Charming Proofs

Charming Proofs: A Journey into Elegant Mathematics

Claudi Alsina
Roger B. Nelsen
Volume: 42
Copyright Date: 2010
Edition: 1
Pages: 320
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  • Book Info
    Charming Proofs
    Book Description:

    Theorems and their proofs lie at the heart of mathematics. In speaking of the purely aesthetic qualities of theorems and proofs, G. H. Hardy wrote that in beautiful proofs 'there is a very high degree of unexpectedness, combined with inevitability and economy.' Charming Proofs present a collection of remarkable proofs in elementary mathematics that are exceptionally elegant, full of ingenuity, and succinct. By means of a surprising argument or a powerful visual representation, the proofs in this 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. Charming Proofs is organized as follows. Following a short introduction about proofs and the process of creating proofs, the authors present, in twelve chapters, a wide and varied selection of proofs they consider charming, Topics include the integers, selected real numbers, points in the plane, triangles, squares, and other polygons, curves, inequalities, plane tilings, origami, colorful proofs, three-dimensional geometry, etc. At the end of each chapter are some challenges that will draw the reader into the process of creating charming proofs. There are over 130 such challenges. Charming Proofs concludes with solutions to all of the challenges, references, and a complete index. As in the authors’ previous books with the MAA (Math Made Visual and When Less Is More), secondary school and college and university teachers may wish to use some of the charming proofs in their classrooms to introduce their students to mathematical elegance. Some may wish to use the book as a supplement in an introductory course on proofs, mathematical reasoning, or problem solving.

    eISBN: 978-1-61444-201-1
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-viii)
  2. Preface
    (pp. ix-xii)
    Claudi Alsina and Roger B. Nelsen
  3. Table of Contents
    (pp. xiii-xviii)
  4. Introduction
    (pp. xix-xxiv)

    This is a book about proofs, focusing on attractive proofs we refer to ascharming. While this is not a definition, we can say that a proof is an argument to convince the reader that a mathematical statement must be true. Beyond mere convincing we hope that many of the proofs in this book will also be fascinating.

    Proofs: The heart of mathematics

    An elegantly executed proof is a poem in all but the form in which it is written.

    Morris Kline

    Mathematics in Western Culture

    As we claimed in the Preface, proofs lie at the heart of mathematics. But...

  5. CHAPTER 1 A Garden of Integers
    (pp. 1-18)

    The positive integers are the numbers used for counting, and their use as such dates back to the dawn of civilization. No one knows who first became aware of the abstract concept of, say, “seven,” that applies to seven goats, seven trees, seven nights, or any set of seven objects. The counting numbers, along with their negatives and zero, constitute the integers and lie at the heart of mathematics. Thus it is appropriate that we begin with some theorems and proofs about them.

    In this chapter we present a variety of results about the integers. Many concern special subsets of...

  6. CHAPTER 2 Distinguished Numbers
    (pp. 19-38)

    Numbers are not only beautiful; some are popular, so popular that they have had their biographies written. Here is a short list of number biographies published since 1994:

    e: The Story of a Number[Maor, 1994];

    The Joy of π[Blatner, 1997];

    The Golden Ratio and Fibonacci Numbers[Dunlap, 1997];

    The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number[Livio, 2002];

    An Imaginary Tale: The Story of i[Nahin, 1998];

    Gamma: Exploring Euler’s Constant[Havil, 2003]; and

    The Square Root of Two[Flannery, 2006].

    In this chapter we prove some basic results about some special numbers such...

  7. CHAPTER 3 Points in the Plane
    (pp. 39-50)

    In this chapter we present some intriguing results, and their delightful proofs, about some of the simplest geometric configurations in the plane. These include figures consisting solely of points and lines, including those constructed from the lattice points in the plane. We will deal with structures such as triangles, quadrilaterals, and circles in later chapters.

    Pick’s theorem is admired for its elegance and its simplicity; it is a gem of elementary geometry. Although it was first published in 1899, it did not attract much attention until seventy years later when Hugo Steinhaus included it in the first edition of his...

  8. CHAPTER 4 The Polygonal Playground
    (pp. 51-70)

    Polygons, along with lines and circles, constitute the earliest collection of geometric figures studied by humans. Polygons, along with their associated star polygons, appear frequently throughout history in symbolic roles, often in religious and mystical settings. Book IV of theElementsof Euclid deals exclusively with the construction of certain regular polygons (triangles, squares, pentagons, hexagons and 15-gons) and how to inscribe and circumscribe them in and about circles.

    In this chapter we consider some remarkable results and their proofs that apply to general polygons. In subsequent chapters we study particular polygons such as triangles, squares, general quadrilaterals, polygons that...

  9. CHAPTER 5 A Treasury of Triangle Theorems
    (pp. 71-90)

    Book I of theElementsof Euclid is devoted to theorems about parallel lines, area, and triangles. Of the 48 propositions in this book, 23 concern the triangle. Proposition 47 in Book I is perhaps the best-known theorem in mathematics, the Pythagorean theorem. Consequently one can justifiably say that triangles lie at the very core of geometry.

    We begin this chapter with several proofs of the Pythagorean theorem, followed by some related results, including Pappus’ generalization of the Pythagorean theorem. Consideration of the inscribed and circumscribed circles for general triangles leads to Heron’s formula and Euler’s inequality. We conclude this...

  10. CHAPTER 6 The Enchantment of the Equilateral Triangle
    (pp. 91-106)

    Equilateral triangles lie at the heart of plane geometry. In fact Euclid’s first proposition—Proposition 1 in Book I of theElements—reads [Joyce, 1996]:To construct an equilateral triangle on a given finite straight line.Equilateral triangles continue to fascinate professional and amateur mathematicians. Many of the theorems about equilateral triangles are striking in their beauty and simplicity.

    Mathematicians strive to find beautiful proofs for beautiful theorems. In this chapter we present a small selection of theorems about equilateral triangles and their proofs.

    The Pythagorean theorem is usually illustrated with squares on the legs and hypotenuse of the triangle,...

  11. CHAPTER 7 The Quadrilaterals′ Corner
    (pp. 107-120)

    Euclid’sElementscontains approximately three dozen propositions concerning properties of triangles, but only about a dozen concerning properties of quadrilaterals, and most of those 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. In this chapter we discuss some intriguing results and their proofs concerning general quadrilaterals as well as some of the special cases just mentioned.

    Triangles may be acute, right, or obtuse and equilateral, isosceles, or scalene. Similarly, quadrilaterals may beplanarorskew(non-planar); planar quadrilaterals may becomplex(self-intersecting)...

  12. CHAPTER 8 Squares Everywhere
    (pp. 121-136)

    Squares have a special place in the world of quadrilaterals, just as equilateral triangles have a special place among all the triangles. We devote this chapter to theorems about squares, both in the geometric and number-theoretic sense. The two are closely related, as you read in Section 3.2 concerning the representation of an integer as the sum of two squares and will see again in Sections 8.2 and 8.3.

    We present our theorems about squares according to the number of squares in the theorem. For example, the Pythagorean theorem can be thought of as a three-square theorem.

    The golden ratio...

  13. CHAPTER 9 Curves Ahead
    (pp. 137-158)

    Many mathematical curves have intriguing properties. In visiting the world of curves, one enjoys three complementary views: some curves arise as geometrical shapes appearing in nature, others come from the observation of dynamic phenomena, and a wide range of curves result from mathematical ingenuity [Wells, 1991].

    Our aim in this chapter is to present a selection of attractive proofs related to various extraordinary properties of some curves. We begin with some theorems about lunes, a shape that appears in nature as phases of the moon.

    As we noted in the previous chapter, the ancient Greeks were concerned with the notion...

  14. CHAPTER 10 Adventures in Tiling and Coloring
    (pp. 159-190)

    There are many lovely proofs in which thetilingof regions and thecoloringof objects appear. We have seen examples in earlier chapters, such as tiling in the proof of Napoleon’s theorem in Section 6.5, and coloring in the proof of the art gallery theorem in Section 4.6. Tiles enable us to compare areas without calculation, and color enables us to distinguish relevant parts of figures easily.

    In this chapter we continue our exploration of elegant proofs that employ these techniques. We begin with a brief survey of basic properties of tilings, followed by some results using tilings with...

  15. CHAPTER 11 Geometry in Three Dimensions
    (pp. 191-208)

    The geometry of three-dimensional Euclidean space is sometimes called solid geometry, since it has traditionally dealt with solids such as spheres, cylinders, cones, and polyhedra, as well as lines and planes in space. Three-dimensional space is the space we live in, and it is the setting for some lovely theorems and charming proofs.

    In this chapter we present three different types of theorems and their proofs. We first consider three-dimensional versions of some two-dimensional theorems (such as the Pythagorean theorem). Second, we look at two-dimensional theorems whose proofs are surprisingly simple when the theorem and its proof are viewed from...

  16. CHAPTER 12 Additional Theorems, Problems, and Proofs
    (pp. 209-238)

    In our final chapter, we present a collection of theorems and problems from various branches of mathematics and their proofs and solutions. We begin by discussing some set theoretic results concerning infinite sets, including the Cantor-Schröder-Bernstein theorem. In the next two sections we present proofs of the Cauchy-Schwarz inequality and the AM-GM inequality for sets of sizen. We then use origami to solve the classical problems of trisecting angles and doubling cubes, followed by a proof that the Peaucellier-Lipkin linkage draws a straight line. We then look at several gems from the theory of functional equations and inequalities. In...

  17. Solutions to the Challenges
    (pp. 239-274)
  18. References
    (pp. 275-288)
  19. Index
    (pp. 289-294)
  20. About the Authors
    (pp. 295-295)