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Geometry and Its Applications

Walter Meyer
Academic Press
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
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OK, I'll admit it. I didn't think I was going to like this book. But it surprised me. It is, in my opinion, just the sort of thing its intended audience needs, and quite well executed.

Most American mathematics departments offer a regular course in geometry, usually aimed mostly at future teachers. Given that students now arrive in college with very little geometrical knowledge, these courses have settled on a fairly standard pattern. First, one does a little synthetic geometry, following Euclid as modified by Hilbert, in more or less detail and at varying levels of rigor. Next comes some (still synthetic) non-Euclidean geometry, usually very lightly done. At that point, coordinates, vectors, and transformations can come in, which creates the opportunity to introduce various other kinds of geometry (especially projective) and/or to spend some time considering symmetries of the plane and related topics (Meyer does the latter). From there on, one is free to consider special topics; Meyer chooses to do a little bit of the theory of polyhedra.

All this is fairly standard, as is the provision, made through a web site, of software support (in this case, using Geometer's Sketchpad). What makes Meyer's book stand out are two things. First, he puts to good use his experience in industry (at Grumman Corporation, where he ran a robotics research program) in order to present applications that, while usually simple, seem real. This includes some fairly important (and non-classical) material, such as a discussion of Voronoi digarams.

The second is harder to pin down; I'd describe it as the book's "voice": a humane, intelligent, reflective way of discussing things that is quite interesting to read. Read his discussion of what axioms are, early in the first chapter, to see what I mean. If we can get students to read the book and think about what they read, they'll learn a lot from this book.

So: this may look fairly traditional (especially from the outside), but it's actually quite creative and very well done. Anyone teaching this kind of geometry course should consider adopting this book.

Fernando Q. Gouvêa is professor of mathematics at Colby College.

 1. The Axiomatic Method in Geometry
2. The Euclidean Heritage
3. Non-Euclidean Geometry
4. Transformation Geometry I: Isometries and Symmetries
5. Vectors in Geometry
6. Transformation Geometry II: Isometries and Matrices
7. Transformation Geometry III: Similarity, Inversion and Projection
8. Graphs, Maps and Polyhedra
Answers to Odd-Numbered Exercises