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The Four Pillars of Geometry (Undergraduate Texts in Mathematics) Hardcover – 9 Aug 2005

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From the reviews:

"This is an introductory book on geometry, easy to read, written in an engaging style. The author’s goal is … to increase one’s overall understanding and appreciation of the subject. … Along the way, he presents elegant proofs of well-known theorems … . The advantage of the author’s approach is clear: in a short space he gives a brief introduction to many sides of geometry and includes many beautiful results, each explained from a perspective that makes it easy to understand." (Robin Hartshorne, SIAM Review, Vol. 48 (2), 2006)

"The pillars of the title are … Euclidean construction and axioms, coordinates and vectors, projective geometry, and transformations and non-Euclidean geometry. … The writing style is both student-friendly and deeply informed. The pleasing brevity of the book … makes the book especially suitable as an instruction to geometry for the large and critically important population of undergraduate mathematics majors … . Each chapter concludes with a well-written discussion section that combines history with glances at further results. There is a good selection of thought-provoking exercises." (R. J. Bumcrot, Mathematical Reviews, Issue 2006 e)

"The author acts on the assumption of four approaches to geometry: The axiomatic way, using linear Algebra, projective geometry and transformation groups. … Each of the chapters closes with a discussion giving hints on further aspects and historical remarks. … The book can be recommended to be used in undergraduate courses on geometry … ." (F. Manhart, Internationale Mathematische Nachrichten, Issue 203, 2006)

"Any new mathematics textbook by John Stillwell is worth a serious look. Stillwell is the prolific author of more than half a dozen textbooks … . I would not hesitate to recommend this text to any professor teaching a course in geometry who is more interested in providing a rapid survey of topics rather than an in-depth, semester-long, examination of any particular one." (Mark Hunacek, The Mathematical Gazette, Vol. 91 (521), 2007)

"The title refers to four different approaches to elementary geometry which according to the author only together show this field in all its splendor: via straightedge and compass constructions, linear algebra, projective geometry and transformation groups. … the book can be recommended warmly to undergraduates to get in touch with geometric thinking." (G. Kowol, Monatshefte für Mathematik, Vol. 150 (3), 2007)

"This book presents a tour on various approaches to a notion of geometry and the relationship between these approaches. … The book shows clearly how useful it is to use various tools in a description of basic geometrical questions to find the simplest and the most intuitive arguments for different problems. The book is a very useful source of ideas for high school teachers." (EMS Newsletter, March, 2007)

“The four pillars of geometry approaches geometry in four different ways, devoting two chapters to each, the first chapter being concrete and introductory, the second more abstract. … The content is quite elementary and is based on lectures given by the author at the University of San Francisco in 2004. … The book of Stillwell is a very good first introduction to geometry especially for the axiomatic and the projective point of view.” (Yves Félix, Bulletin of the Belgian Mathematical Society, Vol. 15 (1), 2008)

From the Back Cover

For two millennia the right way to teach geometry was the Euclidean approach, and in many respects, this is still the case. But in the 1950s the cry "Down with triangles!" was heard in France and new geometry books appeared, packed with linear algebra but with no diagrams. Was this the new right approach? Or was the right approach still something else, perhaps transformation groups?

The Four Pillars of Geometry approaches geometry in four different ways, spending two chapters on each. This makes the subject accessible to readers of all mathematical tastes, from the visual to the algebraic. Not only does each approach offer a different view; the combination of viewpoints yields insights not available in most books at this level. For example, it is shown how algebra emerges from projective geometry, and how the hyperbolic plane emerges from the real projective line.

The author begins with Euclid-style construction and axiomatics, then proceeds to linear algebra when it becomes convenient to replace tortuous arguments with simple calculations. Next, he uses projective geometry to explain why objects look the way they do, as well as to explain why geometry is entangled with algebra. And lastly, the author introduces transformation groups---not only to clarify the differences between geometries, but also to exhibit geometries that are unexpectedly the same.

All readers are sure to find something new in this attractive text, which is abundantly supplemented with figures and exercises. This book will be useful for an undergraduate geometry course, a capstone course, or a course aimed at future high school teachers.

John Stillwell is Professor of Mathematics at the University of San Francisco. He is the author of several highly regarded books published by Springer, including Elements of Number Theory (2003), Mathematics and Its History (Second Edition, 2002), Numbers and Geometry (1998) and Elements of Algebra (1994).

Inside This Book

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For over 2000 years, mathematics was almost synonymous with the geometry of Euclid's Elements, a book written around 300 BCE and used in school mathematics instruction until the 20th century. Read the first page
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46 of 49 people found the following review helpful
A textbook for that geometry course you wish existed 30 Oct. 2005
By Viktor Blasjo - Published on
Format: Hardcover
This nice book contains many things that every mathematics student should know (but don't). Chapters 1-2 are on Euclid. The main ideas are picked out very nicely, in welcome contrast to the usual "let ABCD..."-style books. Chapters 3-4 on linear algebra in geometry will probably be skimmed by most readers. This is perhaps just as well, since it consists of little more than the pearls from the usual linear algebra course and since the presentation feels somewhat forced (e.g., things like the inner product are cheerfully defined out of thin air) compared to how naturally the rest of the book flows. Chapters 5-6 on projective geometry are more interesting. Perspective in art sets the stage, but we quickly move to purer things. Stillwell is very fond of the mysterious and fascinating connection with hypercomplex number systems one obtains by building up an arithmetic from within a projective plane. Then in chapter 7 we learn that transformation groups are a clarifying and unifying idea, and we play around with them a bit, somewhat aimlessly it seems, until we finally get to hyperbolic geometry in chapter 8. This chapter crowns the whole book beautifully, showing how projective geometry explodes into hyperbolic geometry through transformation-group thinking. It ends with an excellent short survey of the history of non-Euclidean geometry.
14 of 15 people found the following review helpful
Makes the connections, simply, readably 26 Oct. 2006
By Phillip I. Good - Published on
Format: Hardcover
This text makes the connections simply, readably among various interrelated branches of mathematics including geometry (four of them), trig, group theory, complex variables, and linear algebra. Recommended for every high school mathematics instructor.
15 of 18 people found the following review helpful
Many Mistakes 20 Sept. 2010
By AmCan - Published on
Format: Paperback Verified Purchase
While this book is a cheaper alternative to the hardcover, there are many mistakes in the printing. There are little dots where there should be parentheses, brackets and square root signs in the problem section. It makes the problems hard to understand if you do not know what symbols are supposed to be there. If you are using this for a class, make sure have someone else's book to compare it to. I know it is not just my book because everyone with the soft cover in my class has the same problem.
6 of 7 people found the following review helpful
Poor man's "David Hilbert's Foundations of Geometry" 15 Aug. 2013
By flashgordon - Published on
Format: Hardcover Verified Purchase
Around 1900, David Hilbert published his "Foundation of Geometry." It was the first book to really make Euclid's Elements obsolete. Humans had developed mathematics for thousands of years - from bone markings to the Summerian/Babylonian algebra. They certainly left some record of their mathematical activities. But, for one, their mathematics was arbitrary; it kind of worked. Only with the Greek discovery of deductive reasoning(as far as everybody can tell) did mathematics become firmly established. It's a subtle point. But, a major point is that with Hippocrates of Chios(not the guy from cos who did the Hippocratic oath) discovery of axiomatics which was an advance beyond Thales and Pythagoras's deductive reasoning discovery(not to be systematized till hundreds of years later with Aristotle; Aristotle's only real lasting contribution), and eventually Euclid's effort at axiomatisation, all that knowledge before and Greek mathematical knowledge was systematized, put in a bottle figuratively speaking, and sent out to the world(most people back then never heard of it or saw a copy). Euclid's "Elements", the works of Archimedes, and Appolloniuses Conics were the only real systematic deductive mathematics all the way up to David Hilbert's work.

That last part isn't strictly true. There was some beginning efforts to axiomatise algebra in the 1800s. There's also Ptolemy's work; i'm really not sure how much of an axiomatic effort that book is. The point is that David Hilbert's "Foundations of Geometry" was the first axiomatic effort of geometry to make Euclid's Elements more or less obsolete(see Thomas Heath's translation of Euclid's Elements; in it, Thomas Heath shows the mathematical, and historical significance of just about every theorem in there.

But, this book is more of a easier version. John Stillwell tries to make an easier approach by relating advanced with ancient mathematics. The truth is that advanced mathematics solves problems the ancient or previous mathematics couldn't solve or did so in a less elegant way. Still, John finds modern fresh versions of those ancient theorems like Pythagoras and Thales. As with new deductive theories that are suppose to be able to derive the old theories and deduce new theorems the old couldn't, one certainly should try to make connections between old and new. One must realize that John Stillwell can only fit so much in one book(even with his putting extra stuff sometimes more advanced stuff as exercises). As far as I can tell, nowhere does John Stillwell feel the need to show Archimedes theorems about pie, and all the great Greek mathematics involved and relations to the new modern mathematics. This is just one example. See Thomas Heath "History of Greek Mathematics" not to mention his translation of Euclid's Elements, and Van Der Waerden's "Science Awakening for a proper technical history of Mathematics(certainly more about ancient mathematics).

I'd like to note that John Stillwell does some things in Geometry that I never got into in two geometry courses(one in High school, and then when I went to college after the Navy, they made me take it again). In those two geometry courses, I never proved the Pythagorean theorem once, much less delt with transformations. Transformations overcome a famous logical bottleneck of Euclid's Elements. So yes, John Stillwell's book is like the geometry class you never got to have.

I wouldn't worry about figuring everything out. I'd worry if you can't figure your way through the main text though.

A big point is that John Stillwell tries to show some great connections between ancient mathematics and modern. He tries to show what little one can understand of advanced mathematics. The stress is on geometry of course, but if you read his other books which do similar things, you'll see that his point is that one shouldn't disregard geometry. So, in some ways this book is a good place to start. He has other geometry books, but I wouldn't get into them till you get through the number theory/abstract algebra, and at least one semester of calculus. Overall, John Stillwell has succeeded in showing people that they too can learn mathematics. I would read the majority of these books before reading his "Mathematics and its History" as well. He's able to show some more stuff there, but his accounts of abstract algebra and number theory are even sketchier there.
4 of 4 people found the following review helpful
The Four Pillars of Geometry by John Stillwell 21 Feb. 2011
By Victor Hughes - Published on
Format: Hardcover Verified Purchase
I found this text to be of excellent quality. Dr. Stillwell's writing is easy to read, yet very informative. He creates many examples that helps in uderstaning his main thesis. I found his reason why 'The Slope is consistent throughout a line' by using similar traingles quite interesting. I highly recommend this book for all who enjoy mathematics, regardless of your degree or interest. I am anxious to read and study more of his books.
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