This book explores 20 geometric diagrams that play crucial roles in visualizing mathematical proofs. Alsina and Nelsen examine the mathematics within these diagrams and the mathematics that can be created from them. Those who teach undergraduate mathematics will find their book useful for problem-solving sessions, and will have in their hands enrichment material for courses on proofs and mathematical reasoning.
Each of the book's icons is covered in a separate chapter, illustrating its presence in life, its primary mathematical characteristics, and how it plays essential roles in visual proofs of a wide range of mathematical facts. Included are classical results from plane geometry; properties of the integers; means and inequalities; trigonometric identities; theorems from calculus; and puzzles from recreational mathematics. Every chapter contains selections of challenges that readers can explore for further properties and applications of the icons.
Twenty Key Icons of Mathematics
1. The Bride's Chair
2. Zhou Bi Suan Jing
3. Garfield's Trapezoid
4. The semicircle
5. Similar Figures
7. The Right Triangle 8. Napoleon's Triangles
9. Arcs and Angles
10. Polygons with Circles
11. Two Circles
12. Venn Diagrams
13. Overlapping Figures
14. Yin and Yang
15. Polygonal Lines
16. Star Polygons
17. Self-similar Figures
19. The Rectangular Hyperbola
Solutions to the Challenges References
Mrs. Miniver is a fictional character created by the British author Joyce Maxtone Graham (1901-1953) who wrote columns under the pen name Jan Struther for The Times of London between 1937 and 1939. In a column entitled “A Country House Visit,” she describes an aspect of real-life relationships in mathematical terms: “She saw every relationship as a pair of intersecting circles. The more they intersected, it would seem at first glance, the better the relationship, but this is not so. Beyond a certain point the law of diminishing returns set in, and there aren't enough private resources left on either side to enrich the life that is shared. Probably perfection is reached when the area of the two outer crescents, added together, is exactly equal to that of the leaf-shaped piece in the middle. On paper there must be some neat mathematical formula for arriving at this: in life, none.” What is the solution to Mrs. Miniver's problem, remembering that two circles are rarely equal?
Claudi Alsina (Universitat Politècnica de Catalunya, Barcelona) and Roger B. Nelsen (Lewis and Clark College, Portland) also collaborated on Math Made Visual, When Less Is More: Visualizing Basic Inequalities, and A Journey into Elegant Mathematics.
Icons of Mathematics is #45 in the Dolciani Mathematical Expositions series, and it certainly lives up to the charge of the Dolciani series, providing highly readable discussions of a selection of 20 images that are as recognizable to the general public as they are to mathematicians. Each chapter focuses on one of the icons and includes a standard set of items: quotes from a wide variety of sources (Zhang Zai to Ralph Waldo Emerson), a short history of the figure including examples of uses of the image in mathematical and non-mathematical settings, examples of the figure in mathematical proofs of theorems or solutions to problems, and problems for the reader to solve. Outlines of the solutions to the problems are in the back of the book. Continued...