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Icons of Mathematics

Icons of Mathematics: An Exploration of Twenty Key Images

Claudi Alsina
Roger B. Nelsen
Volume: 45
Copyright Date: 2011
Edition: 1
Pages: 347
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  • Book Info
    Icons of Mathematics
    Book Description:

    Icons of mathematics are certain geometric diagrams that play a crucial role in visualizing mathematical proofs, and in the book the authors present 20 of them and explore the mathematics that lies within and that can be created. The authors devote a chapter to each icon, illustrating its presence in real life, its primary mathematical characteristics and how it plays a central role in visual proofs of a wide range of mathematical facts. Among these are classical results from plane geometry, properties of the integers, means and inequalities, trigonometric identities, theorems from calculus, and puzzles from recreational mathematics.

    eISBN: 978-0-88385-986-5
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-viii)
  2. Preface
    (pp. ix-x)
    Claudi Alsina and Roger B. Nelsen
  3. Twenty Key Icons of Mathematics
    (pp. xi-xii)
  4. Table of Contents
    (pp. xiii-xviii)
  5. CHAPTER 1 The Bride′s Chair
    (pp. 1-14)

    Perhaps the best-known theorem in mathematics is thePythagorean theorem, Proposition 47 in Book I of theElementsof Euclid (circa 300 bce), which states:In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle. Perhaps one of the most recognizable images in mathematics is the figure that often accompanies the Pythagorean theorem, an icon variously known as thebride’s chair, thepeacock’s tail, thewindmill, and theFranciscan’s cowl. In Figure 1.1 we see the bride’s chair in the bookLos Seis Libros Primeros...

  6. CHAPTER 2 Zhou Bi Suan Jing
    (pp. 15-20)

    TheZhou bi suan jing(周髀算经), “The Arithmetical Classic of the Gnomon and Circular Paths of Heaven,” is a Chinese text dating from the Zhou dynasty (1046–256 bce). Although primarily an astronomy text, it also discusses right triangle geometry. The image in Figure 2.1, called thehsuan-thuin Chinese, appears in this text; however, we will call this icon theZhou bi suan jing, the title of the text in which it appears.

    We begin by examining theZhou bi suan jingproof of the Pythagorean theorem, and then generalize the icon to a rectangular form to prove results...

  7. CHAPTER 3 Garfield′s Trapezoid
    (pp. 21-28)

    In 1876 a new proof of the Pythagorean theorem appeared in theNew England Journal of Education(volume 3, page 161). The author of this proof was James Abram Garfield (1831–1881) of Ohio, a member of the United States House of Representatives. The proof was unusual in that it employed a trapezoid constructed from right triangles, an icon we callGarfield’s trapezoid. In 1880 Garfield was elected the twentieth President of the United States, only to be assassinated four months after his inauguration. He was the last American president to have been born in a log cabin. For more...

  8. CHAPTER 4 The Semicircle
    (pp. 29-44)

    The semicircle has long played an important role in architecture and art. The Romans employed semicircular arches in their constructions such as the Pont du Gard aqueduct near Nîmes, France (see Figure 4.1a). Similar arches can be found in Romanesque architecture throughout Europe. The Moors also used semicircular arches in their most impressive buildings, such as in the interior of the Mezquita mosque in Córdoba, Spain (see Figure 4.1b). Semicircles also appear in paintings of modern artists such as Wassily Kandinsky (Semicircle, 1927) and Robert Mangold (Semi-Circle I–IV, 1995).

    The icon for this chapter is a semicircle, simple and...

  9. CHAPTER 5 Similar Figures
    (pp. 45-60)

    Similar figures not only exist in geometry, they are pervasive. Our icon for this chapter on similar figures is a pair of similar triangles. Equality of ratios of corresponding sides of similar triangles—Thales’ proportionality theorem—is the key to exploring similarity of figures in geometry. Consequences of the theorem include indirect measurement, trigonometric identities, geometric sequences and series, etc.

    But what are “similar” figures? Intuitively, similar objects are objects with the same shape, or which are congruent after suitable scaling. We will need to be more specific when dealing with mathematical (i.e., geometric) objects such as triangles and other...

  10. CHAPTER 6 Cevians
    (pp. 61-76)

    Acevianis a line that connects a vertex of a triangle to a point on the opposite side (extended if necessary) of the triangle. Familiar cevians are the medians, angle-bisectors, and altitudes, but there are many others. The name comes from the Italian mathematician Giovanni Ceva (1647–1734), and in the next section we prove Ceva’s theorem, providing a necessary and sufficient condition for three cevians to intersect at one point.

    Our icon shows a triangle and three concurrent cevians. The point of intersection of the cevians yields acenterof the triangle. Examples of centers for the cevians...

  11. CHAPTER 7 The Right Triangle
    (pp. 77-90)

    The Pythagorean theorem, arguably the best-known theorem in geometry (in spite of the Scarecrow’s misquote), is also the best-known property of right triangles. Right triangles have many other intriguing properties, and have figured prominently in each of the preceding chapters. So we now turn our attention to this important geometrical icon.

    Unlike general triangles, the sides of a right triangle have special names. The sides adjacent the right angle are thelegs(by analogy with the human body), and the third side is thehypotenuse(from the Greekhypo-, “under,” andteinein, “to stretch,” as the hypotenuse stretches from leg...

  12. CHAPTER 8 Napoleon′s Triangles
    (pp. 91-102)

    In Chapter 1 we studied the bride’s chair—a right triangle with squares erected on its sides. In this chapter we consider a similar icon—three equilateral triangles erected on the sides of a triangle. It figures prominently in the statements or proofs of a variety of results, including the one known asNapoleon’s theorem.

    Did Napoleon prove the theorem with his name? No one knows, but we have given his name to the icon for this chapter. After Napoleon’s theorem, we will examine a variety of other results such as the Fermat point of a triangle, Weitzenböck’s inequality, Escher’s...

  13. CHAPTER 9 Arcs and Angles
    (pp. 103-116)

    The icon for this chapter—an angle and a circular arc (or a complete circle)—is omnipresent in mathematics. It appears in a great many problems, theorems, constructions, and applications in geometry, astronomy, surveying, engineering, etc.

    A multitude of historical and modern common objects exhibit angles and circles or circular arcs. Among these are, left to right in Figure 9.1, a pair of compasses, a navigational sextant, the clock on the Big Ben clock tower in London, and a Jeffersonian wind gauge for measuring wind speed and direction. Other such everyday objects include wagon wheels, scissors, protractors, and even a...

  14. CHAPTER 10 Polygons with Circles
    (pp. 117-130)

    Euclid devoted Book IV of theElementsto propositions concerning inscribing and circumscribing polygons in or about circles, and circles in or about polygons. They have had a profound impact on geometry. For example, Archimedes, inMeasurement of the Circle, was able to show that π is approximately 22/7 by inscribing and circumscribing regular polygons with 96 sides and computing the ratios of the perimeters of polygons to their diameters.

    In Leonardo da Vinci’sVitruvian Man, seen in Figure 10.1 from his 15th century drawing and on a one euro coin from Italy, the artist is comparing the proportions of...

  15. CHAPTER 11 Two Circles
    (pp. 131-148)

    This chapter is devoted to properties of a pair of circles in their many forms: (a) disjoint, (b) tangent, (c) intersecting, or (d) concentric, as illustrated in Figure 11.1

    Some of the regions bounded by such circles are sufficiently important to have names in Figure 11.2, such as (a) and (b)lunesorcrescents, (c) alens, (d) the symmetric lens calledvesica piscis, and (e) theannulus.

    We see objects with these shapes every day. For example the flags of Turkey, Algeria, and at least ten other nations employ the crescent, as does the flag of the state of...

  16. CHAPTER 12 Venn Diagrams
    (pp. 149-162)

    John Venn (1834–1923) introduced the diagrams that bear his name in his paper entitledOn the Diagrammatic and Mechanical Representation of Propositions and Reasoningsin 1880 [Venn, 1880]. The diagrams, which Venn called “Eulerian circles,” were used by Venn to represent sets and the relationships among them. They became a common part of the new math movement of the 1960s based on set theory. Venn diagrams usually consist of intersecting circles (as in our icon), although other shapes may be used. Similar diagrams can be found in the work of Ramon Llull (1232–1315), Gottfried Wilhelm Leibniz (1646–1716),...

  17. CHAPTER 13 Overlapping Figures
    (pp. 163-172)

    In many of the preceding chapters we have used icons that consist of a geometric figure partitioned into several other figures. In this chapter we extend that idea to figures that overlap. For example, we interpret the icon above as consisting of two overlapping squares within a larger square rather than a square partitioned into three smaller squares and two L-shaped regions.

    This simple idea has remarkable consequences. In this chapter we first introduce the little-known carpets theorem and present two applications—a proof of the irrationality of$ \sqrt{2} $and a characterization of Pythagorean triples. Overlapping figures also provide a...

  18. CHAPTER 14 Yin and Yang
    (pp. 173-182)

    In Chinese thought yin and yang represent the two great opposite but complementary creative forces at work in the universe. The yin and yang diagram, our icon, is known in Taoism astaijitu(diagram of the supreme ultimate), and enables us to visualize the philosophical idea of complementary opposites within a greater whole (e.g., day and night, feminine and masculine, good and evil, positive and negative, odd and even, etc.). Thetaijitu, a circle partitioned into two differently colored congruent parts by a curve consisting of two semicircles, appeared on the flag of the Kingdom of Korea in 1893, and...

  19. CHAPTER 15 Polygonal Lines
    (pp. 183-200)

    Informally, a polygonal line is a collection of line segments where adjacent segments share endpoints. More formally we have the following definition. Given a finite sequence {P0,P1,P2,…,Pn} ofn+ 1 distinct points in the plane, calledvertices, apolygonal lineconsists of the vertices and the associatededges, the line segmentsP0P1,P1P2,…,Pn–1Pn.

    Examples of polygonal lines in man-made objects include the folding wooden ruler, the data display, and the articulated ladder in Figure 15.1.

    WhenPn=P0we have a closed figure, called apolygon(or ann-gonwhen we wish to...

  20. CHAPTER 16 Star Polygons
    (pp. 201-220)

    Star polygons have long fascinated humans. Since antiquity they have been used as mythical and religious symbols, and they figure prominently in Judaism, Islam, and Christianity. Star-shaped objects are found in nature as well as in manufactured items and as symbols of achievement. In Figure 16.1 we see a starfish, some star-shaped bicycle sprockets, and a star on the Walk of Fame on Hollywood Boulevard.

    In addition to the stars in Figure 16.1, there are star-shaped flowers, stars on flags, and star logos of organizations and businesses. Stars are commonly used to indicate quality (e.g., five star restaurants), and are...

  21. CHAPTER 17 Self-similar Figures
    (pp. 221-232)

    An object is calledself-similarwhen it is similar to a proper subset of itself, such as the partitioned equilateral triangle in the icon for this chapter. Self-similar objects also look the same when magnified or reduced in size. Approximately self-similar natural objects include the romanesco broccoli, the shell of the chambered nautilus, and the flower of the tromsø palm seen in Figure 17.1.

    In mathematics we can produce self-similar figures throughiteration, repeating a process to subdivide a figure into similar parts or grow it by adjoining a region to produce a figure similar to the original. We will...

  22. CHAPTER 18 Tatami
    (pp. 233-242)

    Tatami mats are a type of flooring used in traditional Japanese homes. In the Muromachi period (1332–1573), tatami had a core made of rice straw (today it might be wood chips or styrofoam) covered in woven rush. See Figure 18.1a. Tatami flooring is cool in the summer, warm in the winter, and well suited to the humid months in Japan. The size of a tatami mat varies by region, but they are rectangular, with one dimension twice the other, usually about 1 meter by 2 meters, and about 5-1/2 to 6 cm thick. Sizes of rooms are often measured...

  23. CHAPTER 19 The Rectangular Hyperbola
    (pp. 243-252)

    Hyperbolas, along with circles, ellipses, and parabolas, belong to the family of plane curves known as conic sections. Menaechmus, Euclid, and Aristaeus studied conic sections, and Apollonius of Perga (circa 240-190 bce) wroteConics, an eight-volume work that presented the modern form of characterizing these curves as intersections of cones with planes.

    Hyperbolas appear in nature and in man-made objects. From left to right in Figure 19.1 we see the hyperbolic path taken by some single-apparition comets, three of the six hyperbolic arcs at the tip of a sharpened hexagonal pencil, and the hyperbolic shape of the light beam of...

  24. CHAPTER 20 Tiling
    (pp. 253-260)

    Among the most beautiful of the mathematician’s patterns are tilings. Aplane tilingortessellationis an arrangement of closed shapes that completely covers the plane without overlapping and without leaving gaps. Beautiful plane tilings abound in man-made objects, such as simple patterns for quilts and floors, the intricate mosaics found in Moorish buildings such as the Alhambra in Granada, and the ingenious designs of the Dutch graphic artist Maurits Cornelis Escher (1898–1972).

    Three naturally occurring tilings are shown in Figure 20.1—a honeycomb, the columns of basalt in Giant’s Causeway in Northern Ireland, and mud drying in a...

  25. Solutions to the Challenges
    (pp. 261-308)
  26. References
    (pp. 309-320)
  27. Index
    (pp. 321-326)
  28. About the Authors
    (pp. 327-328)