>> they say beauty is in the eye of the beholder. and in fact, what we consider to be beautiful in nature, art, or music often differs from culture to culture, nationality to nationality, even generation to generation. but somehow there seem to be constants, commonalities in how

we as human beings see beauty. there's a sense of balance, of order, to beautiful things. but where does that sense come from? and what does algebra or geometry have to do with it? can we quantify the beauty of a butterfly? welcome to the world of symmetry. crystals, snowflakes, shells, geometric shapes, flowers, decorative arts such as the tilings of the alhambra palace in spain; they all share something besides what we might call "beauty," and that something that is imicit in the sort of regularity or self-similarity is symmetry, a symmetry we experience with satisfaction and pleasure in our everyday lives. often, that which eerience as symmetry is simply the final picture. but how did that beautiful, symmetric pattern come into being? with some close and careful

inspection, anyone can see that symmetry starts with a basic motif which is then manipulated in time and space. mathematicians call such manipulations "transformations," or, though this might be confusing, we also call them "symmetries," thereby identifying the outcome with the transformation tt got us there. for example, start with a basic shape or pattern, say, the number 4. if we flip it over like it's on a mirror, we're using a kind of transformation called a reflection. and that reflection creates what we call bilateral symmetry. this bilateral symmetry seems fundamental to nature. we see it our bodies and in the form of all sorts of living things. take the butterfly: its folded wings are the basic pattern. opening its wings, it performs a reflection, resulting in one of nature's most beautiful examples of bilateral symmetry. but visual symmetries are only the tip of the mathematical iceberg.

in fact, there are an uncountable number of types of symmetries, some we casee and some that we can't, some having to do with galaxies and others having to do with subatomic particles, some having to do with magic and others having to do with equations. but in the beautiful visual symmetries that nature and art present, we see everything that we need to know. symmetry is as much about the final pattern as it is about the motions that got us to that final pattern. and it's the math of those motions that we're now interested in. remember, we said an object or picture is bilaterally symmetric if we can draw a line through it, what we would call an axis of symmetry, and the two halves are identical in the sense that one half can be overlaid on the other perfectly, with each point finding its exact copy on the other side, each little fleck and curlicue matched up exactly when i fold the one onto the

other. the symmetry that the butterfly reveals and illustrates in opening up her wings is an example of a mathematical and geometric transformation that can be used to both generate and characterize symmetry. it's called a reflection for the way in which it creates an image that would appear by reflecting the original in a mirror. reflections take place outside the plane of the figure. see how we needed to actually flip our little icon or motif "outside" the page in order to make the mirror image? now, there are also symmetries and transformations that take place entirely in the plane of the image. the simplest of these is a translation. here we draw an imaginary line above or below our motif and simply slide across that line without rotating or reflecting it. notice that a translation will be completely described by how far you move and in what direction you move in. another example is given by a rotation. here, notice that the rotation

is characterized by how much you rotate as well the point around which you rotate. the closer the center of rotation, the tighter the curve that the icon traces out. now, with these three kinds of symmetries -- reflections, translations, and rotations -- we can generate other kinds of symmetries, too. a famous one is the glide reflection, a symmetry that combines a reflection and a translation. a flip and a slide equal a glide. now, using these symmetries, we can form all kinds of patterns. let's take a basic motif. we've been using the "r," so let's stay with it for a moment. start with a reflection, then add a rotation, and then maybe a glide. if we follow one symmetry by another symmetry, that is, "compose" one symmetry with another, we get another symmetry. and what we end up with can be, well, beautiful. we can compose symmetries along one dimension and end up with a frieze pattern. we can even expand the

symmetries to cover an entire wall. and if we allowed ourselves to go out to infinity -- and that's something that mathematicians love to do -- we can cover the entire two-dimensional plane with a beautiful and highly symmetric pattern. now, as we said, symmetries are both the motions that generate a beautiful pattern like this as well as the properties of some of these patterns. and when we look at the mesmerizing infinite design, we now see that there are certain transformations in our bag of symmetries that, when applied to the design, bring it right back on itself, like that. so there's a set of symmetry transformations that leave the entire infinite design unchanged, or "invariant," is what we say. we call this set the symmetry group of the design. now, the important word here is "group." a group is a set of symmetries that behave sort of like the good old integers. now, what i mean by that is i can take two numbers and add

them and get another number. similarly, i can take two symmetries, apply them one after another, and get another symmetry. i can take a number and find its negative, and when i add those two numbers, i get 0. and similarly, i can take a symmetry, do its inverse symmetry, and it's as if i did nothing at all. those simple properties define what we mean by a group. one of the most beautiful results in all of mathematics is the characterization, or classification, of the various possible groups of symmetries of these infinite planar or linear designs that we've been looking at. now, the former are called wallpaper groups, speaking to these infinite pieces of wallpaper that we've been creating, and the latter are called frieze groups. and while you might think that there are an infinite number of possibilities for these, the fact is that their structure is really highly constrained and that effectively, there are only seven frieze groups and seventeen wallpaper groups.

now, as we've seen, although these groups are finite in number, their possibilities for creating beauty seem endless. and in fact, they've been inspiring magnificent geometric artistry for centuries. this is the alhambra, a combined mosque, palace, and fortress built in southern spain in the 13th century. in many ways, it stands as a monument to the ways in which geometry and algebra were combined by islamic artists as a means of spiritual expression. the ancient religious commandment, "thou shall not carve idols for yourselves in the shape of anything in the sky above or on the earth below or in the waters beneath the earth" was an injunction against figurative art taken seriously by both arabs and hebrews. as a result, they developed a purely abstract and geometric art like that exemplified in the alhambra.

much of the mathematics we study today owes at least as much to the early islamic mathematicians as it does to the ancient greeks. in fact, many of the ideas developed by european renaissance mathematicians were first developed by islamic mathematicians four centuries earlier. beginning in the late 8th century, at the house of wisdom in baghdad, islamic mathematicians translated the greek texts -- like euclid's elements -- into arabic. islamic mathematicians were interested in both pure and applied mathematics. they used it in astronomy, geography, time-keeping, and even in the legal arena, where they used it to settleroblems in the division of inheritances. after the work of al-khwarizmi in the 9th century, algebra became a unifying theory. it allowed things like rational numbers, irrational numbers, and geometric magnitudes to all be treated as "algebraic objects," that is, abstract symbols that stood in for specific concepts.

it is from al-khwarizmi that we get the word "algorithm," a mathematical process followed by a computer. omar khayyam, born in 1048 and mostly known as a poet, was also a mathematician well aware of the power of combining algebra and geometry. khayyam wrote that "whosoever thinks algebra and geometry are different has thought in vain. algebras are geometries which are proved." whicbrings us back to the alhamb, with its surfaces covered in repeating geometric motifs, beautifully integrated with poetry carved into the walls: "you will say upon seeing it: it is a fortress, and at the same time, a mansion of happiness. it is a dwelling for the peaceful and the warrior. it is an artistic work that produces wisdom." whether the islamic designers of the alhambra understood the mathematics of symmetry and how is a matter of conjecture. what we do know is that the work

of islamic mathematicians in spatial symmetry and algebra foreshadowed the creation of group theory, a core concept of symmetry. the study of symmetry allows us to look at geometric things using algebraic tools. and that lets us both solve difficult problems and connect things that don't seem to have anything to do with each other. how? it's all about symmetry groups and something called invariance. so symmetry, group theory, the math of beauty, actually, is what it is, is ultimately a lot of algebra, and so we're lucky to have with us today rosa orellana, professor of mathematics at dartmouth college, who uses algebra all over in her work. so, rosa, thanks for coming today. >> thank you for having me, dan. >> well, we've been seeing a lot of beautiful geometric objects and trying to make sense of them algebraically, but let's sort of make it a little bit more rigorous with some good examples, shall we?

>> we should start with, like, a rectangle and try to explain the ideas, because it's easy to see it there. so actually, let's have a square. >> okay, squares are good because squares are highly symmetric. >> so now the idea, dan, is that we would want to have a motion in space that brings us back to the exact configuration that we have. >> okay. >> so, for instance, here is the square. you close your eyes and i do something to it, and you open it and you tell me if i did anything. >> okay. >> i can rotate it by 90 degrees, and this brings me actually back to the same configuration. >> that's right, that's right. occupies the same space and looks as if i haven't moved space at all. >> exactly. i can do 180 degrees. if you think about it -- 90, 180, 360 -- you can think 180 is 90 + 90. so if i do 90 and then i do 90 again, it's the same as if i

would have rotated by 180 degrees, right? >> that's right. so you're allowed to compose these moves, is what we would say. there's a composition. >> exactly. so this is kind of like when you are doing algebra, you start to realize, you know, "can i combine these motions so that i get a similar motion?" and in this case, we want a motion that leaves the square invariant, the same. >> yeah, and it's interesting to look -- i mean, in fact, the square specifies that there are only particular transformations that are going to leave it alone. like if i had rotated that thing by 30 degrees, or you did that, i would know you had moved it. now, the square is of course very symmetric, we've said, but you know, you can kind of reduce its symmetry, right? i mean, reduce the symmetry of a quadrilateral. >> so if you, for instance, colored the top half yellow and the bottom half blue, now we can ask the same question: what motion in space will leave it invariant, so that somebody

wouldn't be able to tell that we have moved. and now we notice that, for instance, if we do like a reflection, top to bottom, you can tell that i moved it because the blue went to the top and the yellow went to the bottom. >> of course, of course. >> but so now the only possible choice seems to be that we can reflect it from left to right. >> and this connection of numbers and symmetries is a real one, right? >> exactly. because in mathematics, one thing that we love to do is abstract from what we know. here we have the numbers, and this forms a set. >> so just a collection of entities, right, of things. >> you see, you're combining two things that are the same in the same set and making a new one, right? if you take an integer, like let's take -4, and you add it to 2, we get -2, which is again in

the set. and if you add 2 + 4, you get 6. >> right, so always getting some other thing on the line there. >> so this is an example of a binary operation, binary meaning "two." you take two objects, you combine them, and you get a new object that is back in the set. and this we call closure, like the operation doesn't take you away from the set. >> okay, so we have our binary operation, and it makes sense for this collection of integers in the sense that if i take two of them and do addition, i get another one, right? >> exactly. so now you might ask, what other properties does this operation have? so let's say i give you 1, 2, and 3, and i ask you, "add them, give me the results." if somebody's new to it, will go 1 + 2 and then add the third number to that result of 1 + 2, or go 2 + 3 and add that result to the 1. >> right, to 1, right. >> so, and the question is, do

we get the same answer when we do it in these two ways, right? and this is the property that we call associativity. so another thing that is quite an amazing thing, is there an element such that when i combine it with any other element, it doesn't change the value? >> right, right. >> like, you know, is there an identity element, something that doesn't change? >> an identity element, okay. >> yes, an identity element. if we add 0 to any other integer, we get whatever integer we obtain. so for instance, if you get 4 and you add it to 0, you get 4. >> you still have 4. so, 4 retains its identity. >> so, and even if we do it in either way, 4 + 0 or 0 + 4, you still get 4, and this is the property of having an identity. so now that you have an identity element, you might ask yourself, if i give you an element in the set, is there an element that

would give us 0? so for instance, if i give you 3, you might come up and say, you add -3 and you get 0. >> right, so every element on the line has its negative, and those are sort of specified by the fact that when you combine those two, you get this identity. >> this identity element, and this is the existence of inverses. >> in this case, an additive. >> an additive inverse, exactly, for the integers. so these four axioms have been recognized as being special. you start with a set, you add closure, associativity, you have this identity element, and you have this inverse for every element in the set. and these four properties are what forms a group. this is -- you know, is a set with these four properties. >> and it's sort of amazing that out of those four axioms somehow that you get an incredible richness, you know, from the wallpaper groups that we saw to, you know, groups in physics. we've been talking about groups in a very geometric way: moving

designs around, moving shapes around. but in fact, the birth of group theory was much more formal, right, sort of closer to the algebra that people think of being algebra, isn't that right? >> usually what people think about when you say algebra, they think high school, but the first person to use the term "group" was evariste galois. >> ah, a very famous mathematician, and we're going to hear his fascinating story right now. let's take a look. a firebrand radical republican who lived in the time of king louis-philippe, galois was just 20 years old when he solved one the most hotly contested problems in mathematics: the problem of solvability by radicals. and it goes like this: under what conditions could you find a formula for the roots of any polynomial in terms of the coefficients, using only the usual algebraic operations and application of radicals: square roots, cube roots, and so on. galois's quest was a powerful result: to be able to say for

any polynomial with integer coefficients both when it was possible and impossible to find one of these simple formulas for its roots in terms of the coefficients. galois's epiphany was to consider the symmetries of the roots of the polynomial. the young frenchman's answer became known as galois theory and is the basis of modern group theory and our understanding of the mathematics of symmetry. on may 29, 1832, galois is said to have worked through the night desperately composing his mathematical treatise summarizing six years of work. it was the night before he would be fatally wounded in a duel. with apparent foreboding, galois wrote, "i have no time." while the circumstances remain murky -- the duel may have been the result of romantic entanglement or political intrigue -- galois's legacy is the algebraic basis of our understanding of symmetry.

so galois discovered a set of conditions in terms of the symmetries of the roots of the polynomial that could determine if the polynomial could be solved by radicals. but the roots are numbers, and what could it mean to consider the symmetries of a collection of numbers? well, let's take a step back, all way the back to the simple example of the quadratic formula. really, the quadratic formula tells us two things. the familiar one is that from just the coefficients of the polynomial, i can write down the roots. but it also works in the other direction: knowing the roots, i can figure out the coefficients of a polynomial that they turn into 0. so galois's idea was exactly to adapt this backward point of view and add a very clever twist: he would use these roots as the basic ingredients to start making numbers. he'd take the roots, toss in all the rational numbers, too, and then see what numbers he could make by taking arbitrary sums and products.

like taking a pile of atoms and seeing what sorts of molecules you can make. decades later, scientists would begin to understand the properties of molecules by determining their invariance under symmetries that mix up the atoms. in a similar way, galois discovered that the ability to solve a polynomial built out of some roots depended on the invariance of those numbers created from the roots after permuting the roots. permuting: it means mixing up, or shuffling. and in order to see what we have is a group, let's actually think about shuffling, but card shuffling. here's a familiar old deck of cards, and notice that any shuffle or rearrangement of the deck leaves it invariant. in our new language, it's a symmetry of the deck of cards. notice that if i do two shuffles in a row...i get another shuffle, so closure. notice that for every shuffle... there is an inverse shuffle,

just the one that undoes the shuffle. not doing anything at all gives me the identity shuffle. and take my word, associativity works. so the shuffles of a deck of cards are a group, and it turns out that the structure of the particular "shuffles" that galois considered on his roots are what give the conditions for solvability by radicals. using shuffles to understand polynomials is a pretty good trick, but an even better one is using the algebra of shuffles to actually understand card shuffling, which is something that people have been doing for decades, culminating in the recent mathematical discovery that in order to mix a deck of cards by riffle shuffling... that is, it takes seven riffle shuffles to randomize an ordered deck of cards. this was proved in 1992 by persi diaconis and david bayer. diaconis and bayer identified a key signature of a shuffled deck

of cards: the number of rising sequences. let's look at an example. i start here with an ordered deck of 10 cards, riffle shuffle them once, and take a look. see how it's now composed of two rising sequences, two sub-sequences of increasing cards. and as i continue to shuffle, the order dissolves. after i do another shuffle, i expect four rising sequences, and then after that eight, and so on and so forth. the way in which permutations of a given number of rising sequences compose their algebra provides the key to understanding how the mathematical model of riffle shuffling works. this is just one example of what is in fact a very intriguing intersection of the world of cards, and even card tricks, with mathematics. but here's another trick. did you know that we can use symmetry to uncover the virtually invisible worlds of the atoms in a crystal? well, we can. let's take a look. >> so crystallography is the study of how atoms or molecules

are arranged in the solid form. my name is mary mcgrath. i'm the senior director of structural chemistry here at gilead sciences. our goal at gilead is to get safe and effective drugs to patients sooner. in the field of crystallography, symmetry is very important. symmetry tells you how each molecule in the crystal is related to the other molecules. so we need to know that information in order to get out our final picture of what our protein looks like. it's also very important in just trying to complete our experiments successfully. we need to know those relationships. what we end up doing is using math as a lens to refocus our

image. when we are able to provide a picture in crystallography of the protein target to the chemists, they are able to specifically design for that protein. so what ends up happening is you accelerate the drug discovery process. instead of taking ten years to come up with a potent compound, maybe now it would take three or four years. our process has many parts to it, and one of the things i really love about being a protein crystallographer and doing structure-guided drug design is that we use techniques that have really not changed all that much for close to 100 years, and we use other technology that's just changing all the time. so it's a really fun blend of the traditional and the cutting-edge. >> the uncovering of symmetries in a crystal are just one example of the use of symmetry

in physics. a much more mysterious fact is that laws of nature generally exhibit some sort of invariance, a fact proved by emmy noether, perhaps the greatest woman mathematician of the 20th century. a prime example is the law of conservation of momentum that comes from the spatial invariance of the laws of motion. fundamentally, the world works via symmetry. the symmetries of nature, the patterns of the alhambra tilings, the manipulations of mathematical equations, or a deck of cards. whether we're talking about symmetry as a property or as a transformation, it creates the emotional responses we feel in the presence of beauty, whether it's art, math, or butterflies. captions by lns captioning portland, oregon www.lnscaptioning.com