Theorems and their proofs lie at the center of mathematics. In Charming Proofs, Claudi Alsina and Roger B. Nelsen present proofs involving numbers, geometry, inequalities, functions, plane tilings, origami, polyhedra, and other facets of elementary mathematics. Using surprising arguments or evocative illustrations, the authors offer keen insights into the essence of mathematics. They invite readers to discover and enjoy the beauty of mathematics, while developing the ability to create proofs of their own.
G. H. Hardy observed that, in beautiful proofs, “there is a very high degree of unexpectedness, combined with inevitability and economy. Yuri I. Manin said, "A good proof is one that makes us wise.” Andrew Gleason echoed that sentiment, "Proofs really aren't there to convince you that something is true—they're there to show you why it is true."
This book highlights a collection of remarkable proofs that are elegant, ingenious, and succinct.
1. A Garden of Integers
2. Distinguished Numbers
3. Points in the Plane
4. The Polygonal Playground
5. A Treasury of Triangle Theorems
6. The Enchantment of the Equilateral Triangle
7. The Quadrilaterals' Corner
8. Squares Everywhere
9. Curves Ahead
10. Adventures in Tiling and Coloring
11. Geometry in Three Dimensions
12. Additional Theorems, Problems and Proofs
Solutions to the Challenges
Euclid's Elements contains approximately three dozen propositions concerning properties of triangles, but only about a dozen concerning properties of quadrilaterals, and most of these deal with parallelograms. These statistics belie the richness found in the set of quadrilaterals and its various subsets: cyclic, bicentric, parallelograms, trapezoids, squares, and so on. . . .
Triangles may be acute, right, or obtuse and equilateral, isosceles, or scalene. Similarly, quadrilaterals may be planar or skew (non-planar); planar quadrilaterals may be complex (self-intersecting) or simple (non-self-intersecting); and simple quadrilaterals may be convex (each interior angle less than 180°) or concave (one interior angle more than 180°).
Claudi Alsina (Universitat Politècnica de Catalunya, Barcelona) and Roger B. Nelsen (Lewis & Clark College, Portland, Oregon) also collaborated on Math Made Visual: Creating Images for Understanding Mathematics (2006) and When Less Is More: Visualizing Basic Inequalities (2009), published by the MAA.
As the authors write in the Preface,
"The aim of this book is to present a collection of remarkable proofs in elementary mathematics (numbers, geometry, inequalities, functions, origami, tilings…) that are exceptionally elegant, full of ingenuity, and succinct. By means of surprising argument or a powerful visual representation, we hope the charming proofs in our collection will invite readers to enjoy the beauty of mathematics, to share their discoveries with others, and to become involved in the process of creating new proofs."
I can confidently confirm that the authors have achieved the stated goal admirably. They put together more than 100 mostly elementary mathematics facts — quite curious in their own right — proved by elegant and often surprising arguments. Continued...