Solid geometry is the traditional name for what we call today the geometry of three-dimensional Euclidean space. Courses in solid geometry have largely disappeared from American high schools and colleges. The authors are convinced that a mathematical exploration of three-dimensional geometry merits some attention in today’s curriculum. A Mathematical Space Odyssey: Solid Geometry in the 21st Century is devoted to presenting techniques for proving a variety of mathematical results in three-dimensional space, techniques that may improve one’s ability to think visually.
Special attention is given to the classical icons of solid geometry (prisms, pyramids, platonic solids, cones, cylinders, and spheres) and many new and classical results: Cavalieri’s principle, Commandino’s theorem, de Gua’s theorem, Prince Rupert’s cube, the Menger sponge, the Schwarz lantern, Euler’s rotation theorem, the Loomis-Whitney inequality, Pythagorean theorems in three dimensions, etc. The authors devote a chapter 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. In addition to many figures illustrating theorems and their proofs, a selection of photographs of three-dimensional works of art and architecture are included. Each chapter includes a selection of Challenges for the reader to explore further properties and applications. It concludes with solutions to all the Challenges in the book, references, and a complete index.
Readers should be familiar with high school algebra, plane and analytic geometry, and trigonometry. While brief appearances of calculus do occur, no knowledge of calculus is necessary to enjoy this book.
The authors hope that both secondary school and college and university teachers will use portions of it 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.
5. Plane sections
10. Folding and Unfolding
Solutions to the Challenges
About the Authors
About the Authors
Claudi Alsina was born 30 January 1952 in Barcelona, Spain. He received his BA and PhD in mathematics from the University of Barcelona. His post-doctoral studies were at the University of Massachusetts, Amherst. Claudi, Professor of Mathematics at the Technical University of Catalonia, has developed a wide range of international activities, research papers, publications and hundreds of lectures on mathematics and mathematics education. His latest books include Associative Functions: Triangular Norms and Copulas with M. J. Frank and B. Schweizer, WSP, 2006; Math Made Visual (with Roger Nelsen) MAA, 2006; Vitaminas Mathemáticas and El Club de la Hipotenusa, Ariel, 2008, Geometria para Turistas, Ariel, 2009; When Less is More (with Roger Nelsen) MAA, 2009; Asesinatos Matematicos, Ariel, 2010; Charming Proofs (with Roger Nelsen) MAA, 2010; and Icons of Mathematics (with Roger Nelsen) MAA, 2011.
Roger B. Nelsen was born in Chicago, Illinois. He received his BA in mathematics from DePauw University in 1969. Roger was elected to Phi Beta Kappa and Sigma Xi, and tAught mathematics and statistics at Lewis & Clark College for forty years before his retirement in 2009. His previous books include Proof Without Words, MAA, 1993; An Introduction to Copulas, Springer, 1999 (2nd ed. 2006); Proofs Without Words II, MAA, 2000; Math Made Visual (with Claudi Alsina), MAA, 2006; When Less is More (with Claudi Alsina), MAA, 2009; Charming Proofs (with Claudi Alsina), MAA 2010; The Calculus Collection (with Caren Diefenderfer), MAA, 2010; Icons of Mathematics (with Claudi Alsina), MAA, 2011; and College Calculus (with Michael Boardman), MAA, 2015.