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This is a wonderful book. It isn't for mathematical beginners, but it isn't opaque either. It requires a student to think, experiment and to learn by puzzling things out in one's mind rather than simple memorization and regurgitation. Nor does it follow the all too common modern method of over simplifying things to allow people to pretend they have learned math while only dabbling in a few basic topics.
This book is amply illustrated with many exercises (answers are provided at the back for all the exercises). The book also has some humor and wit with the quotes it distributes throughout the book to help liven things up.
There is also a list of helpful references and an index. When reading the book, don't be afraid of going to a dictionary or the web or some other math books for clarification of some terms or more basic concepts. It is essential to have everything clear in your mind before moving on or you will stumble. As in all math, it is like a building with the next stage being built on the present one which is built on the previous one. You can't skip steps very successfully very often.
This is a great volume to have in your library, but even better to work through.
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on July 22, 2000
A better title for this book would be "Advanced Topics in Geometry". The chapters are pretty much self-contained.
This book presumes a thorough, rigorous knowledge of high school geometry such as you might get in a college geometry course designed for future teachers along with considerable mathematical maturity.
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on June 27, 2000
This is Coxeter best book. Introduction to Geometry covers a wide range of topics and is the first book that I will look at for any geometry topic. It is now a little dated but only in the topics that it does not cover. Like all of Coxeter works each topic is clear and to the point. If you only buy one book on geometry this is it.
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on August 23, 2009
This is the best book I've seen covering geometry at this level. Coxeter was known as an apostle of visualization in geometry; many other books that cover this material just give you page after page of symbols with no diagrams. He motivates all the topics well, and lays out the big picture for the reader rather than just presenting a compendium of facts. This is a survey of a huge field, but he does a great job of focusing on the most important results. As other reviewers have noted, this book is not "introductory" in the sense of high school geometry; it's introductory in the sense of being the kind of book a college math major would use in his/her first upper-division geometry course. It doesn't presuppose a great deal of mathematical knowledge, but it probably isn't a book that one could appreciate without having already developed quite a high level of mathematical maturity.
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on March 10, 2012
This is one of those books that's listed in the bibliography of almost every other geometry text I've read -- and rightly so. Reading through it, you'll find some absolute gems of geometric insight. So why am I giving it only three stars? Primarily because it misrepresents itself as an "Introduction," which it isn't. It's much more like one of those fast-paced "ten countries in five days" package tours offerred by various travel agents. In a mere 412 pages, Coxeter zips through a vast number of topics -- each of them actually a specialty area in the larger field of geometry. It simply isn't possible for a book of this length to give the reader any kind of serious grounding in this material.

In addition, some of the topics are ones at which Coxeter himself admitted he wasn't very skilled. During his career, his main areas of interest were symmetry, n-dimensional Euclidean geometry, projective geometry, and higher-dimensional polygons. Things like topology and differential geometry were outside his territory, so the treatment of these topics in "Introduction" is not as engaging as his discussion of various isometries.

This book originally grew out of a set of lectures that Coxeter gave to college-level math majors and math teachers. By all accounts, Coxeter was a very lively and engaging teacher; I imagine it must have been wonderful to listen to those lectures, and then have Coxeter's own lecture notes (i.e., this book) as a reminder of everything that he said. Unfortunately, I don't think the book stands as well on its own as a teacher; it needs Coxeter himself to fill in the gaps between the words and bring it to life.

So if you already know this material, and you just want to discover some wonderful Coxeterian pearls of wisdom about the subject, then go for it; pay the hundred dollars. As your knowledge deepens, you'll always be able to return to this book and find some new insight that you missed on a previous reading. There's also a wonderful "visual" quality to the way Coxeter thinks about geometry -- something that's missing from many other texts.

On the other hand, if your goal is to learn the material covered in this book, you'll need other books to do it. This is definitely not a good choice for a first exposure to the subject.
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on May 31, 2014
Euclid said "there is no royal road to learning geometry". However geometry for the boomer generation has been easier to learn because of the classic H.S.M. Coxeter "Introduction to Geometry". This 1980 final edition simply perfects the 1961 first edition that helped me start my career in computer graphics. The minor typographical errors have been fixed. One such defect in the first edition for equation 18.21 had stopped my progress in tensor notation for weeks before I could confidently mark it as a printing error. Almost all the exercise answers have been improved. And the four color map proof is mentioned. Returning to Coxeter now in 2014 I see it as a practical review for the serious physics enthusiast to get from the geometry of Euclid to that of the Einstein Field Equations. An alternate more modern and more difficult book covering this mathematics would be Penrose's "Road to Reality" first published in 2004. Everyone must find their own road through geometry, this book might help you.
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on April 14, 2011
The only reason I give it a 4/5 is because the diagrams need to be labeled. It's pretty hard to keep up with the conversation when the author refers to a poorly labeled complex diagram. I think Coxeter's other book does a great job and you can download it for free or buy it for under $10. It is a great introduction to college level geometry; and introduction because it doesn't really go into too much depth, but is not shallow either. I would recommend it, but buy it used because it's just not one of those books you'll constantly look back at for help in the future...it's pretty much a one read and that's it.
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on July 9, 2009
H.S.M. Coxeter is a legend in Geometry, primarily for his work on higher dimensional objects...polytopes, and the Physics applicable, study of Symmetry. The book covers the gamut of Geometry, touching on the peaks and covering the entire range. No book could cover the entire subject, in one volume. This is an "Introduction" for someone who has already made peace with Mathematics....its not an elementary text. The proofs are rigorous and many details are not presented as the volume was written for university students. I reccomend it for true Math Philes. Dr.Pratt.
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on May 26, 2010
And not really to modern geometry, but rather to Coxeter-style geometry. I studied with Coxeter as an undergraduate (he was a very good teacher), and am a professional geometer, but I have never liked this book. Unfortunately, there is no good "Introduction to Geometry", and hence, 2500 years after Archimedes, still no royal road to it. I would very much recommend Thurston's notes (or his book, which is a little easier going, but has a lot less content), or (on a more basic level) Geometric Transformations by Yaglom.
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on January 20, 2013
As usual, Coxeter shows that he is the master of the revival of geometry !

Look at the table of contents to see how "rich" this book is... while every subject is treated in a wonderfully comprehensive way.

This book is part of Coxeter's geometry SUM : Introduction to Geometry, The real Projective Plane, Projective Geometry, Geometry Revisited, Non-Euclidean Geometry... to be included in the collection of anyone interested in mathematics.
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