CHAPTER 1 Introduction A typical Modern Geometry course will focus on some variation of a set of axioms for Euclidean geometry due to Hilbert. At the end of such a course, non-euclidean geometries (always one called hyperbolic geometry and sometimes one called elliptic geometry) are explored as alternatives. The underlying justification for this sort of geometry course lies in the widely assumed history of geometry. An ancient Greek mathematician named Euclid, in a book called The Elements, layed out an axiom system for geometry. This was a great work, but it contained serious flaws and gaps in its logic. A modern mathematician named David Hilbert fixed the flaws and bridged the gaps, and now, everything is great. As far as I m concerned, all of this is sufficiently true. There is a book, which we call The Elements, and the name Euclid is associated with it. While the actual details are a bit sketchy and complicated, I will assume that there was a man named Euclid, and he wrote this book. What is not in doubt is that this book sets the example for how mathematics should be done and what mathematics should be. For that reason, Euclid s Elements must be the most important book in mathematics. All of modern mathematics ultimately depends on something called an axiom system, and while an axiom system is a modern concept, the basic idea comes to us through Euclid. I think Euclid is often unfairly criticized (not necessarily on purpose) for not meeting the modern expectations of an axiom system, since these expectations were not yet established. It is true, however, that our modern expectations were developed in refining Euclid s work. This is where Hilbert comes into the picture. His refinement of Euclid s work is essentially the standard we now look to, and so Hilbert s axiom system for Euclidean geometry is among the most important works in mathematics. It is basically true, therefore, that Hilbert fixed the flaws and bridged the gaps. 1
1. INTRODUCTION 2 It is also true that everything is great, but not in the way you might think. Hilbert s axiom system is not a very good axiom system for Euclidean geometry. At least that s what I think. In fact, it would be very surprising, if it were. The ancient Greeks didn t know what we know now, and because of Euclid, we ve learned a lot about axiom systems and geometry. Hilbert only refined Euclid s work. It seems like we would be able to do a better job, if we started from scratch. Some have done this, but geometry is much grander than that. What I want to do in this course is to get some understanding of what Euclid and Hilbert did, because what they did was fundamentally important. But that does not mean that we must be bound by what they did, so I m going to try to focus on the good stuff that we want to keep, and then move on. There are several important branches of geometry other than the one that ends with Hilbert. One of these is the transformational approach associated with Klein. We re going to focus, however, on another branch that runs through Descartes, Gauss, and Riemann. This is the one that I think is the most important, and this is the one that points towards the work of Einstein and modern cosmology. Much of modern physics, however, especially with regards to symmetry, aligns better with the transformational approach. Just keep in mind that I have a bias (towards manifolds, whatever those are). One assumption that Euclid had was that his geometry was the only one. He was describing the universe as he saw it, and he did that very well. The cool thing is that the particular way Euclid saw the universe allowed us to imagine another possibility. Among Euclid s axioms, generally called postulates, we have one called the Fifth Postulate. The Fifth Postulate is logically equivalent to the statement: Given a line l and a point P not on l, there is exactly one line m that contains P and is parallel to l. This statement (in various forms) has several names. Playfair s Axiom is one, and Hilbert s Parallel Axiom is another. The standard story continues with three mathematicians Gauss, Bolyai, and Lobachevsky discovering that you can take an axiom system like Hilbert s (although Hilbert s axiom system came later) and replace the Parallel Axiom with something like:
1. EUCLID S POSTULATE SYSTEM 3 Given a line l and a point P not on l, there are at least two lines m and n that both contain P and are parallel to l. We ll call this the Hyperbolic Axiom. The result is a perfectly consistent axiom system that describes a seemingly odd geometry, which we usually call hyperbolic geometry. Theoretically, therefore, we have a logical system for a geometry that is non-euclidean. That leads to a really cool question: Can our universe be hyperbolic? Why not? And if our universe doesn t have to be Euclidean, does it have to be hyperbolic? Is there anything else? That s the real question. 0.1. Exercises. 1 Draw a picture illustrating Hilbert s Parallel Axiom. Does it sound like a true statement to you? 2 Draw a picture illustrating the Hyperbolic Axiom. Can this be true? 3 The Hyperbolic Axiom says at least two parallels. How many are there actually? 1. Euclid s postulate system Book I of Euclid s Elements (I ll always refer to the Dover edition of Heath s translation) starts on page 153 with some definitions. Some of the twenty-three that he gives are [Euclid, p 153] 1. A point is that which has no part. 2. A line is breadthless length. 3. The extremeties of a line are points. 4. A straight line is a line which lies evenly with the points on itself. 8. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line. 10. When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands. 11. An obtuse angle is an angle greater than a right angle. 12. An acute angle is an angle less than a right angle.
1. EUCLID S POSTULATE SYSTEM 4 Note that we would probably say curve instead of line in definition 2 and later. Keep that in mind when you re reading. This is somewhat normal usage in older books like Heath s translation from about a hundred years ago. Also keep in mind that Euclid did not say line. I believe the Greek word that Heath translated as line was γραµµή. The letters in this word are gamma-rho-alpha-mu-mu-eta, and if you sound out the initial letters in the names of the Greek letters, it sometimes makes sense. I would guess (probably badly) that Euclid s word for curve sounds like gramme, or maybe even gramma. If you don t believe me (and there is good reason not to), consider the word παραβoλὴ (pi-alpha-rho-alpha-beta-omicron-lambda-eta), which we ll run into later. This word translates as parabola. Euclid then gives his five postulates. These are [Euclid, p 154] 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any centre and distance. 4. That all right angles are equal to one another. 5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. As mentioned earlier, Hilbert refined Euclid s axiom/postulate system to modern standards, and this required four times as many axioms. The two most directly related to Euclid s First Postulate are [Hilbert, p3] I, 1. For every two points A, B there exists a line a that contains each of the points A, B. I, 2. For every two points A, B there exists no more than one line that contains each of the points A, B. 1.1. Exercises. 1 In his First Postulate, Euclid is basically saying that if you have two points, then you can be sure that there is a straight line that passes through them. Which of Hilbert s axioms says the same thing?
2. A MECLIDEAN GEOMETRY 5 2 It is common to say, two points determine a line uniquely. There are two ways in which this can be false. What are the two ways? 3 According to Hilbert, given two points A and B, how many lines contain both? 2. A Meclidean geometry Despite what people say about Columbus, it s a pretty good bet that Euclid knew the earth was round 1. It seems that the ancient Greeks knew a bit more about the geometry of the surface of the earth than we know about the geometry of the universe today, but it is useful to imagine not knowing that the earth is not flat. Figure 1. A simple Meclidean figure. What happens as you extend these lines? Let s imagine an ancient geometer Meclid, who is slightly more naive than Euclid. To this person, geometry is the geometry of the surface of the earth, which as far as Meclid can tell, is flat. The study of Meclidean geometry consists of drawing figures on the ground, and for theoretical purposes, we assume that the earth is a gigantic whiteboard. Consider the Meclidean figure in Figure 1, which has two straight lines intersecting. What is a straight line in Meclidean geometry? Well, imagine walking along one. if it s straight, then you wouldn t curve to the left or right. You would just go straight. 1 If anyone ever corrects you saying, the earth is not round, it s spherical, slap them. Only snobs with a limited knowledge of geometry would make that distinction. A more meaningful distinction was made by one of my mom s elementary school students: No, the earth is not round like a ball. It s round like a plate!
2. A MECLIDEAN GEOMETRY 6 2.1. Exercises. 1 Tell me what happens with these two lines. If you wish, you may assume that this figure is at the north pole. 2 How many lines pass through both the north and south poles? 3 A Meclidean plane is really a sphere, and a Euclidean plane is flat like the xy-plane from calculus. Which one is more real? Does either really exist? 4 If either of the two kinds of plane exists, even approximately, where would they be?
CHAPTER 2 Analytic Geometry Euclid talks about geometry in Elements as if there is only one geometry. Today, some people think of there being several, and others think of there being infinitely many. Hopefully, after you get through this course, you will be in the second group. The people in the first group generally think of geometry as running from Euclid to Hilbert and then branching into Euclidean geometry, hyperbolic geometry and elliptic geometry. People in the second group understand that perfectly well, but include another, earlier brach starting with Rene Descartes (usually pronounced like day KART ). This branch continues through people like Gauss and Riemann, and even people like Albert Einstein. In my mind, two of the major advances in the understanding of geometry were known to Descartes. One of these, was only known to Descartes, but Gauss, it seems, figured it out on his own, and we all followed him. The other you know very well, but you probably don t know much about where it came from. This was the development of analytic geometry, geometry using coordinates. The other is not known very well at all, but Descartes noticed something that tells us that geometry should be built upon a general concept of curvature. What is generally called Modern geometry begins with Euclid and ends with Hilbert. The alternate path, the truly contemporary geometry, begins with Descartes and blossoms with Riemann. Note that, as is typical, modern usually means a long time ago. We ll look at analytic geometry first, and Descartes other piece of insight will come later. You ll often hear people say that Descartes invented analytic geometry, but they usually don t go into much more detail than that. Descartes Discours de la Methode was first published in French, I believe, in 1637, and it is an important book in philosophy. An appendix to this book is known as La Geometrie [Descartes], or in English The Geometry. This, apparently, is where analytic geometry was invented. Many people associate axioms and proofs with the word geometry, and you may think that your only exposure to geometry was in your high school geometry class. On the 7