Michael loves geometry, both the classical ideas we met in high school geometry class, and modern geometrical ideas that lie at the frontiers of mathematics. It's going to be Michael's great pleasure to show us some of these beautiful ideas. Geometric mathematics begins with what we see, yet then extends well beyond. When we look around us, we see shapes, patterns, and forms. Yet what we actually see is not mathematics. We see circular shapes when we look at the moon or sun. We see some what straight lines when we look at bamboo or the horizon.

Yet one of the powerful strategies for understanding the world, has been to let it give us the seeds for mathematical ideas, but then to allow ourselves the freedom to pursue those ideas in the abstract, on their own. The foundational study of our visual world, is Euclidean geometry. On that foundation, mathematicians have built a vast edifice of mathematical realms that include concepts, methods of proof, and strategies of analysis that Euclid never dreamed of.

Often, we find that the abstract exploration returns us back to the real world via unexpected insights, that give us further insights into nature. While mathematicians are exploring these ideas, the motivations that they have are often an aesthetic within the realm of mathematical thought. They seek elegant reasoning and startling connections.

While the geometrical insights that Michael most likes are those where different ideas unexpectedly come together to reveal some sort of relationship that was not obvious at first. For example, the Pythagorean Theorem can help explain Einstein's Theory of Special Relativity, which we'll see in a future lecture.

He also likes visual surprises and mysterious connections. SO he's tried to include as many as possible in this series of lectures. Well lets take a minute to describe the overall structure of the course. The lectures are grouped into four different themes, the first being Euclidean geometry. Yet we'll discuss both classical results from ancient times, and newer insights as well. What delights Michael in this topic is how many unexpected relationships there are to be discovered when we look for them.

One strategy for finding connection is to seek different ways of proving the same theorem. For example, take the Pythagorean Theorem. It's the most famous theorem in all of mathematics, being known for thousands of years. Yet people across the globe and eons, have found new ways to look at that same theorem. So we'll see several proofs of it, including an artistic one by Leonardo da Vinci.

We'll see some insights about this ancient subject of Euclidean geometry that were discovered by mathematicians who are living today. While we're examining the evolving application of Euclidean geometry, we'll also look at its relationship with art. One of the points of intersection between art and mathematics, occurs when artists attempt to render our three-dimensional world on a two-dimensional canvas. That challenge leads to artistic and mathematical explorations of perspective.

During our lectures on Euclidean geometry, we explore basic ideas that apply throughout the whole series of lectures. In the second section of the course, we'll discuss what are non-Euclidean geometries, and we'll understand how we can build, describe, and understand geometries in which no lines are parallel, and where infinitely many lines are all parallel to a given line, through a given point.

Well these non-Euclidean geometries at first seem strange and unreal. However, as we said, mathematical ideas have an uncanny history of finding amazing connections with actual, real nature. So now, non-Euclidean geometries are centrally involved in scientific descriptions of our real universe.

The third section of our course explores mathematical ideas that arise from studying symmetries. We'll see ancient and modern examples of symmetric patterns from around the world. In fact, there are some modern variations in this idea of symmetry, and one occurred recently that produced things called aperiodic tilings. These are bizarre patterns, yet have been used by architects in Australia to create an entire suite of buildings that present us with an aesthetically challenging combination of order on the one hand, and chaos on the other.

We'll conclude the course with the study of graphs, and the somewhat non-geometrical geometry that has grown out of that study. Graphs are apparently rather simple object, just constructed of dots and lines. They help us model connections all over the world, among people, in electronic circuits, in games, maps, everywhere. Well, coloring these graphs and tracing paths around them, presents us with some of the most challenging questions that have ever been considered and resolved, yet others that still remain tantalizingly out of reach.

Well lets put this study of mathematics from the visual world, into a bit of historical perspective. Geometrical idea shave been known and studied around the globe since ancient times. The word geometry means measuring the world, and although it may have begun as a practical subject beginning with land measurement, the term geometry actually encompasses mathematical subjects that abstract our sense of the visual world, and create a coherent mathematical system.

The ancient Egyptians certainly had to have a clear working knowledge of geometrical ideas when they built the great pyramids around 2700 BCE. Geometrical mathematics appears on two ancient Egyptian papyri, the so-called Moscow mathematical papyrus, and the Rhind papyrus, both written from 2000-1650 BCE. They contain things such as an approximation for pi.

Well the earliest mathematician whose name is known to history is Thales of Miletus c.600 BCE, who studied geometry. One thing he understood was things about triangles, which we'll be introduced to in the next lecture, and from that he was able to figure our how to measure the heights of the pyramids. One method he used was to stand near one of them at a moment when his shadow was exactly the same length as he was tall. He then deduced that the shadow of the pyramid at that moment would also be the length of its shadow. So the length of its shadow was the same as its height, and the measured length of the shadow, gave him the height of the pyramid. Well Thales' method was really using similar triangles, and we'll discuss those in the next lecture, though they come up throughout the course.

By the 6th century BCE, Pythagoras and his school, proved his famous Pythagorean Theorem, which we'll explain later. Yet that theorem still has a claim to be called the most famous and most basic theorem in all of mathematics. Geometry and mathematics were central to the Pythagorean sense of how to understand the world and universe, but when we think of geometry, we think of Euclid c.300 BCE. He wrote the most famous textbook in any subject, at anytime in history, called The Elements.

At that time, geometry clearly had to have been well-developed, since he wrote 13 volumes about it. Euclidean geometry has been extremely influential in education and culture for thousands of years, even beyond its role in mathematics. For centuries the seven liberal arts formed a foundation of a classical education. Those were divided into three disciplines called the trivium, and four more advanced subjects known as the quadrivium. Euclidean geometry was one of the four higher liberal arts. Thus for centuries, Euclidean geometry was considered one of the central subjects that any educated person should master. In fact, above the entrance to Plato's Academy, stood the injunction:

"Let no one ignorant of geometry enter here."

Well over the years, the role of geometry in the curriculum has, Michael has to say, diminished somewhat. For many years, students were introduced to the concept of mathematical proof in their geometry classes. Some of that still occurs, however formal proofs are emphasized less in geometry classes now, than they were formerly. That may be a good thing, or may be a bad thing, it's difficult to tell.

The Elements established the gold standard for rigorous argument, and that standard has lasted for millennia, largely still existing today. We're going to describe the Elements in detail a bit later in this lecture, but the Euclidean template for defining terms, then stating unproved assumptions that he called axioms, then proving assertions step by step from the assumptions, that format of reasoning, influenced thinking well beyond geometry.

For example, the structure of the Declaration of Independence is an example. How does it start? It states assumptions or axioms by saying:

"We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness. --That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed..."

(and so on. So those are the axioms. It then describes the grievances and from them deduces the following)

"these United Colonies are, and of Right ought to be Free and Independent States"

Well, in the ancient world, scholars felt that their mathematical ideas were directly describing the way the world is. Until the 19th century, people considered the Euclidean axioms or assumptions, to be statements of absolute truths about the world. However, by the 19th century, mathematicians realized that non-Euclidean geometries could exist. That is, they found that the axioms that Euclid assumed, were not the only ones possible to be assumed. You could assume different axioms that contradicted the Euclidean axioms, yet still produce a coherent self-consistent geometrical system. Well we're going to describe these non-Euclidean geometries in detail in later lectures.

In the present day, we have a different view of mathematics and its relationship to the real world. The current view is that mathematics is an exploration of self-consistent systems, not systems that are absolutely true in the world. Yet we still face the cosmic challenge, shard by ancient thinkers, and ourselves, of describing the shape and nature of the universe. Yet now, both Euclidean and non-Euclidean geometries, figure centrally in that quest.

Now Michael describes what he thinks mathematics is. Einstein described mathematics as merely a refinement of common sense. Michael sees a lot of truth in that. The heart of all mathematics is captured in definitions, examples, theorems, and proofs. A definition captures an essential concept, which often emerges from looking at specific "paradigmatic examples."

Then theorems arise that describe relationships among the concepts that we've already isolated and defined. Theorems can assert some mathematical structure that may be unexpected. Yet after it's proved, we know it's true. A theorem is a statement that has been proved. A proof is a formal argument that verifies a theorem is true, that is that the conclusion follows logically from the premises. Every proof has to confirm the truth of the theorem, that is the defining feature of a proof. Yet some proofs, in addition to that, give some insight into why the theorem is true. To Michael, one of the satisfying aspects of a great proof, occurs when the proof reveals some underlying, often surprising connection or relationship, from which we see some truth that we previously could not fathom. When we see that kind of a proof, we say, "Aha, that's why it's true!" Hopefully we'll have several of those "Aha moments" during these lectures.

Well a great place to start an exploration of geometric mathematics is with Euclid's Elements. So lets begin by looking at the elements of The Elements. They begin with definition, axioms, and "common notions." These are the unproved assumptions from which all of the future theorems are derived. For example, Euclid defines a point as "that which has no part." Well of course that definition doesn't really mean anything, and in fact modern axiom systems admit that we have to start with some terms that don't mean anything, literally undefined terms. So the modern system says that the term point, would be an undefined term.

Yet think about a point as an abstraction of a dot. As being a single location, yet reduced in size to have no extent in any direction. Euclid then went on to define a line as breadthless length. So again, a more modern convention would be to consider this as as undefined term. Yet we should think of it as a straight segment or infinitely extended straight line that goes off in both directions. A line is an abstraction from what you could actually draw on paper, so the concept of of has no thickness.

Then Euclid went on to define a plane surface as that which has length and breadth only, so it's just a flat thing. We should think of it as an infinitely large and flat surface, having no thickness whatsoever. The word plane comes from the Latin word plantum, that means "flat surface."

Well we'll continue with some of Euclid's definitions, but instead of talking about what he said, we'll just give you some definitions of standard terms we'll be using throughout this course. So first of all, an angle is a shape that really consists of two rays that emerge from a common point, a ray being a half-line that starts at a point and then just goes off straight in one direction. An angle can be measured from 0 to 360 degrees, the latter being the measure around an entire circle.

So for example, if you take a line and consider a point in its middle, that line has an angle of 180 degrees, called a straight angle. If we have two lines that meet at right angles, perpendicularly, then a right angle has 90 degrees, so it's exactly half of a straight angle. A circle is a set of points that are equidistant from a fixed point in a plane, as we know. Parallel lines are simply straight lines on a plane, extended infinitely, so they have no ends, and never meet anywhere in the plane.

Well, after defining terms, Euclid proceeded to list some statements that he was going to assume without proof. These are the axioms, or equivalently, the postulates of Euclidean geometry. These two terms mean precisely the same thing. His first postulate says that given two points, it determines a line. His second says that if you have a line segment, you can extend it to create an infinite line. Postulate three says if you're given any point in the plane, and any radius, you can construct a circle with that radius.

His postulate four says that all right angles are equal. Yet once again, that kind of a postulate is hard to know the meaning of, and modern systems of axioms would alter that postulate somewhat. Euclid's fifth postulate is the most famous, the parallel postulate. His formulation of it, is that if you have one straight line, and two lines cross it, and the sum of the angles on that side of the first line are less than 180 degrees in sum, then those tow lines will meet on that side of the line.

So that is a rather cumbersome statement of the parallel postulate, but there's another version of it, that most of us learned in high school, which is an alternative formulation framed by the mathematician John Playfair 1748-1819, in 1795. So he states it by saying that if you were given a straight line, and a point not on the line, then there is one and only one line, through the point, that is parallel to the given line.

Well Euclid's axioms over the years have been discovered to have some deficiencies from the point of view of modern mathematicians. So a modern axiom system for Euclidean geometry often has more than 20 axioms, which are rather complicated axioms, so they're very difficult to remember. Yet to understand geometry, it's much more important and sufficient to accept these kinds of intuitive facts about the plane, and then explore the amazing richness that follows.

So lets begin right now to look at some basic theorems that lie at the foundation of plane geometry. The first theorem we'll explore is that opposite angles of crossing lines are equal. Of course one can see visually that this must be the case, and we can prove this by simply taking the picture of these two straight lines, and rotating the entire picture by 180 degrees. When we do that, each line falls back on itself, and therefor the angle that was at the top, gets rotated to become the angle at the bottom. So that's a basic theorem in geometry, not a very exciting example, but it is a true theorem that we use all the time, so is absolutely fundamental.

The next theorem we'll explore, has to do with parallel lines. Suppose that we have two parallel lines that are cut by a third line, called a transversal. Then, the corresponding angles are equal, and the alternate interior angles are equal. Let's see why this is true. We can use Euclid's formulation of the parallel postulate, since it was actually designed to make this theorem obvious. The reason it would be obvious is because if we consider this transversal and have the two parallel lines it crosses, if the alternate interior angles were not equal, then the sum of two angles on one side of the transversal would sum to less than 180 degrees. Therefor by his postulate, it says that they will meet on that side, and therefor not be parallel. So that was his proof that these alternate interior angles were equal.

However, we can see another way to show why they're equal that maybe is more visually compelling. If we take the bottom parallel line and slide it up, you can just see that the corresponding angles will be equal, and once corresponding angles are equal, and we know that the opposite angles of crossing lines are equal, then that shows us that these alternate interior angles are equal, and likewise with the other pair of alternate interior angles as well.

By the way, the converse is also true, and we'll use it many times in the future. It says that if we have two lines and a transversal, and the alternate interior angles are equal, then in fact, the two lines are parallel.

Now we come to what Michael sees as the first real theorem that we're going to prove, whose statement is not obvious, and we might not even guess that it's true. The statement we'll prove now is that if you take any triangle whatsoever, it has of course, three angles. The fact is that their sum is precisely 180 degrees. Well it's not obvious why this theorem is true. Yet it is true, and this is the way we can prove it.

We take our triangle, ABC, and will use one of the postulates of geometry, namely the parallel postulate, to draw a line parallel to the side BC, through the point A. When we have seen this construction, we can now look at the picture and see what angles must be equal to each other. Well notice that the side AB is a transversal that cuts these two parallel lines, BC and the line parallel to it. Therefor, the angle at B, inside the triangle, is equal to the alternate interior angle at A, next to the top parallel line outside of the triangle.

Likewise, if we look at the angle at C, inside the triangle, it's equal to the other angle at the top at A, next to the top parallel line outside of the triangle. Therefor, we see that the three angles add up, and exactly fill up that straight line angle, which is 180 degrees. Therefor, we have seen that the sum of the angles of any triangle, add up to 180 degrees.

Now to Michael, this is really a clever proof and he likes it since it's a theorem that is not intuitively obvious. Yet we can prove it true. Another theorem has to do with isosceles triangles. It we take a triangle with two sides exactly equal in length to each other, that's the definition of an isosceles triangle, the theorem says that the angles opposite those two equal sides, are also equal. So angles opposite the equal sides of isosceles triangles are equal. Actually the converse is also true, that if you have a triangle with two equal angles, then the opposite sides of those two equal angles, have equal length, so therefor the triangle is isosceles.

Well lets see if we can understand how to prove this theorem. If we take our triangle with two equal sides, we can simply take it, pick it up, and flip it over to exchange the sides, which will exactly fit on top of each other. So what we've done is taken the angle that used to be down at point C, and moved it over to B. So those angles must be equal. To prove the converse of this theorem, suppose we have a triangle with two angles that are equal. Then we can take the edge between those two equal angles, and flip the triangle over the midpoint. So we just take that side, flip it over, so then those two angles will come back to themselves since we assumed they were equal. The triangle is back to itself again, so the triangle must be an isosceles, since one side has become the other, meaning they had equal lengths.

Well this theorem we've just been talking about, has a name of pons asinorum, meaning ass's bridge. The reason is not clear, but it refers to this theorem in Euclid's Elements, his 5th theorem. If you look at the book, which is online, and see the proof of this 5th proposition, you'll see it's very difficult and quite elaborate. It involves adding extra lines and following several steps around, so the term pons asinorum has come to mean "difficult for a beginner" in a phrase used even outside of mathematics.

The conjecture is that the term ass's bridge was meant to suggest a challenge, because it was difficult to get a donkey to go over a bridge! Well Michael is not sure about the exact history of that, and some think it has to do with the picture in Euclid's Elements looks like a bridge. Whatever it is, this is an example of a case where The Elements have impacted culture even outside of mathematics.

Why did his proof have to be so complicated? Well, it's because he didn't clearly have it in mind that you could pick up a triangle and turn it over. If he knew that, he would have done it. Yet he didn't assume that as an axiom, and therefor didn't want to use it as a construction. Yet later in modern views of the axioms of geometry, we came to believe that we had to make an assumption, some sort of an assumption where you could pick up a triangle and move it to a different location, preserving the angles. So that makes the proof much easier in our day. Well in any case, these theorems begin our study of the Euclidean plane.

When king Ptolemy of Egypt asked Euclid whether there was a simpler way to understand geometry, his famous reply was that there was no royal road to geometry. However, Michael hopes we'll find his road to geometrical insights, to be a delightful journey. In the next lecture, we'll discuss congruence, similarity, and the Pythagorean Theorem.