Euclid established the format of axiomatic mathematics and axiomatic reasoning in the Elements, and thereby, he established the gold standard of mathematical and logical rigor. The most controversial of Euclid's axioms was his parallel postulate. Well, more than two millennia after Euclid, the investigation of the parallel postulate led to the important development of non-Euclidean geometries.

So we'll begin our discussion of non-Euclidean geometries in this lecture, starting with a description of what an axiom system actually is. Axioms are not where a mathematical subject starts. Axioms are a way to give a rigorous foundation to a known body of information.

Now, I know that in schools, geometry has traditionally been taught axiomatically, meaning that on day 1, the axioms are listed and then theorems are proved from them. Yet this approach really gives a misleading impression about how mathematics is actually developed. In reality, a lot of mathematical knowledge and knowledge about the world precedes the step of stating axioms.

In writing the Elements, Euclid was trying to capture the essential features of the physical and visual world that we see around us and the mathematics that flows from those features. So his strategy was to isolate a few basic observations about the world that he viewed as so fundamental and so basic, that they were assumed without justification; they were just true.

Those fundamental observations were the axioms, or the postulates (remember, that means the same thing; axiom and postulate mean the same thing), yet those fundamental axioms were the things from which all the further theorems followed as logical consequences. Euclid and the mathematicians who followed him for over 2000 years considered the axioms to be true statements about the world. Yet as we will soon see, that view of the axioms is no longer tenable.

Well, there really are two different kinds of questions about Euclid's axioms that we'll discuss in this lecture. First, before we turn to non-Euclidean geometries, I want to say a few words about the modern view of the adequacy of Euclid's own axioms to describe Euclidean geometry.

Well, perhaps during the last 150 to 200 years, mathematicians have come to realize that Euclid really used some assumptions that he did not state. In other words, Euclid's axioms aren't really completely satisfactory for proving the theorems of Euclidean geometry. So over the centuries, several deficiencies have been noted in his axioms, and I'll give you an example of one.

Euclid assumed that if you take a circle, and from the center of that circle, you draw a ray, then that ray will intersect the circle. Well, this statement is obvious when we imagine we are looking at the intuitive flat, tabletop plane. However, Euclid's axioms didn't actually imply that such intersections must exist.

Another basic kind of assumption that Euclid made, but he didn't state, had to do with the intuitively reasonable idea that if we have two segments that meet in an angle, and then we do things like slide that angle to a different location, then that's a possible thing to do. Well, nowhere among the stated assumptions or axioms of Euclid's presentation of geometry does it say that such a transformation is allowable.

So, correcting that type of defect, resulted in different and alternative axiom systems for Euclidean geometry. David Hilbert developed one of the most famous of such alternative axiom systems for Euclidean geometry in his Foundations of Geometry of 1899. One of the differences between newer axioms systems and Euclid's, is that the new axiom systems have many more axioms. Hilbert's presentation, for example, had about 20. The axioms are far more complicated, and they're far less memorable.

By the way, when I say "about 20," you might be asking, well, why didn't he find out how many there were? The answer is that he actually gave axioms for 3-dimensional geometry and you don't need all of them for plane geometry, and then one of them was discovered to be redundant, and so on. So I just say about 20.

However, the axioms that he presented and other alternatives are probably more correct—they are more correct—than Euclid's presentation. For example, a new axiom in a new statement of Euclidean geometry might state explicitly that a circle divides the whole plane into two pieces, or something equivalent to that. Or that in an axiom system, we might assume that the side-angle-side theorem—you might assume that as an axiom, because it basically captures the idea that you can take an angle and move it from one place to another.

Well, one reality is that different axiom systems can produce the same body of mathematics. So we would say that two axiom systems are the same, if the set of axioms, together with the theorems that they prove, are in some sense the same set of theorems and axioms. Yet maybe in one system what's a theorem might be assumed as an axiom in another system.

For example, the side-angle-side theorem or postulate could be a theorem or could be a postulate. We could choose some set of axioms for which the side-angle-side statement is proved from something else, or alternatively, we could assume that it is just an axiom.

Well, we've talked about the issue of improving Euclid's axioms of Euclidean geometry, but that's not the most interesting part of the story. It was the realization that non-Euclidean geometries were possible, that really required mathematicians to shift their long-held view that Euclidean geometry was the one and only description of the real world. Euclid's axioms, as we've said, tried to capture fundamental truths about the world. Yet one of them was troublesome to mathematicians even in Euclid's own time, and that difficult axiom was the parallel postulate. That axiom is far more convoluted than the other axioms of Euclid. Euclid's statement of the parallel postulate says, as I said in Lecture One:

“If a line segment intersects 2 straight lines forming 2 interior angles on the same side, and that sum is less than 2 right angles, then the 2 lines, if extended indefinitely, meet on that side on which the angle sum is less than 2 right angles.”

We then heard in lecture one that you can rephrase that parallel postulate in a way that's more common now, which is actually Playfair's axiom, which states:

“Given a line and a point not on that line, exactly one line can be drawn through the given point that's parallel to the given line.”

The parallel postulate is the star player in the development of whole new worlds of geometry. One way to think about the parallel postulate and why it's so problematical is that it refers to things that happen far away. Our experience of reality is severely limited. We only see a tiny part of the world, and from that, we extrapolate an imaginary extension that goes on forever.

If we draw two lines that appear to be parallel on some piece of paper, who knows what's going to happen a trillion light-years from here? After all, the entire universe is thought to be only less than 14 billion years old, so we're certainly not talking about distances that are real in any sense. Yet that is where our imaginary extension of a flat tabletop takes us.

The point is that there are different alternatives to what we can imagine to happen in the far distance, all of which look very much the same in our tabletop or nearby, for example, in our own solar system, or our own or nearby galaxies. All geometries, Euclidean geometry and non-Euclidean geometries, locally look the same. They all capture the sense of reality that we have about our immediate surroundings.

For centuries, people tried to prove the parallel postulate from the other axioms. In other words, they wanted to know that unique parallel lines were necessary consequences of having the other postulates. Really there are 2 alternatives to the parallel postulate.

(1): Given a line and a point not on the line, there are more lines than one that are parallel to the given line and go through that given point.

By the way, "parallel," remember, just means they don't ever touch each other.

(2): Given a line and a point not on the line, there are no lines parallel to the given line through the given point.

Well, finally, in the 19th century, mathematicians came to realize that these alternatives to the parallel postulate are compatible with consistent geometries that locally look very much like our familiar Euclidean plane geometry. Having more than one parallel leads to "hyperbolic geometry," which we'll discuss in the next lecture. Geometries that have no parallel lines at all are called "elliptic geometries."

So, Euclid thought his axioms were describing the one and only real world. Yet a couple of thousand years later, his axioms inadvertently led mathematicians to the idea that an axiom system and its consequences form a mathematical entity, so that the relationship between that mathematical system and the real world is not present within the mathematics itself.

Well, today, most mathematicians consider mathematics to be an exploration of internally consistent assumptions and their implications, rather than an exploration of ideas that are somehow true in the world. The amazing thing, though, is that—as I've said before—often, those possible worlds are, in reality, represented in our own world. Or perhaps it would be more accurate to say that our own world has features that are well-modeled by those abstractions.

So, let's begin our exploration of geometries that do not satisfy the Euclidean parallel postulate. When we create a system of axioms, we want to know whether the axioms are self-consistent. That's to say, we want to know that it's not possible to prove contradictory statements within that system. One way to show that a set of axioms is consistent, is to construct what's called a "mathematical model" for the axioms. If the model exists, is self-consistent, and the axioms are true in the model, then the axioms must be self-consistent; otherwise, they would clash in the model.

So we can create a model of an axiom system in which there are no parallel lines by imagining that our entire universe were in the shape of a sphere. Then our concept of what constitutes a line would be altered. You see, we would find that some but not all of the axioms of Euclidean geometry—that he used to describe reality—would seem appropriate for describing our spherical world, while some would not.

By the way, I don't mean to say that we're thinking about this sphere as sitting in some sort of a straight, flat universe. Instead, I'm thinking that the sphere is the totality of existence. So our light beams would actually go around the sphere, and when we walked in what we said is a straight line, we'd actually be staying on the surface of the sphere.

Geometry in this spherical world would differ from plane geometry in several ways. In particular, there'd be no parallel lines, as we'll see in a minute. So the non-Euclidean spherical geometry has commonalities with, and differences from, Euclidean geometry.

Well, let's now proceed to develop the concept of what we will mean by a line, on this spherical world. Of course, the sphere is something that we're familiar with, because the Earth is basically a sphere, and we have an intuitive familiarity with its properties. The first thing is to decide: What are we going to mean by a "line" in this spherical world?

First of all, we'd want the line to have the property that if you take two points, the shortest path between those two points follows a line. And that can be the generative idea of describing what we mean by a line.

So in the case of the Earth, we're actually familiar with that kind of line, because if we've flown in airplanes to distant places, we know that the airlines have an awareness of the shortest distance. They think about these things, and they fly their routes in great circle segments to get from one place to another. So they fly the shortest path, and later I'll tell you why it's a great circle segment. Yet they fly in the shortest distance they can between two points.

Let me give you just a specific example. Here's a globe, and on this globe we have two cities marked; this is Chicago over here, and this is Rome here. Now it turns out that Chicago and Rome happen to lie on exactly the same latitude. Both are 41° north latitude. That is to say, they lie on the same circle; if we cut the plane parallel to the equator, then they lie on that same latitude.

It turns out that if you fly from Chicago to Rome along the latitude, you would go 500 miles more than taking the shortest route! So let's try to figure out what is the shortest route between two points on the globe. Well, the answer is that the shortest route between two points on a sphere is the route that takes you on a great circle segment. A great circle is the biggest circle that can lie on a globe.

Now, the biggest circle that can lie on a globe (it's like the equator ) is the one where you take a plane that cuts through the center of the sphere, and then the way that it intersects the sphere itself is the biggest circle that there is on the sphere.

Now, why is it that a great circle segment is shorter than another kind of a path on the globe? Well, the answer is that a bigger circle looks more like an actual straight line in space, than a smaller circle.

Just think about it. If you take two points, and if I took a very huge circle, that circle between those two fixed points would just come up a little bit; it would deviate from a straight line by a little bit. Whereas, if you took a smaller circle, it would go up and come down.

Therefore, the bigger the circle, the smaller the distance between two points. So that's why we use great circle pathways to fly from place to place. What we have then decided is that great circles are the lines of a sphere, and they are the shortest-distance paths. So that's what we're going to call a line.

Notice, by the way, then, there are no two lines on a sphere that miss each other, because each line is a great circle. If you have a circle—that is, a circle that's from a plane cutting through the center of the sphere—any two of those will meet in two points, just like you have two circles that go through the North and South Poles. So this is definitely a world in which our interpreation of line as great circle, means it does not have any parallel lines.

Well if you lived in just a small part of a sphere—for example, on the Earth, and we're just in a little local environment—it would look exactly like the plane, see? So it is capturing the idea of our local image of geometry.

Well, circles on the sphere are different from circles in the plane—not great circles, but just any circles. For one thing, circles can't be arbitrarily large. Right? Because the whole sphere is only a certain size.

Second, there is no fixed ratio between the radius of a circle and its circumference. Remember that on the plane, there's a fixed ratio. The circumference is always pi times the diameter. See? Yet on a circle on a sphere, the ratio is not fixded.

Suppose you look at the northern polar cap circle, and it has just a short radius. If you then look at the circle that goes from the North Pole to all the way around until you're at the southern polar cap, then that has a huge radius, but it has the same circumference.

Once again, the area of a circle on a sphere, depends on the size of the sphere. If you have a huge sphere, then a circle of radius r looks more like a flat circle, whereas if it's a small sphere, that same radius will have a different size.

Well, we can explore the geometry of the sphere by looking at things like triangles, so let's look at a triangle on the sphere. If you look at a triangle, there are different kinds of triangles on the sphere.

For example, look at a little, tiny triangle on a sphere. That is almost like a flat triangle, and therefore, the sum of the angles, for example, of that triangle, will be close to 180°.

However, suppose that you look at a triangle that goes from the North Pole down a longitude line to the equator, goes around the equator—say, a quarter of the way—and then goes back up to the North Pole. Then look—it's interesting. There are actually three right angles on that triangle, three right angles. So the sum of the angles of that triangle is 270°. You see?

In fact, by the way, you could have a triangle whose sum of angles is even bigger than that. Suppose that you take a little, tiny triangle and think about the exterior triangle, the one that's outside. The little, tiny triangle might be like a little equilateral triangle near, let's say, the South Pole.

Yet the exterior angles—because you see, it's a triangle on both sides—the exterior angles at each angle are very close to 300°! So we have 300 + 300 + 300. It's almost 900° for that exterior triangle.

So you see that the sum of the angles of a triangle varies, and it turns out there's a very special relationship between the amount of a sphere that's enclosed in a triangle and the sum of the angles of that triangle. So that's what we're going to talk about next.

There's a theorem that's named after a mathematician, Albert Girard, who proved this theorem in 1626. What he said is that the fraction of the area of a sphere contained within a spherical triangle (whenever I say "triangle," I mean that the three sides of the triangle are created by great circle segments, because that's what a line is on a sphere) is directly proportional to the excess angle sum in the triangle The "excess angle sum" means the amount of degrees more than 180, that that triangle contains. Every triangle on a sphere contains more than 180° as the sum of the three angles.

So specifically, here's what he said. He said, suppose you draw a triangle on a sphere. Then the area of this triangle is computed in the following way: You take the 3 angles of that triangle, and then say that 180° is less than that sum, and that the sum is 180 + e (for excess). Then the area of the triangle is e/720, times the area of the whole sphere.

Let's show that it works for the triangles that we've already looked at. For example, remember that triangle that went from the North Pole down to the equator, and then it went around the equator a quarter of the way, and then back up to the North Pole? The sum of its angles was 90 + 90 + 90, which is 270°. So its excess was 90°. That's 90 more than 180. Then 90 over 720 is 1/8th. So indeed, that triangle is 1/8th of the area of the whole sphere, because it's a quarter of the upper hemisphere. Right?

Then a little, tiny triangle that's in a little, tiny area, its angles add up to almost exactly 180, and therefore, its area is very small.

So let me go ahead and give you the proof of Girard's theorem, because this is a wonderful proof. It's just a delight and I think you'll enjoy it. Here we go.

We have an example of a triangle on the sphere, and it has three angles: this angle, this angle, and this angle. What we're going to do is extend each of the sides of that triangle to be a great circle, because we know that every side of a triangle is a straight line on a sphere (which we saw was a great circle segment), so we just continue them around the sphere.

Look at one of these angles. For example, this is the sharpest of these angles. I want you to notice, first of all, that we have symmetry here. Due to the fact that these are all great circles, this triangle on this side also appears on the other side. You see? It's exactly the same triangle, except that it's—if you looked right through the sphere, as you can see on this graphic (as you look through the sphere), you see that it's upside down and backwards. Right? Since every point corresponds to just going straight through the center of the sphere to the other side, that turns everything upside down and backwards. So we have a picture of this triangle here, and then we have another copy of it on the other side.

Now here is the way that Girard's theorem works. It's very clever. Here's what we do: We look at this, what's called a lune, that's created when we look at one of our angles of our triangle and we just look at this area between these two great circles. If this angle is a, we can compute what the area is inside this lune. The reason we can compute what's inside this lune, is because it's just a certain fraction of the whole sphere. Right? Namely, what fraction? The fraction that this angle is, of the whole sphere.

If this angle is a, for example, then this lune will contain a divided by 360° of the sphere. That's this lune. By the way, the same thing is true of the opposite lune, the lune that goes the other way around. Then you see on its other end, it's the corresponding angle of the triangle that's on the back side.

Then if we look at the two lunes at angle a, we get these two stripes around the sphere. Then we go to another angle, and we do the same thing. We take the two lunes around there, and we know exactly how much area there is in each lune. If this is angle b, it's b divided by 360 times the area of the sphere.

Same thing over here. We have two of those lunes, and then at angle c, we have two of those lunes. If we add those up, we first of all see that we've covered the sphere entirely, but we've actually covered it a little more. Namely, this triangle has been covered three times, because the lune here at angle a covered this triangle, as well as more stuff. The lune here covered this triangle again, and the lune here covered it again. So this was covered three times,

Yet a point outside this triangle was only covered one time, except for the points in this other triangle on the back. They also were covered three times instead of once. Therefore, the sum of all of those lunes—that is, two of the a-type lunes, two of the b's, two of the c's—is going to cover the whole sphere. So it's equal to the area of the sphere, plus you've covered this triangle's area four times, two excess times for this triangle and two excess times for this triangle.

So we have a little formula which says that 2a/360 (that's the area in the lunes of a) times S, plus 2b/360 times the size of the sphere (S), plus 2c/360 times the size of the sphere (S), all equals the entire area of the sphere plus four times the area of the triangle.

Now all we do is some very simple arithmetic, and we just solve for t, and you can see the solution here. The solution is that t is equal to 1/720, times the area of the sphere, times the sum of the angles minus 180°. So this is a really neat proof of Girard's theorem.

Notice that one of the implications of this theorem is that on a sphere, you cannot have two triangles that have the same angles but have a different area. So you can't have two similar triangles. It just can't happen. There are no similar triangles on a sphere unless they are actually congruent. That's an interesting feature.

Well, one of the issues with spherical geometry is that lines meet in two points. They do so because if you have great circles, they meet at both ends. It turns out that you can correct that defect—because you don't want lines to meet in two points, really. It's fine, but it's not what you expect from lines!

It turns out that you can create another geometry called "projective geometry," which is also an elliptic geometry (one in which there are no parallel lines), but in which every pair of lines meets in exactly one point.

The way that projective geometry arises is from our concepts of perspective. So here's the way you can think of it: You take a plane and put your eye at the origin, and suppose that the plane doesn't contain the origin. So here's a plane, like this table, and you look at it. Then every point in the plane corresponds to a line through your eye, through the origin. Then associating every line through the origin with a point, captures the concept of the plane.

Except there are also parallel lines—lines that are parallel to the plane. Those additional lines are sort of added-into the plane. You can think of them as lines at infinity, points at infinity, so that the points of projective space are lines through the origin in 3-space. That creates yet another elliptic geometry.

Well, in this lecture, we saw some theorems about Euclidean geometry and how we began our exploration of non-Euclidean geometry. Then we talked about spherical geometry as a specific example of non-Euclidean geometry. We then proved a famous theorem, Girard's theorem, that said that we can compute the area of a triangle based on the sum of the angles of the triangle.

In the next lecture, we're going to discuss hyperbolic geometry. I'll see you then.

So we'll begin our discussion of non-Euclidean geometries in this lecture, starting with a description of what an axiom system actually is. Axioms are not where a mathematical subject starts. Axioms are a way to give a rigorous foundation to a known body of information.

Now, I know that in schools, geometry has traditionally been taught axiomatically, meaning that on day 1, the axioms are listed and then theorems are proved from them. Yet this approach really gives a misleading impression about how mathematics is actually developed. In reality, a lot of mathematical knowledge and knowledge about the world precedes the step of stating axioms.

In writing the Elements, Euclid was trying to capture the essential features of the physical and visual world that we see around us and the mathematics that flows from those features. So his strategy was to isolate a few basic observations about the world that he viewed as so fundamental and so basic, that they were assumed without justification; they were just true.

Those fundamental observations were the axioms, or the postulates (remember, that means the same thing; axiom and postulate mean the same thing), yet those fundamental axioms were the things from which all the further theorems followed as logical consequences. Euclid and the mathematicians who followed him for over 2000 years considered the axioms to be true statements about the world. Yet as we will soon see, that view of the axioms is no longer tenable.

Well, there really are two different kinds of questions about Euclid's axioms that we'll discuss in this lecture. First, before we turn to non-Euclidean geometries, I want to say a few words about the modern view of the adequacy of Euclid's own axioms to describe Euclidean geometry.

Well, perhaps during the last 150 to 200 years, mathematicians have come to realize that Euclid really used some assumptions that he did not state. In other words, Euclid's axioms aren't really completely satisfactory for proving the theorems of Euclidean geometry. So over the centuries, several deficiencies have been noted in his axioms, and I'll give you an example of one.

Euclid assumed that if you take a circle, and from the center of that circle, you draw a ray, then that ray will intersect the circle. Well, this statement is obvious when we imagine we are looking at the intuitive flat, tabletop plane. However, Euclid's axioms didn't actually imply that such intersections must exist.

Another basic kind of assumption that Euclid made, but he didn't state, had to do with the intuitively reasonable idea that if we have two segments that meet in an angle, and then we do things like slide that angle to a different location, then that's a possible thing to do. Well, nowhere among the stated assumptions or axioms of Euclid's presentation of geometry does it say that such a transformation is allowable.

So, correcting that type of defect, resulted in different and alternative axiom systems for Euclidean geometry. David Hilbert developed one of the most famous of such alternative axiom systems for Euclidean geometry in his Foundations of Geometry of 1899. One of the differences between newer axioms systems and Euclid's, is that the new axiom systems have many more axioms. Hilbert's presentation, for example, had about 20. The axioms are far more complicated, and they're far less memorable.

By the way, when I say "about 20," you might be asking, well, why didn't he find out how many there were? The answer is that he actually gave axioms for 3-dimensional geometry and you don't need all of them for plane geometry, and then one of them was discovered to be redundant, and so on. So I just say about 20.

However, the axioms that he presented and other alternatives are probably more correct—they are more correct—than Euclid's presentation. For example, a new axiom in a new statement of Euclidean geometry might state explicitly that a circle divides the whole plane into two pieces, or something equivalent to that. Or that in an axiom system, we might assume that the side-angle-side theorem—you might assume that as an axiom, because it basically captures the idea that you can take an angle and move it from one place to another.

Well, one reality is that different axiom systems can produce the same body of mathematics. So we would say that two axiom systems are the same, if the set of axioms, together with the theorems that they prove, are in some sense the same set of theorems and axioms. Yet maybe in one system what's a theorem might be assumed as an axiom in another system.

For example, the side-angle-side theorem or postulate could be a theorem or could be a postulate. We could choose some set of axioms for which the side-angle-side statement is proved from something else, or alternatively, we could assume that it is just an axiom.

Well, we've talked about the issue of improving Euclid's axioms of Euclidean geometry, but that's not the most interesting part of the story. It was the realization that non-Euclidean geometries were possible, that really required mathematicians to shift their long-held view that Euclidean geometry was the one and only description of the real world. Euclid's axioms, as we've said, tried to capture fundamental truths about the world. Yet one of them was troublesome to mathematicians even in Euclid's own time, and that difficult axiom was the parallel postulate. That axiom is far more convoluted than the other axioms of Euclid. Euclid's statement of the parallel postulate says, as I said in Lecture One:

“If a line segment intersects 2 straight lines forming 2 interior angles on the same side, and that sum is less than 2 right angles, then the 2 lines, if extended indefinitely, meet on that side on which the angle sum is less than 2 right angles.”

We then heard in lecture one that you can rephrase that parallel postulate in a way that's more common now, which is actually Playfair's axiom, which states:

“Given a line and a point not on that line, exactly one line can be drawn through the given point that's parallel to the given line.”

The parallel postulate is the star player in the development of whole new worlds of geometry. One way to think about the parallel postulate and why it's so problematical is that it refers to things that happen far away. Our experience of reality is severely limited. We only see a tiny part of the world, and from that, we extrapolate an imaginary extension that goes on forever.

If we draw two lines that appear to be parallel on some piece of paper, who knows what's going to happen a trillion light-years from here? After all, the entire universe is thought to be only less than 14 billion years old, so we're certainly not talking about distances that are real in any sense. Yet that is where our imaginary extension of a flat tabletop takes us.

The point is that there are different alternatives to what we can imagine to happen in the far distance, all of which look very much the same in our tabletop or nearby, for example, in our own solar system, or our own or nearby galaxies. All geometries, Euclidean geometry and non-Euclidean geometries, locally look the same. They all capture the sense of reality that we have about our immediate surroundings.

For centuries, people tried to prove the parallel postulate from the other axioms. In other words, they wanted to know that unique parallel lines were necessary consequences of having the other postulates. Really there are 2 alternatives to the parallel postulate.

(1): Given a line and a point not on the line, there are more lines than one that are parallel to the given line and go through that given point.

By the way, "parallel," remember, just means they don't ever touch each other.

(2): Given a line and a point not on the line, there are no lines parallel to the given line through the given point.

Well, finally, in the 19th century, mathematicians came to realize that these alternatives to the parallel postulate are compatible with consistent geometries that locally look very much like our familiar Euclidean plane geometry. Having more than one parallel leads to "hyperbolic geometry," which we'll discuss in the next lecture. Geometries that have no parallel lines at all are called "elliptic geometries."

So, Euclid thought his axioms were describing the one and only real world. Yet a couple of thousand years later, his axioms inadvertently led mathematicians to the idea that an axiom system and its consequences form a mathematical entity, so that the relationship between that mathematical system and the real world is not present within the mathematics itself.

Well, today, most mathematicians consider mathematics to be an exploration of internally consistent assumptions and their implications, rather than an exploration of ideas that are somehow true in the world. The amazing thing, though, is that—as I've said before—often, those possible worlds are, in reality, represented in our own world. Or perhaps it would be more accurate to say that our own world has features that are well-modeled by those abstractions.

So, let's begin our exploration of geometries that do not satisfy the Euclidean parallel postulate. When we create a system of axioms, we want to know whether the axioms are self-consistent. That's to say, we want to know that it's not possible to prove contradictory statements within that system. One way to show that a set of axioms is consistent, is to construct what's called a "mathematical model" for the axioms. If the model exists, is self-consistent, and the axioms are true in the model, then the axioms must be self-consistent; otherwise, they would clash in the model.

So we can create a model of an axiom system in which there are no parallel lines by imagining that our entire universe were in the shape of a sphere. Then our concept of what constitutes a line would be altered. You see, we would find that some but not all of the axioms of Euclidean geometry—that he used to describe reality—would seem appropriate for describing our spherical world, while some would not.

By the way, I don't mean to say that we're thinking about this sphere as sitting in some sort of a straight, flat universe. Instead, I'm thinking that the sphere is the totality of existence. So our light beams would actually go around the sphere, and when we walked in what we said is a straight line, we'd actually be staying on the surface of the sphere.

Geometry in this spherical world would differ from plane geometry in several ways. In particular, there'd be no parallel lines, as we'll see in a minute. So the non-Euclidean spherical geometry has commonalities with, and differences from, Euclidean geometry.

Well, let's now proceed to develop the concept of what we will mean by a line, on this spherical world. Of course, the sphere is something that we're familiar with, because the Earth is basically a sphere, and we have an intuitive familiarity with its properties. The first thing is to decide: What are we going to mean by a "line" in this spherical world?

First of all, we'd want the line to have the property that if you take two points, the shortest path between those two points follows a line. And that can be the generative idea of describing what we mean by a line.

So in the case of the Earth, we're actually familiar with that kind of line, because if we've flown in airplanes to distant places, we know that the airlines have an awareness of the shortest distance. They think about these things, and they fly their routes in great circle segments to get from one place to another. So they fly the shortest path, and later I'll tell you why it's a great circle segment. Yet they fly in the shortest distance they can between two points.

Let me give you just a specific example. Here's a globe, and on this globe we have two cities marked; this is Chicago over here, and this is Rome here. Now it turns out that Chicago and Rome happen to lie on exactly the same latitude. Both are 41° north latitude. That is to say, they lie on the same circle; if we cut the plane parallel to the equator, then they lie on that same latitude.

It turns out that if you fly from Chicago to Rome along the latitude, you would go 500 miles more than taking the shortest route! So let's try to figure out what is the shortest route between two points on the globe. Well, the answer is that the shortest route between two points on a sphere is the route that takes you on a great circle segment. A great circle is the biggest circle that can lie on a globe.

Now, the biggest circle that can lie on a globe (it's like the equator ) is the one where you take a plane that cuts through the center of the sphere, and then the way that it intersects the sphere itself is the biggest circle that there is on the sphere.

Now, why is it that a great circle segment is shorter than another kind of a path on the globe? Well, the answer is that a bigger circle looks more like an actual straight line in space, than a smaller circle.

Just think about it. If you take two points, and if I took a very huge circle, that circle between those two fixed points would just come up a little bit; it would deviate from a straight line by a little bit. Whereas, if you took a smaller circle, it would go up and come down.

Therefore, the bigger the circle, the smaller the distance between two points. So that's why we use great circle pathways to fly from place to place. What we have then decided is that great circles are the lines of a sphere, and they are the shortest-distance paths. So that's what we're going to call a line.

Notice, by the way, then, there are no two lines on a sphere that miss each other, because each line is a great circle. If you have a circle—that is, a circle that's from a plane cutting through the center of the sphere—any two of those will meet in two points, just like you have two circles that go through the North and South Poles. So this is definitely a world in which our interpreation of line as great circle, means it does not have any parallel lines.

Well if you lived in just a small part of a sphere—for example, on the Earth, and we're just in a little local environment—it would look exactly like the plane, see? So it is capturing the idea of our local image of geometry.

Well, circles on the sphere are different from circles in the plane—not great circles, but just any circles. For one thing, circles can't be arbitrarily large. Right? Because the whole sphere is only a certain size.

Second, there is no fixed ratio between the radius of a circle and its circumference. Remember that on the plane, there's a fixed ratio. The circumference is always pi times the diameter. See? Yet on a circle on a sphere, the ratio is not fixded.

Suppose you look at the northern polar cap circle, and it has just a short radius. If you then look at the circle that goes from the North Pole to all the way around until you're at the southern polar cap, then that has a huge radius, but it has the same circumference.

Once again, the area of a circle on a sphere, depends on the size of the sphere. If you have a huge sphere, then a circle of radius r looks more like a flat circle, whereas if it's a small sphere, that same radius will have a different size.

Well, we can explore the geometry of the sphere by looking at things like triangles, so let's look at a triangle on the sphere. If you look at a triangle, there are different kinds of triangles on the sphere.

For example, look at a little, tiny triangle on a sphere. That is almost like a flat triangle, and therefore, the sum of the angles, for example, of that triangle, will be close to 180°.

However, suppose that you look at a triangle that goes from the North Pole down a longitude line to the equator, goes around the equator—say, a quarter of the way—and then goes back up to the North Pole. Then look—it's interesting. There are actually three right angles on that triangle, three right angles. So the sum of the angles of that triangle is 270°. You see?

In fact, by the way, you could have a triangle whose sum of angles is even bigger than that. Suppose that you take a little, tiny triangle and think about the exterior triangle, the one that's outside. The little, tiny triangle might be like a little equilateral triangle near, let's say, the South Pole.

Yet the exterior angles—because you see, it's a triangle on both sides—the exterior angles at each angle are very close to 300°! So we have 300 + 300 + 300. It's almost 900° for that exterior triangle.

So you see that the sum of the angles of a triangle varies, and it turns out there's a very special relationship between the amount of a sphere that's enclosed in a triangle and the sum of the angles of that triangle. So that's what we're going to talk about next.

There's a theorem that's named after a mathematician, Albert Girard, who proved this theorem in 1626. What he said is that the fraction of the area of a sphere contained within a spherical triangle (whenever I say "triangle," I mean that the three sides of the triangle are created by great circle segments, because that's what a line is on a sphere) is directly proportional to the excess angle sum in the triangle The "excess angle sum" means the amount of degrees more than 180, that that triangle contains. Every triangle on a sphere contains more than 180° as the sum of the three angles.

So specifically, here's what he said. He said, suppose you draw a triangle on a sphere. Then the area of this triangle is computed in the following way: You take the 3 angles of that triangle, and then say that 180° is less than that sum, and that the sum is 180 + e (for excess). Then the area of the triangle is e/720, times the area of the whole sphere.

Let's show that it works for the triangles that we've already looked at. For example, remember that triangle that went from the North Pole down to the equator, and then it went around the equator a quarter of the way, and then back up to the North Pole? The sum of its angles was 90 + 90 + 90, which is 270°. So its excess was 90°. That's 90 more than 180. Then 90 over 720 is 1/8th. So indeed, that triangle is 1/8th of the area of the whole sphere, because it's a quarter of the upper hemisphere. Right?

Then a little, tiny triangle that's in a little, tiny area, its angles add up to almost exactly 180, and therefore, its area is very small.

So let me go ahead and give you the proof of Girard's theorem, because this is a wonderful proof. It's just a delight and I think you'll enjoy it. Here we go.

We have an example of a triangle on the sphere, and it has three angles: this angle, this angle, and this angle. What we're going to do is extend each of the sides of that triangle to be a great circle, because we know that every side of a triangle is a straight line on a sphere (which we saw was a great circle segment), so we just continue them around the sphere.

Look at one of these angles. For example, this is the sharpest of these angles. I want you to notice, first of all, that we have symmetry here. Due to the fact that these are all great circles, this triangle on this side also appears on the other side. You see? It's exactly the same triangle, except that it's—if you looked right through the sphere, as you can see on this graphic (as you look through the sphere), you see that it's upside down and backwards. Right? Since every point corresponds to just going straight through the center of the sphere to the other side, that turns everything upside down and backwards. So we have a picture of this triangle here, and then we have another copy of it on the other side.

Now here is the way that Girard's theorem works. It's very clever. Here's what we do: We look at this, what's called a lune, that's created when we look at one of our angles of our triangle and we just look at this area between these two great circles. If this angle is a, we can compute what the area is inside this lune. The reason we can compute what's inside this lune, is because it's just a certain fraction of the whole sphere. Right? Namely, what fraction? The fraction that this angle is, of the whole sphere.

If this angle is a, for example, then this lune will contain a divided by 360° of the sphere. That's this lune. By the way, the same thing is true of the opposite lune, the lune that goes the other way around. Then you see on its other end, it's the corresponding angle of the triangle that's on the back side.

Then if we look at the two lunes at angle a, we get these two stripes around the sphere. Then we go to another angle, and we do the same thing. We take the two lunes around there, and we know exactly how much area there is in each lune. If this is angle b, it's b divided by 360 times the area of the sphere.

Same thing over here. We have two of those lunes, and then at angle c, we have two of those lunes. If we add those up, we first of all see that we've covered the sphere entirely, but we've actually covered it a little more. Namely, this triangle has been covered three times, because the lune here at angle a covered this triangle, as well as more stuff. The lune here covered this triangle again, and the lune here covered it again. So this was covered three times,

Yet a point outside this triangle was only covered one time, except for the points in this other triangle on the back. They also were covered three times instead of once. Therefore, the sum of all of those lunes—that is, two of the a-type lunes, two of the b's, two of the c's—is going to cover the whole sphere. So it's equal to the area of the sphere, plus you've covered this triangle's area four times, two excess times for this triangle and two excess times for this triangle.

So we have a little formula which says that 2a/360 (that's the area in the lunes of a) times S, plus 2b/360 times the size of the sphere (S), plus 2c/360 times the size of the sphere (S), all equals the entire area of the sphere plus four times the area of the triangle.

Now all we do is some very simple arithmetic, and we just solve for t, and you can see the solution here. The solution is that t is equal to 1/720, times the area of the sphere, times the sum of the angles minus 180°. So this is a really neat proof of Girard's theorem.

Notice that one of the implications of this theorem is that on a sphere, you cannot have two triangles that have the same angles but have a different area. So you can't have two similar triangles. It just can't happen. There are no similar triangles on a sphere unless they are actually congruent. That's an interesting feature.

Well, one of the issues with spherical geometry is that lines meet in two points. They do so because if you have great circles, they meet at both ends. It turns out that you can correct that defect—because you don't want lines to meet in two points, really. It's fine, but it's not what you expect from lines!

It turns out that you can create another geometry called "projective geometry," which is also an elliptic geometry (one in which there are no parallel lines), but in which every pair of lines meets in exactly one point.

The way that projective geometry arises is from our concepts of perspective. So here's the way you can think of it: You take a plane and put your eye at the origin, and suppose that the plane doesn't contain the origin. So here's a plane, like this table, and you look at it. Then every point in the plane corresponds to a line through your eye, through the origin. Then associating every line through the origin with a point, captures the concept of the plane.

Except there are also parallel lines—lines that are parallel to the plane. Those additional lines are sort of added-into the plane. You can think of them as lines at infinity, points at infinity, so that the points of projective space are lines through the origin in 3-space. That creates yet another elliptic geometry.

Well, in this lecture, we saw some theorems about Euclidean geometry and how we began our exploration of non-Euclidean geometry. Then we talked about spherical geometry as a specific example of non-Euclidean geometry. We then proved a famous theorem, Girard's theorem, that said that we can compute the area of a triangle based on the sum of the angles of the triangle.

In the next lecture, we're going to discuss hyperbolic geometry. I'll see you then.