 Mathematics is one of the oldest and most fundamental branches of learning. 20,000 BC, there were already evidences of mathematical thinking in animal bones with tallies which turned out to be counting marks, probably even evidences of prime numbers. 4,000 years ago, the ancient Babylonians were working with Pythagorean numbers. These are the sides that make up a right triangle. Mathematics is really one of humankind's greatest intellectual achievements. Today, mathematicians use mathematical theory, computational techniques, algorithms, and the latest technology to solve problems in economics, science, engineering, environment, and business. But the problem is that many people think of math as a scary subject. It doesn't help when this negative stereotype is shared by parents and reinforced by media. Bad teaching also contributes much to the students' disenchantment with math. But I think that the capacity to understand and learn math is innate in us all. Didn't we all learn counting at an early age and enjoyed playing with shapes and numbers and patterns? Poor teaching and poor public perception of math are problems we need to address. I think one big reason is because many people don't really understand the real nature and practice of math. Albert Einstein said, how can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Today, I hope I can help answer this question and reveal to you some powerful and delightful ideas in mathematics. What is mathematics really? Math is not just a collection of formulas and techniques for solving problems. It is the language of science, a tool. It is a creative activity, an art. It is a way of thinking, a way of looking at the world. More importantly, math is not confined to the classroom. Math is everywhere. We'll live in a world of mathematical patterns, shooting stars in the dark sky, patterns and positions of birds flying without colliding in the air, snowflakes, concentric circles formed by a drop of water on a still surface. Patterns can be natural like those. It can also be man-made, like beautiful structures like the Taj Mahal in India. Galileo said that to learn and appreciate nature, it is necessary to understand mathematics, the language in which nature speaks to us. Through the years, human culture has developed a formal system of thought for recognizing, classifying, making sense of patterns. Mathematicians call this mathematics. So what do mathematicians do? Of course, we study numbers, numbers that come up when we vote for our president, numbers when we line up to buy lotto tickets, numbers of spirals in plants, numbers of digits in the non-repeating, non-terminating expansion of the irrational number pi. We also study movement, change, infinity. These are the ideas of analysis or the calculus, if you want. We also study structure and symmetry. These ideas now belong to branch of mathematics called algebra. Algebra which also studies symmetry. But today, I want to talk to you about another branch of mathematics, one of its main ones, and that is geometry. Geometers study shapes, some of them very ordinary shapes like circles, spheres, donuts, coffee cups. But these turn out to be both fascinating, all fascinating and deep. Let me tell you about geometry. Let me start many years ago. Let's go back to ancient Babylonia, ancient Egypt, where geometry was used for very practical purposes, like surveying, for navigation, for building structures. You must remember that geometry comes from the Greek words, jail and metri, which means earth measure. So geometry then was really used for very practical day-to-day purposes. 4000 years ago, the Babylonians already knew how to use the Pythagorean theorem. And so did the ancient Egyptians to build, say, the Great Pyramids. But the geometry as a field of knowledge blossomed in ancient Greece. It is said that inscribed on the gates of Plato's academy in Athens, where the famous words let no one enter who is ignorant of geometry. The Greeks place a lot of importance on mathematics, especially geometry. The seat of learning eventually moved to Alexandria, where Ptolemy, one of Alexander's generals, founded the library and museum of Alexandria. The museum and the library became important centers of learning, similar to Plato's academy. And the most famous product of Alexandria was the great mathematician and geometer Euclid. Euclid is one of the most prominent mathematicians of antiquity. We don't know much about Euclid, his life as a person. Some people even claim that no actual person existed. But Euclid left a lot. And we remember his work. It is called The Elements. The Elements is the second most widely published book in the world after the Bible. It is still being used as textbooks in many schools up to this time. The Elements is actually not a book but a total of 13 volumes covering ideas and numbers and shapes. But what is important about the Elements is that Euclid introduced the so-called actionmatic system, the way we do math and study math now. In an actionmatic system, we start with undefined terms which allow us to make assumptions what we now call actions or postulates. These are things that we assume to be true without no need of proof or verification. And from these undefined terms and assumptions, we can come up with definitions and then use deductive logic to come up with propositions or theorems. Within an actionmatic system, statements that are correctly deduced or proved to be true will forever be true. That is the power of Euclid's actionmatic system and mathematics actionmatic systems. 2000 years after Euclid, mathematicians realized that other types of geometry are possible. Those whose actions differ from those of Euclid. And these new geometries are equally valid. These new geometries have strange and powerful properties such as parallel lines, meeting in infinity. Think of railroad cracks. You know they're parallel, but when you view them from a distance, they seem to be meeting at a point. This is non-Euclidean geometry. Or think about triangles. We know from elementary school that the sum of the angles of a triangle add up to 180 degrees. But in a different kind of geometry, they don't have to be 180 degrees. They can be fatter triangles or they can be skinny ones with angles sums less than 180 degrees. Non-Euclidean geometry, the geometry different from Euclid is visible around us. Think of our earth. Our earth is a sphere. But we don't feel that when we're standing on ground. It looks flat. But viewed from a distance, it is round and curved. So what's the shortest distance on the sphere? It is not the straight line. You can't go through the earth. You have to go on the surface of the earth which is curved. So the shortest distance on the sphere passes through a circle. This is the idea that airplane pilots use when they navigate or by ships crossing the oceans. It was this non-Euclidean geometry, for example, that Albert Einstein needed to formulate the principles of general relativity. So there are many shapes and objects that are fun and intriguing and the mathematical ideas behind them are deep and powerful. I want to share them with you. In the university, before exams, there are two structures in the Dilliman campus where people flock. The Roman Catholic Chapel and the Protestant Chapel. Before exams. But I talk to you about these structures because of their shapes. The Catholic Chapel is like a half sphere, a hemisphere. On the other hand, the Protestant Chapel, although you don't notice it, if you look at it closely, it looks like a saddle-shaped surface. These strange surfaces are what we study in math. Even in calculus, powerful shapes influence many of our ideas. The great religions of the world like the Christian religion, the Jewish religion, Taoist religions. The star of David. Yin and nyan because of their physical symmetry, their nice shapes. Geometry is very important in the physical sciences. Physics. Many principles of physics are governed by geometry. The movement of particles. The shape of the universe which astronomers and astrophysicists study. Study the molecular structure of crystals. And these crystals are nicely shaped on nice structures. Even in the life sciences or in biology or medicine, shapes are very important. Let me tell you about a fantastic result. Gall stones and kidney stones were in the past always treated by surgery. You open up a patient, take out stones. It's invasive. But now we can avoid this invasive surgery by using a principle of mathematics. There is a procedure called extracorporeal shock wave light atripsy. What is that? Well, it's a machine. A machine that eventually breaks down your gall and kidney stones using shock waves. But what principle of math do they use? Well, I hope you remember what an ellipsis. It's like an oval and inside an ellipse are two important points called focus. Two foci. If you bounce a ball, for instance on an elliptical table from one focus, it hits the edge and always returns to the other point. So if you have a billiard table on focus and a ball at another, your ball will always fall into the hole. So this is what medical doctors use to model this principle of light atripsy. They model the kidney or the gallbladder as an ellipse and instead of directly bombarding your stones with waves, they have to bounce it through the edge, through the focus. So a simple mathematical principle saves a lot of time. Patients now come in and 30 minutes later can leave the hospital. No surgeries needed. Medical doctors and life scientists also study of course our genes. But where do mathematicians come in? Well, if you look at the DNA under an electron microscope, they're actually knotted things. What are knots? There's pieces of string which are tied together and that's how DNA looks like. And the way the DNA is knotted says something about how they replicate, how they mutate. So if biologists know something about knots then they might be able to say more about the DNA and that's why they go to mathematicians to reveal to them properties of these knots. Really, there are many questions about nature and the environment and about life that are answered by the shapes and structures we see around us. I want to ask you some and end this talk by talking about one of my favorite shapes. For instance, I think if I ask you to choose which tool is more stable one with four legs or one with three I think any grade school student would say the one with three legs because everyone knows euclidean geometry. Three points corresponding to three legs determine a plane. So a stool with three legs would be more stable than one with four because in a four-legged stool might cause the statement to be unbalanced in balance. So that's easy. Another question why are manhole covers round? We hardly see a square manhole cover. They're usually round like a cover. Well, one explanation would be that you don't want the cover to fall inside the hole and if you have a square hole say one meter along the sides each think of the diagonal what we call the hypotenuse that would be longer than one meter. So the cover would fit pass through the diagonal and fall inside the hole. But if your cover if your hole is circular then because the circle has constant diameter the circle will not fall inside the hole. So maybe that's why manhole covers are round. Plus round objects are easier to roll easier to move around. So talking about the circle what's another common shape? Tires or wheels wheels around. We know that our cars and our bicycles move slowly and smoothly because the wheels are round. Just imagine a square wheel so that would be a very unstable ride. But my question is should wheels always be round? Let me tell you now that they don't have to be. I'll tell you about a shape which will also produce a smooth ride and that's something we call a rollo triangle. It's a special kind of triangle it's almost like an equilateral triangle except that the sides are rounded. Think of a guitar pick that kind of shape will make your ride smooth because it's not really the roundness of the shape that makes the ride smooth it's the constant width of the shape. A circle always has the same diameter but so will this so called rollo triangle. So I can have a car with a triangular wheel and just experience a safe ride as possible. What's another shape that we notice? On the way here I was eating some chips and if you're familiar with this famous potato chips with strange shape in a cylindrical can why do you think shape is like that? Why are the shapes not flat? Well the shape is like the saddle surface because that surface is least resistant to stress so making shapes in this curved shape will lessen or make the chips less prone to breaking or crushing into bits. And finally, let me end here by telling you about one of my favorite results in math really coming from a very simple object which is a ball. So my first question is nothing to do with math something to do with our country why do we have typhoons all the time? Well I'll tell you that a shape has something to do with the answer and that shape has to do with the so called hairy ball theorem. Think about a hairy ball. Do you know how a hairy ball looks like? Well imagine your head covered with hairs all over or if you want imagine a hairy coconut. What does the hairy ball theorem say? Well technically if you want the formal theorem it says there is no non vanishing continuous tangent vector field on the sphere very heavy words may be understood only by mathematicians but the statement of the theorem is very simple the hairy ball theorem says that a hairy ball cannot be combed smoothly What do I mean by that? Well imagine again your head and try to comb the hairs on your head. No matter what direction you take you cannot comb the hairs around your head in one direction only. At some point there's always that spot in the Philippines we call puyo or a world where the hairs seem to stand up or go around that's what the hairy ball theorem says you cannot comb it only in one direction. At some point the hairs will either part or will go around this world. Now what does that have to do with typhoons? Well think of the earth. Earth is like a ball and think of the system of winds around the earth as hairs. What does the hairy ball theorem tell us? You cannot comb the system of winds smoothly so there is no one flow of winds at any time somewhere on the surface of the earth there is this puyo or this coral where the winds will go either up or around that point of the cyclone. So we can see without even much knowledge of meteorology that mathematics assures us that we will have cyclones all the time. Of course these cyclones don't have to be violent. They can just be disruptions in the flow of air. So maybe you think these are frivolous silly ideas of mathematician but they are very deep. If you want for example to create nuclear fusion you have to do it in an object which can be combed smoothly. What's an example of an object which can become smoothly? Think of a hairy donut. If you have a donut then there are two ways to comb this smoothly. You can comb the hairs this way or you can go wrap the hairs around this donut surface. And that makes the donut different from the ball because you cannot comb the ball smoothly but you can comb a donut smoothly. I talk to you about shapes balls and donuts ordinary shapes but I hope I showed to you by trying to understand the mathematics behind these shapes we learn what other roles they play in our lives. I hope that you learned things from what I mentioned today. I hope that you will value these objects try to notice them more as you walk day to day I hope you notice them so that you will learn or try to study them more the same way that we hope that you will study mathematics more learn more about math learn more about geometry and really realize that we are in a wonderful world because disciplines like math can make our lives a lot nicer a lot more pleasant a lot safer a lot more interesting.