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Geometry at Work: Papers in Applied Geometry

Catherine A. Gorini, editor
Mathematical Association of America
Publication Date: 
Number of Pages: 
MAA Notes 53
[Reviewed by
Randall J. Swift
, on

Geometry is the mathematical study of shape and form. Its ancient origins are found in such practical applications as surveying and architecture, while today, geometry is applied to such diverse areas as DNA analysis, quantum physics, and decision theory.

Geometry at Work is a collection of highly readable papers in applied geometry. The papers are organized according to area of application. The collection begins with a very interesting and well-written introduction on the nature of applications of knowledge. This philosophical discussion provides a framework and sets the tone for this broad collection of papers.

The first part of the collection contains papers on Art and Architecture, where several nice articles on the geometry of architectural ornamentation can be found. The paper by P. Calter on Trigonometry and Façade Measurement has several nice elementary applications of trigonometry, which can be used to supplement a course in trigonometry.

The collection then proceeds with some interesting historical work on the geometry of the Vedic civilization. David Henderson's work on square roots in the Sulbasutras is very enlightening for its historical context.

Part 3 of the collection consists of classroom applications of geometry. There one finds an interesting paper on Ethnomathematics by M. Ascher, as well as a nice paper on some applications of descriptive geometry by M. Pokrovskaya.

The papers on the applications of geometry and engineering in Part 4 are outstanding. Especially notable is the paper by R. Shahidi on the geometry of magnetic resonance imaging (MRI) and computed tomography (CT). The detailed description of these imaging methods and how geometry is applied to provide surgeons with "surgical" coordinates is clear and readable and is simply fascinating.

The paper by B. Mishra on the geometry of robotic hands and the use of convexity in the theory of grasping is a very detailed introduction to the subject.

Part 5 of the collection consists of papers that apply geometry to decision-making processes. Donald Saari's paper on voting methods provides a readable introduction to this important branch of applied social science.

The final part of the collection has papers on the applications of geometry in mathematics and science. Here you will find articles on convexity, Penrose tiling and statistical symmetry. As well as quantum physics and topology, graph theory and combinatorial optimization. These are all excellent, well-written papers on these applications of geometry; however, these papers are of a distinctly higher mathematical level than the previous parts of the collection. None the less, they round out and complement the collection nicely.

According to the introduction of the collection, the intention of Geometry at Work is to provide a resource for students and teachers of geometry who are interested in the practical applications of the subject. Each of the papers in the collection is well written and has been carefully edited. From the above discussion, one has the sense that the collection is very broad and should indeed appeal to students of varying backgrounds, as well as teachers of geometry. It could be profitably used as a supplement to geometry courses at the high school through undergraduate levels. Many of the papers could be assigned as student projects and some of the articles could also be used to supplement courses in applied mathematics.

The text begins with a quote of N.I. Lobachevsky: "There is no branch of mathematics, however abstract, that will not eventually be applied to the phenomena of the real world." Lobachevsky is right and Geometry at Work succeeds in its aim of showing the richness and beauty of the applications of geometry.

After nine years in the department of mathematics at Western Kentucky University, Randall Swift will return to his home state of California, where this fall he will be an associate professor of mathematics at California State Polytechnic University, Pomona.

His research interests include nonstationary stochastic processes, probability theory and mathematical modeling. He is a co-author of the MAA text A Course in Mathematical Modeling.

His non-mathematical interests are mainly focused on his wife and three young daughters, but, when he has the time, he enjoys collecting R&L cereal premiums, science fiction, listening to public radio, classical rock and the Blues, cooking and baseball.