Rucker steps the reader through all of the background material necessary to understand relativity and four-dimensional space. With few exceptions, the accounts are understandable to anyone with an understanding of college algebra.
The best book ever in its field.......2007-04-19
This book has presented the most difficult topics of our world with the easiest words. After reading this book many of my questions that I had in my mind for a long time were answered. It's worth thousands more than its price.
Congratulation to Mr. Rudolf Rucker for his great book.
explain dimensions very well.......2007-03-31
it is published years before but it is almost new for today and it explain dimensions and shape of space well and clearly .thanx to amazon for sending me timely.
See what's outside the box.......2007-03-30
Over two millenia ago, Euclid wrote his masterpiece Elements and stated in his fifth postulate that only one perpendicular line could pass through any one point adjacent to another line.
One hundred fifty years ago, it was proven that yet another geometry could be described by asserting that more than one parallel line could pass through such a point.
Building on these ideas, Rucker briefly yet thoroughly surveys the relevant mathematics outside the box of Euclidian geometry.
It's a fascinating place too because it involves considerations of hyperspace, four dimensional travels and ultimately Einstein's theory of relativity.
Copiously filled with illustrations to help drive home his points, Rucker has produced a book that meaningful helps one visualize and better understand the fourth dimension.
This book is an excellent read along with Choas, Coincidences and All that Math Jazz, The Fourth Dimension Simply Explained, Einstein's own Relativity and Hyperspace by Michio Kaku which discusses all these ideas as well as contemporary string theory (which purports to pull it all together).
excellent book, fascinating author, start your exploration! :).......2005-12-31
Mr. Rucker is a 'genius educator' in my opinion :) he can open your mind and get you started - no matter what direction you wanna take! :) whether it be philosophy, math, physics, or even spirtual things - Mr. Rucker can get you going! to me, he is one of the great men of these modern times :) ah, i remember! pay particular attention to visualizing hyper-dimensional objects .. it can be done! good luck and may god bless all of you! :)
Book 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.
Customer Reviews:
Wonderfully clear and complete.......2006-06-26
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.
Average customer rating:
- The beauty of geometry is captured
|
Non-Euclidean Geometry (Spectrum)
H. S. M. Coxeter
Manufacturer: The Mathematical Association of America
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ASIN: 0883855224 |
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.
Customer Reviews:
The beauty of geometry is captured.......2000-02-05
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.
Book Description
In his classic work of geometry, Euclid focused on the properties of flat surfaces. In the age of exploration, mapmakers such as Mercator had to concern themselves with the properties of spherical surfaces. The study of curved surfaces, or non-Euclidean geometry, flowered in the late nineteenth century, as mathematicians such as Riemann increasingly questioned Euclid's parallel postulate, and by relaxing this constraint derived a wealth of new results. These seemingly abstract properties found immediate application in physics upon Einstein's introduction of the general theory of relativity.
In this book, Eisenhart succinctly surveys the key concepts of Riemannian geometry, addressing mathematicians and theoretical physicists alike.
Customer Reviews:
The best classical-style exposition of Riemannian Geometry........1998-11-03
I bought the Russian translation of this book in 1954 and found that this is the best source of the Riemannian geometry, not only for a beginner (as I was at that time), but also for every specialist. Some items fully discussed there by L.P. Eisenhart were even rediscovered decades later --- and published another time as new results. This book is, of course, written in the old good traditional style, one will not find here, e.g., Cartan's forms, but it is really an everlasting treasure. Look also for the Continuous Groups of Transformations by the same author.
Book Description
Fascinating, accessible introduction to unusual mathematical system in which distance is not measured by straight lines. Illustrated. Index.
Customer Reviews:
Excellent for high school teachers and students.......2002-06-19
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.......2001-07-15
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.......2000-07-18
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........2000-07-07
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.......2000-07-05
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.
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Complex Hyperbolic Geometry (Oxford Mathematical Monographs)
William M. Goldman
Manufacturer: Oxford University Press, USA
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ASIN: 019853793X |
Book Description
Complex hyperbolic geometry is a particularly rich area of study, enhanced by the confluence of several areas of research including Riemannian geometry, complex analysis, symplectic and contact geometry, Lie group theory, and harmonic analysis. The boundary of complex hyperbolic geometry, known as spherical CR or Heisenberg geometry, is equally rich, and although there exist accounts of analysis in such spaces there is currently no account of their geometry. This book redresses the balance and provides an overview of the geometry of both the complex hyperbolic space and its boundary. Motivated by applications of the theory to geometric structures, moduli spaces and discrete groups, it is designed to provide an introduction to this fascinating and important area and invite further research and development.
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. "Charming and exciting." — Saturday Review of Literature. 169 figures.
Customer Reviews:
Math Puzzles That Challenges the Brain.......2006-11-10
I am thoroughly enjoying the challenges to my brain that I am finding in this book. it was well worth the price I paid
Enjoyable.......2006-11-06
If you like mathematics and how numbers and formulas work, it's worth having a look.
Great for high schoolers with interest.......2003-12-06
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.......2001-11-05
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.......2001-06-13
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.
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