 Hili-significant developments in math are almost always those in which ideas and techniques from one set of mathematical sources impinge fruitfully on the same thing from another set of mathematical sources. These abstract ideas often find unanticipated applications inside and outside mathematics. I want to talk to you about one of these powerful and ubiquitous concepts, that of symmetry. The word symmetry goes back to the ancient Latin and Greek and refers to a sense of balance or proportion. But mathematicians define symmetry in a more technical or precise way. Symmetry is an intrinsic property of a mathematical object that causes it to remain invariant under certain classes of transformations. So when you see sa a picture that's symmetric, it's because a certain transformation, for instance, a reflection leaves the picture exactly in the same way as the original. The mathematical study of symmetry is formalized in the extremely powerful and beautiful area of modern algebra called group theory. Mathematicians recognize the power of symmetry, but so did physicists. According to the German mathematician Hermann Veil, all a priori statements in physics have their origins in symmetry. Really, symmetry is at the heart of relativity and quantum physics. For Veil and his contemporaries, laws of physics have to possess certain symmetry properties. Surely, this is what Einstein had in mind when he said that the only physical theories that we are willing to accept are the beautiful ones. In physics, the female mathematician Emmy Noether is well known. There are only a few mathematicians with female mathematicians who have made a big mark in mathematics development. Emmy Noether was one of them. Noether's principle states that each symmetry of a system leads to a physically conserved quantity. For instance, when you talk about the conservation of momentum, you're really referring to symmetry under translations. If you want the conservation of angular momentum, then you're talking about symmetry under rotations. Symmetry in time corresponds to energy conservation. According to the physics Nobel laureate Steven Weinberg, symmetry properties dictate the very existence of all physical forces. For instance, in physics, there's this huge collider in Switzerland called the Large Hadron Collider from CERN. It's the largest and most expensive scientific and technological project in the whole history of science. What are they doing at CERN in the collider? They're sending out particles at very fast speeds, close to the speed of light, hoping that when these particles collide, they will understand the underlying symmetries behind those particles. And this has been a hugely successful project. In chemistry, when you study the molecules of a crystal, you will find that they're arranged in very symmetrical configurations. The formal development and origins of symmetry, the study of symmetry, dates back to close to 200 years ago. Let's go back to a day May 30, 1832. A young man was killed in a gunfight. The day after, this young man died, and this young man was Everest Galwa, who did not even reach his 21st birthday. Galwa developed the abstract branch of mathematics called group theory, which is now recognized as the mathematical study of symmetry. Now you have to understand, Galwa was 20 years at that time. And more than that, he wrote all his work and results on symmetry and this new branch of mathematics while in prison. The years that Galwa lived was during the decades after the French Revolution. And people were fighting over the monarchy and the civilian government. You could think of Galwa as a student activist who was jailed two times at least for his revolutionary activities. But Galwa was a mathematician and he was working not really about symmetry, but he was solving equations just like most mathematicians at that time. What was Galwa doing? He was trying to solve the fifth degree polynomial equation. Now remember, 4,000 years ago, the Babylonians already discovered the quadratic formula. This is the formula which gives us solutions for second degree equations. The Italians in the 16th century discovered solutions for the third and fourth degree equations. So mathematicians at that time were naturally working on the fifth degree of polynomials which begin with x to the 5. And what Galwa did along with another mathematician, another young mathematician named Abel, was to show that no formula could exist that could give you the solutions for a fifth degree equation. Now what has that got to do with symmetry? Well, Galwa realized that the roots or the solutions of an equation are symmetric in the sense that they're not random but they're governed by certain transformations that can send one root or one solution to the other. So they're symmetry in the zeros of a polynomial. And that is the beginning of this branch of mathematics called group theory. Now today we can apply group theory to various areas of math. For instance, most people will think of symmetry when they see pictures. A triangle and a hexagon. These are really symmetric figures. Triangle with three sides and a hexagon with six sides. But what makes those objects different for mathematicians? It's because their symmetries are different. How different are these symmetries? Well, let's look at the triangle. What can you do with a triangle? Mathematicians say that an object has symmetry if the object remains unchanged after certain transformations. What transformations are these? Well, you can rotate a triangle by 120 degrees by 240 degrees or by 360 degrees bringing you the triangle back to its original position. On the other hand, you can reflect the triangle. For instance, if I fold the triangle across the vertical line passing through its top vertex, then I get exactly the same picture. So all in all, this triangle has six symmetries unlike a hexagon which will have 12 six rotations this time and six reflections. So in that sense, while both objects are symmetric, they are different mathematically. When you consider those six symmetries of the triangle, you can actually combine them. For instance, I can do a rotation by 120 degrees and then I can fold across the vertical axis. And then you'll see that the triangle still remains the same. And this combination of two rotation and refraction just gives you another one of those six movements. In this sense, those six movements form a structure called a group. And that's the origin of the term group theory. Group is just a set with a certain operation satisfying some nice properties. Think of the numbers. You can add numbers, the sum is again a number. You can add a number to zero, it gives you back the original number, giving you a so-called identity element. You can add a number to its inverse. For instance, number five added to minus five gives you zero. So these are the rules that are required for a structure to be called a group. Eventually, the abstract ideas of Galawah led to this modern notion of a group. Today, students of mathematics in the university study at least two semesters of group theory and continue studying the same subject matter when they go to graduate school. Group theory is now a fundamental course in any mathematicians' education and training. And not only mathematicians but chemists, cryptographers, statisticians also all use group theory. I want to continue this presentation by talking about various facets of symmetry. In geometry, for instance, we can discuss the types of movements which cause an object to look the same. So in euclidean geometry, symmetries are expressed by isometries. These are just functions that preserve distance between points on the plane. So if I have a picture of a cup, I can translate that cup to another position on the plane. No distances are changed, no angles are changed, essentially preserving the shape of the cup. So that's what happens because we applied a translation which is one of those allowable transformations that preserve symmetries. But there's also reflections about lines, rotations about points, and you can even combine a reflection and a translation and a reflection to get a so-called glide reflection. A glide reflection is what you get, for instance, when your left footprint transforms to your right footprint. So the left foot moves forward by translation and then gets flipped to the right side by a reflection. In mathematics, we can study symmetric patterns. And these patterns usually turn up in art, in designs. For instance, finite patterns are used when companies create logos or designs. And even religious symbols are very symmetric. On the other hand, we have so-called freeze patterns. We can think of freeze patterns as one-dimensional patterns moving either to the left or to the right, to infinity, repeating a certain pattern, just like a long wall or a long carpet with the same pattern repeated. Mathematicians know that there are seven distinct freeze patterns. Anytime you see a one-dimensional pattern going left or right where patterns are repeated, it falls into one of seven different classes of freeze patterns. Now, as you go to the next dimension, two-dimensional patterns are what are commonly known as wallpaper patterns. You find wallpaper patterns, of course, in your houses, on the floor, in the form of tiles. And these are just patterns that are repeated in two directions. Maybe going up and to the right or maybe at a certain angle. Mathematicians have classified all the wallpaper patterns. There are only 17 wallpaper patterns. And these ideas of using patterns to decorate have been used throughout history. The ancient Romans started decorating their baths and their temples and their palaces using tiles. Another word for tiling is tessellation, coming from the Roman word tessella. There are 17 different wallpaper patterns in the sense that a particular pattern is different from the other because there's a certain symmetry caused by a rotation or a translation, a reflection that is not present in the other one. So try to look around you and try to look for those 17 patterns When our local math weavers in the north, in the Cordillera, or in the south, or in other places of the Philippines do their bunnies and their carpets and their fabrics, they are actually using ideas of symmetries, just like the 17 wallpaper patterns or the seven freeze patterns. In the Philippines, some Philippine mathematicians study the mathematical properties of these works of art. And unfortunately, however, I don't think the seven freeze patterns appear in our carpets or our rugs. I think they found about four or five. There is a place, however, in Spain where all 17 wallpaper patterns are found. All paper patterns are mathematical because you can study the various geometric shapes that get repeated, that get moved around and even culturally, for instance, you can look at how those patterns are conceptualized or designed. Studying the Philippine mats or fabrics show that our weavers, our local weavers, actually incorporate many elements of the world around them and of their culture. You see patterns depicting mountains, insects, plants in all these local mats and fabrics. In Spain, there is a place that we call the Temple of Symmetry. And this is the Alhambra in Sevilla in southern Spain. The Alhambra is a structure which was occupied by the Moorish conquerors of Spain and Western Europe. And they stayed in that place for around 800 years. And in that structure, you will find all sorts of tiles and patterns on the walls, on the floors, on the ceilings. All 17 patterns have been found on the walls and grounds of the Alhambra. The Muslims, the Islamic artists, developed these designs because in the Koran, it is not... You cannot depict the human face on paintings and that's why many representations come in the form of geometric patterns. Also, any animals cannot be depicted. So, try to visit the Alhambra and try to discover the many patterns there. Today, the idea of patterns and art have been developed. Not only flat walls, but also curved spaces can be tiled. For instance, if I go out of euclidean space and go to something called a hyperbolic plane, then I can also tile them. The Dutch artist Moritz Escher used this technique very skillfully. He was not a mathematician, but he incorporated many ideas of mathematics in his artwork. I want to tell you about a certain pattern that's very interesting. Patterns or wallpaper patterns in particular follow a certain principle called the crystallographic restriction. Scientists and mathematicians know that in crystals, that patterns can only admit rotations by 180 degrees, 120 degrees, 90 degrees, or 60 degrees. This is the restriction. It says that a crystal cannot have rotations of other angles other than these. For instance, you cannot have rotations by 72 degrees or five-fold rotations. However, there are patterns that can be constructed which display this five-fold rotation. Penrose tiles are some type of repetitive patterns which can cover a plane. The difference between a penrose tile and a wallpaper pattern is that a penrose tile may use more than one motif or design or pattern which is repeated. Now, penrose tiles admit five-fold rotations. So this is not a wallpaper pattern. And scientists ask if such a pattern such as a penrose tile exists, is there a physical object that corresponds to it in the same way that a three-dimensional crystal corresponds to a three-dimensional wallpaper pattern? Well, it turns out that the answer is yes. And these objects are now known as quasi crystals or quasi crystals. So quasi crystals are materials that are found in nature. Initially, they were synthesized in the lab, but now they are known to be naturally occurring. Its existence has been confirmed by x-rays and by chemists. So quasi crystals can be used some sort of coating material just like Teflon. Now, why am I telling you about quasi crystals? Because, well, when the scientists who discovered quasi crystals announced his discovery, that scientist was Dan Schechtman. He announced his discovery to his colleagues in 1982. He was ridiculed by many of his colleagues and was even asked to leave his research group when he announced this no idea or discovery of quasi crystals. Well, it turned out that Dan Schechtman was correct. Even just a few years after he was humiliated by his peers, people started realizing that quasi crystals were indeed correct. They were objects that displayed this five-fold rotational symmetries. Schechtman's vindication really came in 2011 when he received the Nobel Prize in Chemistry for this discovery. So science is like that. Ideas sometimes, new ideas tend to be surprising, sometimes even looks foolish, but in the end could be real important discoveries. In the arts, you see many manifestations of symmetry, of course, in paintings dating back to the early Persians, Egyptians, down to the Renaissance artists like Michelangelo, Raphael, Leonardo da Vinci who framed their works using highly symmetrical patterns. In literature, poems display a lot of symmetry in many literary forms like the Japanese Haiku or Tanka and either our own Dalit or Tanaga. You see a lot of symmetry in the sense of number count and rhyme and rhythm. If you've read the works by Louis Carroll, Alice in Wonderland in particular, you'll find many evidences of math and symmetry. Twiddle D and twiddle Dumbered like mirror images of each other repeating and doing the same things. Well, that's not surprising because Louis Carroll was a professor of mathematics and understand symmetry very well. In architecture, ancient structures were highly symmetric like the Colosseum in ancient Rome and the Parthenon in ancient Greece. In biology, symmetry is very important. Let's talk about viruses. Viruses are actually contained in a certain capsule called the viral capsid. And classification of capsids is now can be done by looking of capsids tell you about the different properties of the virus. And these capsids turn out to be highly symmetric structures like an icosahedron. Or a ragel capsule or a spiral shape capsule. If you understand the geometry of those capsids, then you'll also learn a lot about the properties of the virus. In medicine, a lot of principles of symmetry are employed. Instead of cutting open in an invasive procedure to get gall stones or kidney stones out of person, doctors now use lasers and shock waves to blast those gall stones. And they do that by using the reflective principle of an ellipse. They bounce the shock waves to the sides of your bladder so that it comes back hitting exactly your gall stones. When a doctor inspects a lesion or a mole, sometimes they will say that those which are very symmetrical usually tend out to be benign. But when a lesion is asymmetrical or irregular shape, maybe one needs to be more cautious or concerned. Millenia of years of evolution has led to many life forms which are highly symmetric. You have starfishes that are radially symmetric. You have the wings of a butterfly or the shape of a beetle and even the human form which are bilaterally symmetric. Maybe many of our life forms are symmetric because they're pleasing to the eye. But sometimes it's not, it's more than that. There's research for instance that bees prefer more symmetric objects, more symmetric flowers in particular. Not really just for the shape but for some properties due to that symmetry. There's a lot of research claiming that maybe humans have this bilateral symmetric form because, well, evolution wants you to choose maybe mates or partners that are healthier, that can cope with the environment better and a symmetric shape or symmetric features are evidences of these better capacity to survive. There's another side of symmetry and that's the opposite. The absence or the violations of symmetry. Some artists know that for instance when they take pictures of objects they don't necessarily put them in the middle. They put them at an angle or at a side giving the picture more depth. The asymmetric picture is sometimes more beautiful. Talk about the leaning tower of Pisa. If the Pisa were straight and not leaning then it would be interesting or some churches or structures in Europe display a lot of asymmetrical shapes. There was even a period before the Renaissance when such asymmetric shapes were in vogue. Although the scientists like Einstein and mathematicians and physicists really recognized the importance of symmetry the great biologist Louis Pasteur had a different idea. He said the universe is more asymmetric than symmetric. He said the physical forces he called them cosmic forces at that time really preside in the formation of these theories these molecules these evidences of asymmetry in the universe. And maybe Pasteur was also correct because today there are many things in nature that are asymmetric. A fiddler crab has one big claw and a small claw. That's how evolution has produced this crab. Maybe the big claw to pound objects. Think of eating crabs. You need something to crush the shell but you also need a smaller claw to pick out the little pieces of meat. Flatfish we have this in the Philippines. Flatfish tend to they're probably lazy I don't know but they tend to go to the bottom of the ocean. Their eyes of a fish are on both sides of their face but flatfish eyes move to one side of their face. So it's a very strange fish. I don't know how it swims but well it's not like other fishes. There's an animal a sea animal or sea mammal called a narwhal. It has when you see it it looks like it has horns spiral horns on the top of its head. Well a closer look will show that it's not really a horn it's a tooth it's a cuspid or an incisor and not only that it's just on the left side of the face. It can grow to 10 feet in length. Scientists do not know what this left incisor this long horn is used for. It's not used for hunting or spearing or killing. So it's just one of those mysteries that we can see in nature and this mystery involves a very asymmetric property. There's another principle of asymmetry which we call symmetry breaking. Sometimes a certain physical state is so symmetric or even that you want to destroy or break its symmetry. This is very important in some physical phenomena but you can think of even say a fetus a fetus which starts from maybe a ball which eventually will grow arms and digits. This is some form of symmetry breaking. Fractals which you see in many designs really exhibit symmetry in the sense of patterns being repeated getting smaller or bigger but still looking the same way. Fractals are now being used to study these changes for instance interest changes currency changes we still know I have to learn a lot about this strange objects called fractals. I tried to give you an idea of what how mathematician study symmetry. Understanding symmetry its presence absence or violations present theoretical and practical issues that lead to a deeper understanding of nature. When we study symmetry we can continue to know more about the world and its many mysteries. Let me end this presentation by quoting from the great philosopher Ari Stotel who said the mathematical sciences exhibit order symmetry and limitation and these are the greatest forms of the beautiful. I hope you enjoyed my talk today about symmetry try to look for it try to study it try to understand it thank you very much.