The Four Pillars of Geometry (Undergraduate Texts in Mathematics) [Hardcover]

John Stillwell (Author)

Book Description

Publication Date: August 9, 2005 | ISBN-10: 0387255303 | ISBN-13: 978-0387255309 | Edition: 1

This book is unique in that it looks at geometry from 4 different viewpoints - Euclid-style axioms, linear algebra, projective geometry, and groups and their invariants Approach makes the subject accessible to readers of all mathematical tastes, from the visual to the algebraic Abundantly supplemented with figures and exercises

Editorial Reviews

Review

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)

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).

Product Details

• Hardcover: 241 pages
• Publisher: Springer; 1 edition (August 9, 2005)
• Language: English
• ISBN-10: 0387255303
• ISBN-13: 978-0387255309
• Product Dimensions: 9.3 x 6.3 x 0.7 inches
• Shipping Weight: 1 pounds (View shipping rates and policies)
• Average Customer Review: 4.3 out of 5 stars  See all reviews (6 customer reviews)
• Amazon Best Sellers Rank: #152,661 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

40 of 42 people found the following review helpful:

5.0 out of 5 stars A textbook for that geometry course you wish existed, October 30, 2005

By 

Viktor Blasjo - See all my reviews
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This review is from: The Four Pillars of Geometry (Undergraduate Texts in Mathematics) (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.

11 of 12 people found the following review helpful:

5.0 out of 5 stars Makes the connections, simply, readably, October 25, 2006

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Phillip I. Good (Huntington Beach CA) - See all my reviews
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This review is from: The Four Pillars of Geometry (Undergraduate Texts in Mathematics) (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.

8 of 10 people found the following review helpful:

2.0 out of 5 stars Many Mistakes, September 20, 2010

By 

AmCan - See all my reviews

Amazon Verified Purchase

This review is from: The Four Pillars of Geometry (Undergraduate Texts in Mathematics) (Paperback)

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.