A Mathematical Space Odyssey: Solid Geometry in the 21st Century (Dolciani Mathematical Expositions, 50) - Hardcover

Alsina, Claudi; Nelsen, Roger

Hardcover

ISBN 10: 0883853582 ISBN 13: 9780883853580

Publisher: American Mathematical Society, 2015

Synopsis

Solid geometry is the traditional name for what we call today the geometry of three-dimensional Euclidean space. This book presents techniques for proving a variety of geometric results in three dimensions. Special attention is given to prisms, pyramids, platonic solids, cones, cylinders and spheres, as well as many new and classical results. A chapter is devoted to each of the following basic techniques for exploring space and proving theorems: enumeration, representation, dissection, plane sections, intersection, iteration, motion, projection, and folding and unfolding. The book includes a selection of Challenges for each chapter with solutions, references and a complete index. The text is aimed at secondary school and college and university teachers as an introduction to solid geometry, as a supplement in problem solving sessions, as enrichment material in a course on proofs and mathematical reasoning, or in a mathematics course for liberal arts students.

About this title

Book Description

This book presents techniques for proving a variety of mathematical results in three-dimensional Euclidean space, the field traditionally known as solid geometry. The text is aimed at secondary school and college and university teachers as an introduction to solid geometry, or in a mathematics course for liberal arts students.

Review

My preference for "keeping things planar" has persisted to this day, and affects the way I do certain things. For the last several years, for example, I have taught a two-semester course at the senior undergraduate level in geometry, and although in those two semesters I teach geometry from a variety of perspectives(analytic and synthetic, Euclidean and non-Euclidean, transformational, etc.) I have never given any real thought to doing solid geometry at all. As a result of looking at this book, though that may change. If I had had access to this text in my formative years, my entire attitude about multidimensional geometry might well be different now.

The book offers a smorgasbord of topics in solid geometry, arranged in chapters (except for the first one, which is introductory) each center on a useful technique: enumeration, representation (of numbers as volumes), dissection, taking plane sections, intersection, iteration, motion, projection and folding/unfolding. Each chapter provides several examples of the technique that is the subject of that chapter; some are quite modern, but others are classical.

The writing seemed to me to be clear, though succinct and efficient. The authors have managed to keep the prerequisites for reading this book to a minimum; certainly a year of calculus is sufficient for just about everything done here, and even a college freshman without any calculus would find almost all of the text comprehensible, since calculus is used in only a handful of circumstances.

The book employs a number of nice pedagogical features. Each chapter contains lots of photographs and drawings; hardly a page goes by without some kind of picture on it. Many of these are drawings that illustrate the mathematics, but there are also a lot of photographs of works of art and architecture around the world, accompanied by useful discussion. In addition, each chapter ends with quite a few (ranging from ten to twenty) exercises, called "challenges"; solutions or hints to all of these are provided in the back of the book, which may enhance its value for self-study but make it less convenient for use as a text. There is a good six-page list of references, including both books and journal articles, including a few that are not in English.

Not many university mathematics departments offer courses on solid geometry, so I don't imagine that this book will find a a lot of use as a text for a regularly offered course. However, I can certainly see this book being used with profit as a text for an honors seminar or as supplemental reading in a geometry course. It's certainly something that anybody interested in geometry would want to take a look at--even if, like me, you didn't think you were all that interested in solid geometry. --Mark Hunacek MAA Reviews

It is unfortunate that the teaching of solid geometry has largely ceased in American high schools and colleges. It would seem to be a natural topic, for it is one of the areas of mathematics where the student can be exposed to understandable diagrams that reinforce the subject matter. The authors lament this loss and by writing this book, are doing something about it.

Although calculus is occasionally used when it is necessary to make the appropriate point, in general a knowledge of high school algebra is all the background needed. The one critical skill is an ability to think and maneuver in three dimensions. A large number of three-dimensional figures are described and analyzed, the reader needs to be able to conceptualize the shapes as well as what it means to pass lines and planes through them.

This is a text where a set of companion manipulatives would make for a class that the mathematically balanced art student could take. Specifically the art student interested in sculpture or architecture, there are several images that reference structures and other practical applications. There have been many times when a student in a math class has come to me with difficulties and has said, "I am a visual learner."

Geometry is almost certainly the oldest area of abstract mathematics, for it grew out of the most practical of operations, interacting with and manipulating the physical world. Solid geometry and the rules defining and describing the objects are used in many areas, from machining custom parts to the construction of massive buildings. This book provides a valuable resource for the study of solid geometry that could be used by teachers all the way from late elementary school through college.

A set of challenge problems are given at the end of each chapter and solutions to all appear at the end. --Charles Ashbacher

A Mathematical Space Odyssey is an excellent and thorough introduction to the basic ideas of solid geometry. An interesting feature is the inclusion of "challenges" instead of exercises at the end of each section, making the book suitable for both classroom us and self-study. The liberal use of diagrams and figures provides good support for the mathematics presented. The book is written in such a way that it is suitable for anyone with an understanding of basic algebra, with plenty of figures to support the mathematics presented.

A Mathematical Space Odyssey goes well beyond the introductory topics of volume and surface area that are typically taught in high school. It includes basic topics from dissection of three-dimensional solids, projective geometry in three dimensions, and nets of solids. It would be an excellent textbook for an introductory college course in solid geometry. The book covers a number of different topics such as plane sections and intersections of solids that would be relevant to individuals that are interested in three-dimensional printing. --Hilary Smith Risser Mathematics Teacher

Edition details

• Publisher: American Mathematical Society
• Publication date: 2015
• Language: English
• ISBN 10: 0883853582
• ISBN 13: 9780883853580
• Binding: Hardcover
• Number of pages: 286