Euler's Gem

Euler's Gem: The Polyhedron Formula and the Birth of Topology

David S. Richeson
Copyright Date: 2008
Pages: 336
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    Euler's Gem
    Book Description:

    Leonhard Euler's polyhedron formula describes the structure of many objects--from soccer balls and gemstones to Buckminster Fuller's buildings and giant all-carbon molecules. Yet Euler's formula is so simple it can be explained to a child.Euler's Gemtells the illuminating story of this indispensable mathematical idea.

    From ancient Greek geometry to today's cutting-edge research, Euler's Gem celebrates the discovery of Euler's beloved polyhedron formula and its far-reaching impact on topology, the study of shapes. In 1750, Euler observed that any polyhedron composed ofVvertices,Eedges, andFfaces satisfies the equationV-E+F=2. David Richeson tells how the Greeks missed the formula entirely; how Descartes almost discovered it but fell short; how nineteenth-century mathematicians widened the formula's scope in ways that Euler never envisioned by adapting it for use with doughnut shapes, smooth surfaces, and higher dimensional shapes; and how twentieth-century mathematicians discovered that every shape has its own Euler's formula. Using wonderful examples and numerous illustrations, Richeson presents the formula's many elegant and unexpected applications, such as showing why there is always some windless spot on earth, how to measure the acreage of a tree farm by counting trees, and how many crayons are needed to color any map.

    Filled with a who's who of brilliant mathematicians who questioned, refined, and contributed to a remarkable theorem's development,Euler's Gemwill fascinate every mathematics enthusiast.

    eISBN: 978-1-4008-3856-1
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-viii)
  3. Preface
    (pp. ix-xiv)
    Dave Richeson
  4. Introduction
    (pp. 1-9)

    They all missed it. The ancient Greeks—mathematical luminaries such as Pythagoras, Theaetetus, Plato, Euclid, and Archimedes, who were infatuated with polyhedra—missed it. Johannes Kepler, the great astronomer, so in awe of the beauty of polyhedra that he based an early model of the solar system on them, missed it. In his investigation of polyhedra the mathematician and philosopher René Descartes was but a few logical steps away from discovering it, yet he too missed it. These mathematicians, and so many others, missed a relationship that is so simple that it can be explained to any schoolchild, yet is...

  5. Chapter 1 Leonhard Euler and His Three “Great” Friends
    (pp. 10-26)

    We have become accustomed to hyperbole. Television commercials, billboards, sportscasters, and popular musicians regularly throw around sensational words such as greatest, best, brightest, fastest, and shiniest. Such words have lost their literal meaning—they are employed in the normal process of selling a product or entertaining a viewer. So, when we say that Leonhard Euler was one of the most influential and prolific mathematicians the world has ever seen, the reader’s eyes may glaze over. We are not overselling the truth. Euler is widely considered, along with Archimedes (287–211 BCE), Isaac Newton (1643–1727), and Carl Friedrich Gauss (1777...

  6. Chapter 2 What Is a Polyhedron?
    (pp. 27-30)

    According to theOxford English Dictionary,the first appearance of the term

    “polyhedron” in English was in Sir Henry Billingsley’s 1570 translation of Euclid’s (c. 300 BCE)Elements.“Polyhedron” comes from the Greek rootspoly,meaning many, andhedra,meaning seat. A polyhedron has many seats on which it can be set down. Although the termhedraoriginally meant seat, it has been the standard term for the face of a polyhedron since at least Archimedes.² Thus a reasonable translation of polyhedron is “many faces.” By the time of Euler, the transliteration ofhedrainto Latin was well established.


  7. Chapter 3 The Five Perfect Bodies
    (pp. 31-35)

    Modern geometry, and indeed much of modern mathematics, can trace its roots back to the work of the Greeks. During the period from Thales (c. 624–547 BCE) to the death of Apollonius (c. 262–190 BCE), the Greeks produced an astonishing body of mathematical works, and the names of many of the scholars of this era are familiar to schoolchildren everywhere: Pythagoras, Plato, Euclid, Archimedes, Zeno, and so on.

    Although the mathematics from Egypt, Mesopotamia, China, and India may have influenced the Greeks, they quickly made the discipline their own. As Plato wrote inEpinomis,

    ”Whenever Greeks borrow anything...

  8. Chapter 4 The Pythagorean Brotherhood and Plato’s Atomic Theory
    (pp. 36-43)

    The early history of Greek mathematics is full of apocrypha, speculations, contradictory evidence, secondhand accounts, and just enough verifiable truths to create a fascinating puzzle. There are very few surviving records of Greek mathematics, and this scarcity of information makes it challenging to reconstruct the historical truth. Primary sources survived for several centuries after their creation, but nearly all were destroyed or lost during the Dark or Middle Ages. Much of what we know is not drawn from primary sources but from secondary sources written hundreds of years later.

    We know little with certainty about Pythagoras (c. 560—480 BCE)...

  9. Chapter 5 Euclid and His “Elements”
    (pp. 44-50)

    When one thinks of Greek geometry, one thinks of Euclid and of his masterwork, theElements.In antiquity Euclid was often referred to simply as “the Geometer.” It is disappointing that so little is known about his life. We cannot identify the place of his birth or even a reasonably accurate birth or death year. Most books on the history of mathematics do not venture a guess at his exact dates, saying instead that he was alive during the year 300 BCE.

    Euclid learned mathematics and discovered the great works of Theaetetus and the other Platonists at Plato’s Academy in...

  10. Chapter 6 Kepler’s Polyhedral Universe
    (pp. 51-62)

    During the period of Arabic mathematics, Europe was experiencing the darkness of the Middle Ages. Very few Europeans received a formal education; the great works of classical antiquity were all but forgotten; mathematical scholarship was almost nonexistent. Only the minimal teaching of geometry and arithmetic remained in monastic schools. For hundreds of years there were virtually no significant contributions made to the body of mathematics.

    It was not until the European Renaissance in the fifteenth century that mathematical activity began to reemerge. The rise of the humanist movement brought a renewed interest in the Greek classics—first Greek literature, then...

  11. Chapter 7 Euler’s Gem
    (pp. 63-74)

    On November 14, 1750, the newspaper headlines should have read “Mathematician discovers edge of polyhedron!”

    On that day Euler wrote from Berlin to his friend Christian Goldbach in St. Petersburg. In a phrase seemingly devoid of interesting mathematics, Euler described “the junctures where two faces come together along their sides, which, for lack of an accepted term, I call ’edges.’”² In reality, this empty-sounding definition was the first important stone laid in the foundation that would become a grand theory.

    One of Euler’s great gifts was his ability to consolidate isolated mathematical results and create a theoretical framework into which...

  12. Chapter 8 Platonic Solids, Golf Balls, Fullerenes, and Geodesic Domes
    (pp. 75-80)

    “That’s great, but what’s it good for?” the skeptical student asks, sarcasm dripping from his voice. Beauty is a wonderful trait, but some say usefulness is a more important measure of the worth of a theorem. What is Euler’s formula good for?

    That is a fair question to ask of any mathematical theorem. Euler’s formula is more than just an elegant theorem. In the chapters that follow we will present many applications of Euler’s formula. Most will require constructing the appropriate framework to understand the application. To whet the reader’s appetite, we pause now and give two quick applications. First,...

  13. Chapter 9 Scooped by Descartes?
    (pp. 81-86)

    In 1860, over a century after Euler presented his proof of the polyhedron formula, evidence surfaced that René Descartes, the famous philosopher, scientist, and mathematician, had known of this remarkable relationship in 1630, more than one hundred years before Euler. The evidence was found in a long-lost manuscript. The story is fascinating, as is the debate over whose name should accompany the polyhedron formula.

    Descartes was born to a noble, if not wealthy, family in 1596 in La Haye, France, just outside of Tours. His mother died a few days after his birth, and his father, although supportive of his...

  14. Chapter 10 Legendre Gets It Right
    (pp. 87-99)

    The second published proof of Euler’s polyhedron formula, and the first to meet today’s rigorous standards, was given by Adrien-Marie Legendre. Legendre was a French mathematician who belonged to both the Académie des Sciences in Paris and the Royal Society of London. He published in several areas, but his most important contributions were to number theory and the theory of elliptic functions. His legacy also includes an extremely popular textbook on elementary geometry that he wrote in 1794,Élements de géométrie (Elements of Geometry).In many ways Legendre’sÉlementsreplaced Euclid’sElements,becoming the primary geometry text for the next...

  15. Chapter 11 A Stroll through Königsberg
    (pp. 100-111)

    In order to place Euler’s formula in a modern context, we must discuss a mathematical field calledgraph theory. This is not the study of graphs of functions that we encountered in high school precalculus ($y\; = \;mx\; + \;b$is a line,$y = {x^2}$is a parabola, and so on.). It is the study of graphs such as those shown in figure 11.1. They are made of points, calledvertices, and lines joining these points, callededges.*

    In 1736, during his first stay in St. Petersburg, Euler tackled the now famous problem of the seven bridges of Königsberg. His contribution to this problem is...

  16. Chapter 12 Cauchy’s Flattened Polyhedra
    (pp. 112-118)

    In the hundred years after Euler’s proof of his polyhedron formula, there were many new proofs and a variety of generalizations to exotic polyhedral shapes. The first significant generalization came from Augustin-Louis Cauchy, who also gave an ingenious new proof.

    Cauchy was born in Paris in 1789. He was the eldest son of a senior government administrator. Although his family was displaced during the Reign of Terror, his father saw that he received a good education. As a youth he became acquainted with the mathematicians Pierre-Simon Laplace (1749—1827) and Joseph-Louis Lagrange, and the chemist Claude Louis Berthollet (1748 1822),...

  17. Chapter 13 Planar Graphs, Geoboards, and Brussels Sprouts
    (pp. 119-129)

    In the previous chapter we saw Cauchy’s clever technique for proving Euler’s formula. He took a polyhedron, removed a face, and projected the rest down onto the plane. Then he proved that$V - E + F = 1$for this figure, so$V - E + F = 2$for the polyhedron. The connection to graph theory should be obvious. At first glance it appears that it would be trivial to generalize Euler’s formula to graphs that are not projections of polyhedra and which may possess edges that are curved.

    The difficulty with extending Euler’s formula is that it does not apply to every graph. Counting vertices and edges is easy—...

  18. Chapter 14 It’s a Colorful World
    (pp. 130-144)

    The mathematician Charles Lutwidge Dodgson (1832—1898), better known as Lewis Caroll, author ofAlice in Wonderland,invented the following two-person game. Person A draws a map of a continent with any number of countries. Then person B colors the map so that neighboring countries (they have to share a border, touching at a corner does not count) have different colors. The object of the game is for A to draw a map so complicated that it forces B to use many colors. B, meanwhile, has to figure out how to color the map with as few colors as possible....

  19. Chapter 15 New Problems and New Proofs
    (pp. 145-155)

    Suppose you were asked: what trees change color and lose their leaves in the autumn? If you replied, “Maple trees do,” then you would have given a correct answer. However, anyone who has driven through the Pennsylvania countryside in October knows that there are also radiantly colored oak, birch, and beech trees standing amid beds of fallen leaves. So, although the answer was correct, it did not give a complete account of all such trees. Could you say that all trees change color in the autumn? No. Pine, fir, spruce, and cedar trees have no leaves to lose. In order...

  20. Chapter 16 Rubber Sheets, Hollow Doughnuts, and Crazy Bottles
    (pp. 156-172)

    By the middle of the nineteenth century mathematicians had a much better understanding of how Euler’s formula applied to polyhedra. It was during this time that they began to ask whether it applied to other objects. What if the figure was not a polyhedron made of flat faces, but instead was a curved surface like a sphere or a torus? If so, what must the partitions look like? Recall that in 1794 Legendre used a partition of a sphere by geodesic polygons to prove Euler’s formula, and Cayley showed that when we apply Euler’s formula to graphs, the edges need...

  21. Chapter 17 Are They the Same, or Are They Different?
    (pp. 173-185)

    One of the most important recurring questions in mathematics is: are the two mathematical objectsXandYthe same? In different contexts we have different criteria for what “the same” means. Often, when we say the same we mean equal, such as the expression$5 \cdot 4 + 6 - {2^3}$and the number 18, or the polynomials${x^2} + 3x + 2$and$(x + 2)\,(x + 1)$. In other circumstances the same may not mean equal. For a sailor navigating by compass, two angles are the same if they differ by 360º (30º is the same as 390º). A geometer may say that two triangles are the same if they are...

  22. Chapter 18 A Knotty Problem
    (pp. 186-201)

    One of the earliest topological investigations was the study ofknots.We are all familiar with knots. They keep our boats secured to shore, our shoes snug on our feet, and the cables and wires hopelessly tangled behind our computers. These are not, strictly speaking, mathematical knots. A mathematical knot has no free ends; it is a topological circle living in 3-dimensional Euclidean space. (To turn an electrical extension cord into a mathematical knot, simply plug the two ends together.)

    In figure 18.1 we see the projections of six mathematical knots: theunknot, trefoil knot, figure eight knot, pentafoil knot,...

  23. Chapter 19 Combing the Hair on a Coconut
    (pp. 202-218)

    Many scientists use mathematics as a tool to predict behavior. A scientist may have an equation or a system of equations that describes the interactions of the quantities in their model. Then they use mathematics to draw conclusions from these equations.

    Often the mathematical models aredifferential equations.These describe the rates of change of various quantities as a function of time. For example, an ecologist might use a system of differential equations to model the population dynamics of rabbits and foxes living in a wildlife refuge driven by their predator-prey relationship. When the number of rabbits is large, the...

  24. Chapter 20 When Topology Controls Geometry
    (pp. 219-230)

    For most of this book we have been moving away from the rigid confines of geometry, working instead in the much more fluid environment of topology. In this chapter and the next we return to geometry. We will examine polygons, polyhedra, curves, and surfaces, made not of rubber, but of the hardest steel. However, these geometric objects can still be viewed with a topological eye—the polygons and curves are homeomorphic to a circle, and the polyhedra and surfaces are homeomorphic to a sphere or a g-holed torus.

    We will present a collection of theorems that shows the surprising relationship...

  25. Chapter 21 The Topology of Curvy Surfaces
    (pp. 231-240)

    One of the most fundamental topics in the geometry of planar curves is curvature. Thecurvatureat a pointxis a number,k,that measures the “sharpness” of the turn atx—it measures how quickly the tangent vectors change direction. Roughly speaking, given a normal vector$\vec n$to a curve atx, if the curve bends in the direction of$\vec n$, then$k > 0$, if it bends away, then$k'0$, otherwise$k = 0$(see figure 21.1). The more sharply it bends, the larger (in absolute value)kis.

    By the Jordan curve theorem a simple closed curve in the plane...

  26. Chapter 22 Navigating in n Dimensions
    (pp. 241-252)

    Thus far, all of our topological objects have been curves or surfaces—objects that are locally 1- or 2-dimensional, and which live in 2-, 3-, or 4-dimensional space. Surfaces were the topological generalization of polyhedra, and Euler’s polyhedron formula generalized nicely to the Euler number for surfaces. At this point, it is natural to ask what we can say about higher-dimensional topological shapes. What are they, and is there an Euler number for them too?

    As we will see in chapter 23, Poincaré defined the Euler number for higher-dimensional topological spaces and proved that it is a topological invariant. But...

  27. Chapter 23 Henri Poincaré and the Ascendance of Topology
    (pp. 253-264)

    If Euler’s theorems on the bridges of Königsberg and polyhedra mark the birth of topology, and the contributions of Listing, Möbius, Riemann, Klein, and other nineteenth-century mathematicians signify topology’s adolescent years, then its coming of age was signaled by the work of Henri Poincaré. Before this there were theorems that we now categorize as topological, but it was not until the waning years of the nineteenth century that Poincaré systematized the field.

    Looking back at his complete body of work, we see a common theme: a topological view of mathematics. Perhaps this qualitative approach to the subject came from his...

  28. Epilogue: The Million-Dollar Question
    (pp. 265-270)

    In the twentieth century topology rose to become one of the pillars of mathematics, sitting side by side with algebra and analysis. Many mathematicians who do not consider themselves topologists use topology on a daily basis. It is inescapable. Today, most first-year graduate students in mathematics are required take a full-year course in topology.

    One way to gauge the importance of an academic field is to see the trail of awards for accomplishments in the discipline. There is no Nobel Prize in mathematics. The mathematical equivalent is the Fields Medal. Fields Medals have been awarded once every four years since...

  29. Acknowledgments
    (pp. 271-272)
  30. Appendix A Build Your Own Polyhedra and Surfaces
    (pp. 273-282)
  31. Appendix B Recommended Readings
    (pp. 283-286)
  32. Notes
    (pp. 287-294)
  33. References
    (pp. 295-308)
  34. Illustration Credits
    (pp. 309-310)
  35. Index
    (pp. 311-317)