## Metadata - Author: [[David S. Richeson]] - Full Title: Euler's Gem - Category: #books ## Highlights - The objects a topologist studies need not be rigid or geometric. Topologists are interested in determining connectedness, detecting holes, and investigating twistedness. ([Location 223](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=223)) - The sum V − E + F is a quantity intrinsically associated with the shape. In the lingo of topologists, we say that it is an invariant of the surface. Because of this powerful property of invariance, we call the number V − E + F the Euler number of the surface. The Euler number of a sphere is 2 and the Euler number of a torus is 0. ([Location 257](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=257)) - We can prove that there is always at least one point on the surface of the earth where there is no wind. This follows not from an understanding of meteorology, but from an understanding of topology. ([Location 272](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=272)) - In his life of seventy-six years, Euler created enough mathematics to fill seventy-four substantial volumes, the most total pages of any mathematician. ([Location 343](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=343)) - Euler introduced e as the base of the natural logarithm; he made popular the use of the symbol π; at the end of his life he used i to denote (this notation was made popular by Gauss); he used a, b, and c to denote sides of a typical triangle with A, B, and C the angles opposite; he used Σ for sums; he denoted finite differences by ρx; and he began the use of f (x) for a function. ([Location 637](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=637)) - A recent survey of mathematicians showed that in their eyes, Euler’s polyhedron formula is the second-most beautiful theorem in all of mathematics. The theorem voted most beautiful was Euler’s formula eπi + 1 = 0. ([Location 648](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=648)) - Madame, it is an old word and each one takes it new and wears it out himself. It is a word that fills with meaning as a bladder with air and the meaning goes out of it as quickly. It may be punctured as a bladder is punctured and patched and blown up again. —Ernest Hemingway, Death in the Afternoon ([Location 655](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=655)) - “Polyhedron” comes from the Greek roots poly, meaning many, and hedra, meaning seat. A polyhedron has many seats on which it can be set down. Although the term hedra originally meant seat, it has been the standard term for the face of a polyhedron since at least Archimedes2. Thus a reasonable translation of polyhedron is “many faces. ([Location 659](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=659)) - Although never stated explicitly, the historical assumption was that a polyhedron is convex. A convex polyhedron is an object that satisfies our naive definition (given above) and, in addition, has the property that any two points in the object can be joined by a straight line segment that is completely contained in the polyhedron. That is, a convex polyhedron cannot have any indentations. ([Location 700](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=700)) - There are always antecedent causes. A beginning is an artifice, and what recommends one over another is how much sense it makes of what follows. —Ian McEwan, Enduring Love ([Location 719](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=719)) - [Pythagoras] was also the first to open up the enduring gulf of incomprehension between the scientific spirit, which hopes that the universe is ultimately understandable, and the mystical spirit, which hopes—perhaps unconsciously—that it is not. —George Simmons ([Location 796](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=796)) - What distinguished the Pythagoreans was their means of purifying the mind. They did not achieve purity by meditating, but by studying mathematics and science. ([Location 825](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=825)) - The Pythagoreans believed that God ordered the universe with numbers, and that every number could be expressed as a ratio of two whole numbers (every number could be written as a fraction). Using modern terminology, the Pythagoreans believed that all numbers are rational. ([Location 829](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=829)) - The Pythagoreans followed a communal lifestyle; they ate meals, exercised, and studied together. This way of living, along with their tradition of sharing knowledge orally, their enforced secrecy, and their adoration of Pythagoras, makes it difficult to discern which Pythagorean made what contribution to mathematics. In fact, since mathematics was part of their religion, and Pythagoras was their spiritual leader, any mathematical result obtained by his followers was “the word of the master” and was attributed to him. ([Location 836](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=836)) - Plato held Theaetetus in highest regard, second only to his teacher, Socrates ([Location 874](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=874)) - It was from Theaetetus that Plato learned of the five regular solids. Plato recognized their mathematical importance and their beauty. Like many future thinkers, he believed that there must be cosmic significance for this magnificent collection of five objects. ([Location 901](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=901)) - The Greek atomic model was so influential that it was universally accepted until the birth of modern chemistry two thousand years later. It was not until the Irish scientist Robert Boyle (1627–1691) published his book The Sceptical Chymist in 1661 that this chemical model began to crumble. ([Location 926](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=926)) - Euclid is not known for his new contributions to mathematics; much, if not most, of the material found in the Elements was first proved by others. ([Location 950](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=950)) - Mathematicians focused exclusively on metric properties of polyhedra—properties that could be measured. They were interested in finding lengths of sides and diagonals; computing areas of faces; measuring plane angles; and determining volumes. Euler’s fist step was not in this metric tradition. He hoped to discover a way to group together, or classify, all polyhedra by counting their features. ([Location 1228](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=1228)) - It took many years for mathematicians to see the importance of something that was apparent to Euler—that this theorem was about dimension and the rules for building mathematical objects. Euler’s formula and its generalizations became the cornerstone for the field of topology. ([Location 1420](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=1420)) - Mathematics is concerned only with the enumeration and comparison of relations. —Carl Friedrich Gauss1 Mathematicians do not study objects, but relations between objects. —Henri Poincaré2 ([Location 1428](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=1428)) - Although by no means a recluse, Descartes was always yearning for time alone to work on his scientific and philosophical pursuits. His motto illustrated this desire: bene vixit qui bene latuit (he has lived well who has hidden well). ([Location 1533](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=1533)) - Even today, we find books that state emphatically that Descartes did or did not pre-discover Euler’s formula. Of course, we should keep in mind the words of the eminent philosopher Thomas Kuhn (1922–1996), who wrote, “The fact that [the priority question] is asked . . . is a symptom of something askew in the image of science that gives discovery so fundamental a role. ([Location 1590](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=1590)) - There is a widely-held, mistaken belief that the objects in mathematics are named after their discoverers, and that when this does not happen, it is akin to plagiarism or falsification of history. Using this standard, Euler has been wronged repeatedly, for many of his discoveries bear the names of others. (There is a oft-repeated quip that “objects in mathematics are named after the first person after Euler to discover them. ([Location 1615](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=1615)) - The second published proof of Euler’s polyhedron formula, and the first to meet today’s rigorous standards, was given by Adrien-Marie Legendre. ([Location 1634](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=1634)) - A great circle is any circle on a sphere that has the same radius as the sphere, or, equivalently, it is any circle on the sphere of maximum possible radius. Examples of great circles on the Earth are the equator and lines of longitude. ([Location 1646](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=1646)) - Great circles are not straight, but they are as close to straight as is possible on a sphere. They have the noteworthy property that they are length-minimizing. That is, the shortest path between two points on a sphere follows a great circle. ([Location 1649](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=1649)) - We define a triangle on a sphere to be a region bounded by three great circles (as shown in figure 10.3). In mathematical language a great circle is known as a geodesic, so a more precise name for a spherical triangle is a geodesic triangle ([Location 1661](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=1661)) - HARRIOT-GIRARD THEOREM The area of a geodesic triangle on the unit sphere with interior angles a, b, and c is a + b + c − π. In other words, area = (angle sum) − π. Because the sum of the interior angles of every planar triangle is π, we can restate the formula in yet another way: area = (angle sum) − (angle sum for a planar triangle). That is, the area of a spherical triangle is precisely the amount that the angle sum exceeds the angle sum of a planar triangle. As we will see, this remarkable formula generalizes to spherical polygons with more than three sides. By the way, this is our first concrete example showing why it is advantageous to measure angles in radians; it is invalid when angles are measured in degrees. ([Location 1684](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=1684)) - The British scholar Thomas Harriot is somewhat of an enigma. He was a talented and active researcher, but he never published any of his work. When he died, he left ten thousand pages of unpublished manuscripts, diagrams, collections of measurements, and calculations. One biographer wrote that Harriot’s aversion to publishing “may largely be explained by adverse external circumstances, procrastination, and his reluctance to publish a tract when he thought that further work might improve it. ([Location 1701](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=1701)) - Harriot, like Leibniz and Euler, had a reputation for introducing new and elegant mathematical notation. Unfortunately, because of the difficulty of typesetting his nonstandard notation, not all of it appeared in print and was thus not widely adopted. Two symbols that did survive are the signs < for “less than” and > for “greater than. ([Location 1706](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=1706)) - The French mathematician Albert Girard resided in Holland, most likely because he was uncomfortable living in his childhood home of Lorraine, France, as a Protestant. Today he is known for his work in algebra and trigonometry. He has the distinction of being the first to use the abbreviations sin, tan, and sec for the trigonometric functions sine, tangent, and secant, and for introducing the notation for cube root. ([Location 1710](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=1710)) - In his proof, Legendre employed a common mathematical trick. He computed the same quantity—in this case the area of the unit sphere—in two different ways, thereby deriving an equality. ([Location 1796](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=1796)) - Legendre introduced seemingly unrelated concepts— spheres, angles, and area—to prove the theorem. His approach is valid and very clever, but it does not illustrate why the theorem is true—at least not in a transparent way. Nevertheless, Legendre’s proof gave the first indication that there is more here than simply a combinatorial theorem. The fact that we can prove the theorem using metric geometry suggests an important relationship between Euler’s formula and geometry. ([Location 1819](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=1819)) - the degree of a vertex is the number of edges emanating from it. ([Location 1923](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=1923)) - A graph is connected if it is possible to get from any vertex to any other vertex by following a sequence of edges. ([Location 1926](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=1926)) - A tracing of a graph that begins at one vertex and ends at another vertex is called a walk ([Location 1927](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=1927)) - a very special class of walks, ones that visit each edge exactly one time: this is called an Euler walk ([Location 1928](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=1928)) - If the Euler walk begins and ends at the same vertex, then it is called an Euler circuit. ([Location 1928](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=1928)) - A graph has an Euler walk precisely when it is connected and there are zero or two vertices of odd degree. If there is a pair of vertices of odd degree, then the walk must start at one of these vertices; otherwise the walk may begin anywhere. ([Location 1936](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=1936)) - Cauchy was extremely prolific. His voluminous output was second only to Euler ([Location 2047](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=2047)) - Cauchy made deep and substantial contributions to many areas of mathematics, including complex analysis, real analysis, algebra, differential equations, probability, determinants, and mathematical physics. He was one of the early champions of the need for rigor in mathematics. Many of the fundamental ideas of calculus introduced by Newton, Leibniz, Euler, and others were finally placed on a firm theoretical foundation by Cauchy. We can thank him for what are essentially the modern definitions of continuity, limit, derivative, and definite integral. ([Location 2050](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=2050)) - It is a testament to Cauchy’s influence that there are so many theorems, properties, and concepts named after him—perhaps more than for any other mathematician, even Euler. ([Location 2057](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=2057)) - “In nearly all cases he left the final form of his discoveries to the next generation. In all that Cauchy achieved there is an unusual lack of profundity . . . He was the most superficial of the great mathematicians, the one who had a sure feeling for what was simple and fundamental without realizing it. ([Location 2060](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=2060)) - The first notable feature that distinguishes Cauchy’s proof of Euler’s formula from those of his predecessors is that the polyhedra in his proof are hollow, not solid. ([Location 2076](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=2076)) - The first step in Cauchy’s proof is to divide the graph into triangular regions by adding diagonals to all nontriangular faces (see figure 12.4). This procedure is called triangulating a graph. ([Location 2104](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=2104)) - A graph that can be drawn without edges crossing is called planar. ([Location 2171](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=2171)) - G. H. Hardy wrote, “Reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons. It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game. ([Location 2245](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=2245)) - In a typical map there may be countries with many neighbors, but this is impossible for all countries. In any map there must be some country that has five or fewer neighbors. We call this important fact the five neighbors theorem. The proof of the five neighbors theorem uses Euler’s formula and a little counting. ([Location 2481](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=2481)) - An unavoidable set is a collection of configurations, at least one of which must be present in every adjacency graph. ([Location 2575](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=2575)) - a reducible configuration is a collection of vertices and edges that cannot appear in a minimal criminal. ([Location 2578](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=2578)) - in the lingo of the topologists, two shapes are the same whenever they are homeomorphic ([Location 3141](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=3141)) - A topological invariant is a property or mathematical entity associated with a surface that depends only on the topology of the surface. A topological invariant may take the form of a number, such as the number of boundary components. If two surfaces are homeomorphic, then they must have the same number of boundaries. In practice, the contrapositive is more useful: if two surfaces have a different number of boundaries, then they cannot be topologically the same. Because a cylinder has two boundary components and a Möbius band has one, they are not homeomorphic. Intrinsic dimension is another topological invariant: it allows us to distinguish a (2-dimensional) sphere from a (1-dimensional) circle. ([Location 3151](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=3151)) - Orientability is another topological invariant, or more specifically, a topological property. Two surfaces that are topologically the same are both orientable or both nonorientable. Said another way, if one surface is orientable and another is not, then they must not be homeomorphic. ([Location 3158](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=3158)) - Orientablility, dimension, and the number of boundary components are three important topological invariants. Another topological invariant, arguably the most important one, is the quantity V − E + F. Given a surface S partitioned into V vertices, E edges, and F faces (of course, we still need to avoid ring-shaped faces), we define the Euler number of S to be V − E + F ([Location 3166](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=3166)) - In the previous chapter we stressed that topologists are usually interested in the intrinsic properties of topological objects, not the extrinsic properties. Knot theory is one notable exception. What is interesting about a knot is how the circle is placed in space—its extrinsic configuration. Intrinsically, they are all identical—every knot is homeomorphic to a circle. So in the study of knots, “the same” does not mean homeomorphic. Instead, two knots are the same if one can be continuously deformed into the other. That is, two knots are the same if there is an isotopy between them. ([Location 3334](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=3334)) - As Poincaré so eloquently wrote, “The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living. ([Location 4883](https://readwise.io/to_kindle?action=open&asin=B0053YM0HO&location=4883))