S1: this lecture uh, we are very pleased that the Undergraduate Mathematics Society, has uh sponsored uh this lecture uh to uh, uh explain about the solution of what is perhaps the, oldest unsolved problem in mathematics, and uh i want to remind everyone that there is a reception following the, lecture in the mathematics atrium which is uh in the south atrium in uh, this building <SS LAUGH> that way and up one left, so i'd like to introduce to you uh Polly McMahon the President of the Undergraduate Mathematics Society who will introduce our colleague Tom Hales.
S2: a few articles published in the last few months, <READING>packing challenge mastered at last. U-of-M professor solves four-hundred-year-old math riddle. mathematics proves what the grocer always knew. Kepler's orange stacking problem squashed. mathematician proves that shops know how to stack fruit. Hales solves the oldest problem in discrete geometry.</READING> who would have thought that a mathematical proof could receive so much media coverage? well the coverage may be unusual but it's far from undeserved. an expert in representation theory, analysis, algebra and physics, Dr Hales received his M-S and B-S degrees at Stanford in nineteen eighty-two. his PhD at Princeton in eighty-six under the Harold W Dodds Honorific Fellowship, he then went on to the Mathematical Sciences Research Institute to do post-doctoral research. and then to Harvard, where he was an assistant professor for two years under the National Science Foundation Fellowship. he completed the post-doctoral research fellowship at the Institute for Advanced Study in the following year. he was an assistant professor at the University of Chicago from nineteen ninety to nineteen ninety-three. in ninety-three he came to the University of Michigan as an associate professor. here he's a member of the math department Executive Committee, and chair of the Undergraduate Scholarship Committee. he has won the College of Literature Science and the Arts Excellence In Education Award and the Henry Russell Award for nineteen ninety-nine. he's here today with the proof of a problem that has stumped mathematics for four hundred years. a problem that seems simple at first glance. examined more closely, one can see that a proof of such a problem, actually involves over five hundred possible spher- over five thousand possible spherical configurations. thousands of crucial details. two hundred and fifty pages of formal proof. one hundred and fifty variables, combined into one equation. ten years of work. <SS LAUGH> a five-step strategy. three gigabytes of computer memory. a talented, graduate student assistant. and one brilliant associate professor. ladies and gentlemen, Mr Tom Hales.
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S3: uh, last week, the New York Times, ran an article about mathematics, with the title <READING>math emerges blinking into the glare of the pop, world.</READING> this article describes a new booming market for popular math books. uh it quotes an editorial claiming, that the present time is a golden age for popularization of mathematics. it attributes a surge in interest, to mathematics to the British mathe- mathematician Andrew Wiles, who made, mathematics headlines around the world, uh for solving, the most famous problem in the history of mathematics Fermat's last theorem. this mathematical work was popularized by the science journalist Simon Singh, who wrote a popular book, Fermat's Last Theorem, and also produced a T-V program on the subject that you might have seen. about two years ago, Andrew Wiles, collected a prize of, thirty thousand pounds, for the solution this old prize this is a prize that was um, established in the nineteenth century. uh, i personally have benefitted enormously over the past year from this recent rise in interest in popular mathematics. Andrew Wiles collected his prize on June twenty-seventh nineteen ninety-seven. the next day Simon Singh, the one who wrote the book about Andrew Wiles' work, wrote an article for the English science magazine the New Scientist. but as it turns out most of the article was not about the most famous problem in mathematics or its prize, instead the article opens with the words, <READING>with the most notorious problem in mathematics now solved an even older po- puzzle is drawing the crowds.</READING> in that article Simon Singh argues that a worthy successor for Fermat's last theorem must match its charm and allure, Kepler's sphere packing problem is just such a problem. <READING>it looks simple at first sight but it reveals its subtle horrors to those who try to solve it.</READING> of course i was delighted by the problem because this is the problem i've been working on for the last, several years, uh i have no doubt that it's the exaggerations of a journalist trying to popularize a new problem, um but, i was pleased with the article. <LAUGH SS> uh we_ Simon Singh and i uh corresponded a few more times, in August of nineteen ninety-eight last summer, i wrote him an email saying that i had, a packet of papers that i wanted to send him, um he wrote back saying if this is what i think it is, i can have something in the news within twenty-four hours. and so i told him to go ahead with it and um he, um, put it on B-B-C International, Radio and the Daily Telegraph. um so that was last August. um, today i'd like to talk about the solution to this problem in sphere packings the Kepler Conjecture, and i also want to talk about some related problems, in geometry, many of them unsolved. uh i've tried to prepare a lecture that doesn't involve any complicated equations rather i'm going to explain these different models that we have, here and by the end of the lecture you should understand, uh what each of these models is about and what they have to do with sphere packings... um, one of the remarkable aspects about this problem is how quickly, you can move to research-level problems, uh in in many fields you have to talk as fast as you can talk for an hour using very advanced mathematics to get to the frontier of the subject in this subject uh it's quite different um, i'll take as an illustration uh this last month's Orbit, Magazine they have a column Truther Than Strange, uh where it reports <READING>after ten years of research Thomas Har- Hales a professor at the University of Michigan finally figured out that the best way to stack fruit is in a pyramid</READING> and here's um <SS LAUGH> illustration... of that configuration. um, this configuration, of spheres it's often called the cannonball arrangement because it's what you see at war memorials it's also what you see at fruit stands. it fills about seventy-four percent, of space. so seventy-four percent is the amount filled up, by the cannonballs or the fruit or whatever it is that you're stacking, and then the holes in between the balls, fills the remaining space. so if there's one number to remember today it's the number seventy-four percent because that's approximately how well, this packing does, and the problem is to show that nothing can do, any better. um, in the introduction um the complexity of the proof was described, a bit, uh the fact that it takes a couple of months of computers running non-stop to check through all the possibilities, um let me say a bit, about the publication of this paper, um, after, so last August i said i finished but that's not at all the end of the process. uh before it's accepted by the mathematical community it needs to go through a long process of refereeing i submit it to a journal and, uh referees check over it. the journal where i submitted it made the unusual move, of organizing a workshop around this problem, to, orient the referees, to the way this proof works so there was a week-long workshop in Princeton in January where uh a number of experts were brought in and halfway through the conference the participants were deputized as referees and <SS LAUGH> uh i think there're seven or eight referees working on it, another couple of people working on the computer aspects of the proof and the hope is that sometime by about July, uh they'll be back with a verdict as to whether my argument is correct or not. so enough of that let's move to the frontier of, mathematics <STACKING TUBES> um this is about how long it takes to move to an unsolved problem in mathematics, um so here's, an unsolved problem... um i think everybody will agree that this is the best way to stack tubes, um at least, i bought these tubes at Home Depot and this is how they were stacked. um <SS LAUGH> so unless an unemployed mathematician is working there there's um <SS LAUGH> (xx) intuitive solution, um let me show a couple of slides here... so here we have the, fruit at the fruit stand... uh, here's the tube problem, <READING>show that the best packing of tubes is the same size as the honeycomb, arrangement.</READING> this is called the honeycomb arrangement because you see, that around each tube, there're six other tubes and you get a hexagonal structure that shows up, in honeycombs... um... let me move, to a couple of um misconceptions, that you might see, so let me talk about this model now. um, what this model is is it's taking, the pyramid stacking and it's replacing all the spheres by points so that you just get, the skeleton of the cannonball arrangement, and you see that it forms a pyramid. um, a lot of people think that since this is a pyramid the pieces also are all... pyramids or tetrahedra, but you start taking it apart and sure enough there are lots of pieces like that, but in the middle, is this other, type of piece that's hidden from view, when you start out. now this, if i put a sphere at each corner in this arrangement i get an ex- extremely good packing, but this piece hidden in the middle is an extremely poor, packing. and i think one reason that people think, that it's obvious that this is the best arrangement, is that you just see these pieces when it's stacked... together and you don't notice that this_ there's this big piece in the middle that is a very poor, type of arrangement so the thing to remember about this is there's this other type of piece hidden in the middle. um... let me um, describe, this particular arrangement. so this is what's called a tile. you have tiles that fill, say the, you know bathroom tiles, that are two-dimensional this is a three-dimensional tile, this is a piece, if i put these together i can fill all of space by stacking one right next to another. and there's not very much known about packings in general, but one thing that you do know is that if you can completely fill space that's the best, you can do. um, <SS LAUGH> and this completely fills space and let me try to, uh explain that, uh the best way to show you would be to have a whole collection of these and and, fill the room with these and show you that they fit together. but you can see it without, actually having a lot of pieces. uh if you look straight on... you see a square. okay, now if we take... uh here we have a number of squares and you see the squares can be packed together, to fill space. so i can put these, pieces this way and make an array of squares like i have in the slide here, and they fit together except you'll have something that looks sorta like an egg crate where it bumps up and down and up and down on the top level, but the peaks from one level will fit, into the valleys of the next, and you can fill layer upon layer with these things to completely fill space. if you're not convinced by that let me give you a second, way of packing these, if i don't look a- if i look at a different angle you see a hexagon. there are six sides, here, to this figure... okay does everybody see that? and so i can pack those together side by side to fill a layer and again the peaks fit into the valleys, and you can fill space with these things packed one against the other. okay so, what i want to explain is what these models have to do with the cannonball arrangement and sphere packings. what does this have to do with the cannonball arrangement? well if i, put a cannonball in each box, and then pack them and then take away, the boxes, i'm left with the cannonball packing. okay? so that's the significance of this shape. i put a cannonball in each one then i fill space with the boxes and then take away the boxes and i'm left, with, the cannonball packing. so this is called a rhombic dodecahedron but i think of it as a cannonball box. <SS LAUGH> <P :07> um, Kepler, made the observation, that the square does not exist, without the triangle and the triangle does not exist without a square. in connection with close packings. and what he means is this observation that there are two ways to put together this arrangement i can put it together as squares, or i can put it together, as hexagons which are made out of triangles and those, two arrangements, exist side by side. some people ask me, there're two types of pyramid, arrangements and which one is best? this one i said fills about seventy-four percent of space, but what about this one? does it do better or is it worse? well, Kepler's principle is that the triangle doesn't exist without the square the square, without the triangle, this has a base that's a trian- a square this has a base that's, a triangle. but one does not exist without the other, and you find that you can put these two arrangements together and they fit together one is really the same packing as the other just viewed from a different, angle. so you have the square packing, and the triangular packing, they both give about seventy-four percent of space filled... okay let me... get a few more slides here. um, so i went through all my old photo albums and most people come back from Africa with pictur- uh, Kenya with pictures of elephants and giraffes I come back with, fruit <SS LAUGH> uh this is to show you that, <SS LAUGH> people really do pack uh, <SS LAUGH> fruit, in this arrangement. in a market in Paris. <P :05> uh, i wanted to show this slide, because of the artichokes um <SS LAUGH> shortly after the proof of Kepler's Conjecture was announced i started getting, phone calls on my answering machine from Ann Arbor's, Farmer Market <SS LAUGH> saying that we know how to deal with oranges but we're having trouble with artichokes. uh here's an arrangement, that's just a piece of the cannonball arrangement. the thing to learn from this picture and this model, and this also is a piece of the cannonball arrangement it's where the balls have been made smaller so that you can see the structure. uh you might have to come up afterwards to count but if i take any one of these, balls in the middle and i count how many, neighboring balls that there are around it, the answer's always twelve. and that's one way to recognize this structure is that every one always has twelve around it and that's a characteristic property, of these close packings there's actually a number of ways to pack things with density seventy-four percent but they all have this property that there are, twelve around every one and if you count, uh it's a little hard to count in this picture but you see three... you see three here in front, and then there are six going around in a hexagon and then there are three hidden in back so that every one has twelve, around it. <P :05> um, i promised to say a word about applications of sphere packings, uh i don't know what applications there will be to my work if any but my work on sphere packings is just one small, bit of a really big research endeavor. uh people have estimated there have been about eight hundred research articles written in the last ten years on sphere packings and a related topic of error-correcting codes. and an application not of my work particularly but of the work of others is to the relationship between sphere packings and error-correcting codes. so i've, got a slide here that, illustrates how that works, uh at the top i have, um a sphere packing, and in the end we're going to get some binary information that's going to be useful for computers. so i take each sphere um, it's a little dark i've i've written in, the positions of each of the spheres but it's a little hard to see. so if we call this one point-zero, um i go over one unit and that will be position one and this will be over position two, and so i look at each case at how far, to the right the sphere is and how far up and i get a pair of numbers. describing the position, of the center, of that sphere. if i then look at, those two numbers, and ask whether those numbers are even or odd and each time it's an even number i'll write a zero and each time it's an odd number, i'll write a one. uh i get in this case this array, of numbers describing, the parity of the centers of these things so i get zeros or ones and then i look at the list of all the possibilities that i have, and in this case i either get O and zero zero, or one one. and that, is a very very simple example of what's called an error-correcting code. an error-correcting code, is a binary sequence of numbers, with a certain amount of redundancy built into it, so that... uh if you lose, the zero, in transmission, you still have this other zero that tells you, what it was that you were trying to transmit or if you lose one of the ones then you still have another one that tells you, what the information was so you build redundancy into this code, and that's the error-correcting property. and that's very useful whenever you have, um you're trying to transmit binary information, where there's noise so that you lose part of the information you can reconstruct the original, binary, input, because of this error-correcting property. in this case we just repeat everything twice. but if you take more complicated sphere packings and i mean, really much more complicated, sphere packings, you can get very interesting error-correcting codes, in this manner and, these are extremely useful in everything from modem design to encryption of data on compact discs. <P :06> okay so uh this is just a bit of an outline of some of the topics, that i want to cover, um i've given an introduction, and i'm going to talk a little bit about the history, some more about these geometric models, gonna say a few words about why this problem is difficult to solve, and a few words about the solution, and then towards the end i'm gonna talk about a second topic of foam... so here's, Kepler um, the history starts actually a little bit before Kepler, with the mathematician Thomas Hariot who, was, Walter Raleigh's, mathematical assistant and Walter Raleigh gave him, the homework problem of calculating the number of, cannonballs in piles, for him, uh but this sparked, Thomas Hariot's imagination. and he started thinking about not just cannonballs but he started thinking about atoms and what the implications for a- an atomic, theory would be, and he also started thinking about sphere packings and he was thinking of these as atoms more or less and, uh trying to understand nature, through the possible arrangements that you can have, for spheres... so this is Thomas Hariot um, we are indebted to him in many ways, he invented the less than and greater than, symbol, he popularized the equals sign, he was the first, <SS LAUGH> to use binary numbers, um he popularized the atomic, theory, in Europe at the time, and he started the mathematical theory of sphere packings. <P :04> uh so this is, Hariot's solution to, Walter Raleigh's problem, uh we start with one sphere at the top, and then one and one and then we have one and one and so we put two. we add the number of spheres above and we get two. i have one and two one plus two is three so i put three here i have two plus one, and i put below it three. in each case i add the two numbers three plus three, gives six, one plus three gives four, three plus one gives four, so he built up, a big triangle in this way, that gives the number, of spheres in a packing. and so here we see a cannon- this very small cannonball stack we'd have to extend it on down to get interesting formulas. but this triangle, um, well he was doing this long before Pascal but this is called Pascal's Triangle. and so Pascal's Triangle gives the solution, to, this problem, of counting cannonballs in a stack... uh here's Kepler's book, he wrote a book after corresponding with Hariot for a while Hariot, kept telling Kepler that he needed to adopt an atomic, theory to understand objects properly, and Kepler didn't believe in the atomic theory and he kept saying no and Har- Hariot kept writing back and i- insisting that he should, study the atomic theory, uh and finally he did in this book published in sixteen eleven... the question that Kepler asks, in this booklet, is why six? why does the number six keep popping up in nature, in various places? uh for instance he discusses the honeycomb, which is a six-sided arrangement he discusses snowflakes the, title of the book is The Six-cornered snowflake. or you have uh, naturally occurring crystals here's a big waterfall with hexagonal columns. why does this hexagonal, structure keep showing up again and again in nature? and the answer, that he was considering was that it was some sort of crystallographic explanation or atomic, theory that somehow the fact that you can fit six, tubes, or six balls around one perfectly had something to do with the six that keeps, showing up, in places like this. so he didn't answer these questions to his satisfaction, but this book was extremely influential in crystallography for the next two centuries. so in this booklet he's_ to try to understand this number six in in an atomic sort of way he starts making all sorts of sphere packings, and he makes this conjecture, that, the packing meaning the cannonball packing will be the tightest possible so that in no other arrangement could more pellets be stuffed into the same container. <P :06> uh people have worked on this problem, for a long time. uh people haven't been able to prove, this number seventy-four percent, but people have tried, uh the first results, that I know of were from a mathematician named Glickfield in nineteen nineteen who showed that you can fill at most eighty-eight percent, of space, and then somebody el- then he, proved eighty-three percent and then Rankin in forty-seven did eighty-two percent and then Rodgers in fifty-five seventy-seven-point-nine percent, and then there have been a few more small improvements, since then. <P :05> um, this problem took a very long time for me to solve, i had, two big fears in working on a problem that, would take a long time. uh the first is this that i would find out after thirty-seven years <SS LAUGH> that i was, trying the wrong approach. my other big fear, was, that when i was very close to a solution a day away a week away a month a way, somebody else would come along and say i have discovered a truly remarkable way, <SS LAUGH> to solve this problem, and here it is in this paragraph. <SS LAUGH> well that almost happened to me. um, i told you about, this popular article, that, Simon Singh wrote about my research the day after Andrew Wiles collected his prize. uh here's the picture from the art- article. uh, the helicopters represent the computers that i used. uh i guess this is one of these must be me i don't know, <SS LAUGH> (which) one, and then these hills, represent, all possible sphere packings that you can, imagine, and this computer technology is coming down with these linear hyperplanes, to produce bounds on the density of of packings. so, a week after this article was published, i got an email from a mathematician, at Oxford, who said i found a truly remarkable way <SS LAUGH> to prove the Kepler Conjecture. our file formats weren't compatible so it took about a week to find out, what, this idea was, <SS LAUGH> but it turned out, that the mathematician was proposing, let's see if i have the... that you start with one sphere in the center and then as i said you can fit twelve around it, in a tight way like this, and then, the idea is, well, this is the best thing you can do with twelve spheres around one and then you do another layer and you show that that's the best thing you can do and then inductively you build out this big, sphere packing, where you show, that at every layer this is the best possible, solution. so i was quite relieved when i heard the strategy because the strategy is completely wrong. um, <SS LAUGH> for one thing there's more than one way to pack twelve spheres around one in the center so you'll ha- you'll hafta look at this afterwards to see that these are really different, but there's more than one way to pack to- twelve spheres around one in the center so this induction argument doesn't, get going. but, there are more serious problems, and that is, there ar- so here again i have twelve around one in the center one on the top one on the bottom and then two rings of five in this arrangement. but there're not just, a finite number of arrangements, but there're an infinite number there_ you can deform, the spheres that touch the one in the center, in such a way that the twelve spheres are, touching at every moment. so there're infinitely many ways. of getting these to touch. and the best way, of arranging twelve around the one in the center, is not the cannonball arrangement as it turns out. there's, a smaller box, that you can make, uh so, this is a smaller box, than this box. but they hold a cannonball of exactly the same size. and, a student of mine Sean McLaughlin, he's a undergraduate here, he proved that this is the smallest possible box that you can build, not this. and so there's no way to start out_ the problem is that these things don't fit together into good packings but, this, and and not the cannonball is the best way to start out and this is the flaw of this Oxford mathematician's reasoning. so some people wondered, in fact Newton wondered, it's a famous controversy well if there's so much room when you have twelve why can't you get thirteen to fit, around, the one in the center? with these packings you only have twelve around, so here i have an arrangement with thirteen, spheres fitting around the one in the center. there's one in the mi- middle and then uh i've got, uh four here and then the rest down here. got another model of it here, is a model, of thirteen. uh let's see, so i've got one on the top and then i've got three rings of four for a total of thirteen touching the one... in the center. and here's another picture of it, the one thing you learn in mathematics is not, to trust, models until you have a proof, Newton said that it was only possible to get twelve to touch and his_ uh he was debating this with ma- the astronomer Gregory who said that it was possible, to get thirteen to touch, well it turns out that Newton was right and Gregory was wrong it is not possible to get thirteen to touch these are all illusions this is just a computer graphic fuzzy picture uh <SS LAUGH> this, is a styrofoam model, where the styrofoam balls have been crushed <SS LAUGH> so they fit in and touch the one in the middle, this is a wood model, but <DEMONSTRATES> if you shake it <SS LAUGH> you you can hear the rattle of the ball in the center so it's, possible to get twelve to touch, it's almost possible to get thirteen to touch but it was finally settled in the nineteen fifties that it's impossible to actually get this thirteenth sphere to touch, the one, in the middle. um let me say why this problem is hard... um if i start out, with a, a random sphere packing you have to show that the cannonball arrangement's the best one. which means that you have, to show, that all possible other, arrangements are worse. and if you take, this and pack 'em down so they can't move anymore you get one arrangement, and if you mix 'em up again... and pack 'em down again you'll get another good arrangement, and i don't i don't have any conception of how many possible, good arrangements that you can get by mixing 'em up and pushing 'em down until they don't move anymore i don't know whether it's millions or billions or what even with just a small number of balls like a hundred. but... let me say that, in calculus... you learned to do an optimization <P :05> like this you have something that you're trying to optimize and you're given certain, what you do is you take the derivative and set it equal to zero you find_ and then you check these points and you find out which one's best. but when i have something like this fish tank full of balls, there are hundreds of_ well there's a hundred balls or so and who knows how many possibilities, so the optimization problem is going to look something more like this and what you, are taught to do in calculus is just completely ineffective when, you're faced with a large number of possibilities like this... let me say a few words about the solution. um... in my mind there are two main ingredients, to the solution that made it possible. uh the first was designing a new kind of box, to hold the cannonballs, uh this, right here, is one example of the type of box, that i used, in my solution. uh you shou- i didn't finish this model th- you should actually imagine, one of these things, stuck in the middle of this so, just big enough to hold, the cannonball, and this is one possible, puzzle piece or box and, one of the main steps of the proof, was to dine(sic) design about five thousand, different boxes, with the property that any good packing no matter how strange, it looked had to contain, uh lots and lots of these boxes. okay? so the design of the new type of box was, one of the most difficult parts, of the solution. uh the other part... that i think was very critical was finding that this solution was governed by linear, equations. you_ when you start designing boxes you replace... very rapidly you don't talk about the cannonballs anymore you start talking about these tiles and these different pieces that fit, together. and... what you find is that when you when you no longer have spheres, but shapes like this all the equations become linear. nonlinear problems are extremely difficult to solve but, you have a bunch of linear equations that describe these shapes, and you have constraints showing how these pieces fit together, those are all linear equations. and you can solve linear equations very rapidly, uh let me just show you one example i promise no equations but here's, um, here's a typical file i have maybe a hundred and fifty variables and two thousand equations describing one possible box and there are all these thousands of boxes and then i do a linear optimization because these are all linear equations this is about a twenty page file and a computer the computer on my desktop can solve this in uh a few seconds without any difficulty so once you replace the nonlinear equations by linear equations, the solution can come very rapidly. <P :05> of course when you turn to computers you worry about errors, uh the referees are still checking my proof, i received the following email, <READING>there is an error in your proof on page two hundred and forty-nine and a bug in your code in gigabyte sector five. other than that i think your proof is more credible than the Berkeley proof.</READING> um, there's a, there's a Berkeley professor who claimed to have proved this problem in nineteen ninety, the experts never accepted his solution, uh but he's now writing a book on his proof and he's never backed down so there's a bit of a a controversy there... um, so i've, finished now with the main part, of my talk the part on cannonballs, uh i didn't want people to think that all these old conjectures have the solution that you expect, and so i'm gonna say a little bit about another problem foam, here at the end, and, if you take, these boxes, and you pack 'em together and take away the boxes then you're left with the cannonball packing. but, if you take... these boxes without the cannonballs in them, and pack them together, you just have, the boundaries of these pieces filling up space, and if you make these spaces soft then you have something that looks very much like a foam, rather than a cannonball packing so you've gone from a very hard problem to a very soft, problem <SS LAUGH> that's very closely related to the one that you start out with... so my, final topics i'm gonna talk a little bit about the honeycomb problem, which is the two-dimensional problem the Kelvin problem which is the three-dimensional problem, i'm gonna describe an example and then i'll move to a conclusion. <P :06> um so, this, honeycomb shape, i can think of it as a sphere packing problem, and this is the best possible sphere packing, but if i look at the boxes around the spheres, instead, then i have a hexagonal, grid, and you get quite a different problem there, the problem there is to show, that, the way, to divide up, the plane, into regions of equal area, is obtained by this hexagonal, pattern the best way to do it, meaning, the smallest perimeter is obtained, in this way... okay so the honeycomb problem, is to show that the honeycomb is the best way to divide the plane into sve- cells of equal area. as i understand it this is actually an unsolved problem, in mathematics i_ it's just incredible to me that it hasn't been solved but they've solved it under very restrictive hypotheses, but the general problem is still unsolved and it goes back to antiquity in in certain respects people were talking already about, packing together triangles and and squares and hexagons and saying that the hexagonal, packing was best, so here's a good unsolved, problem that i i don't know feel people must have just forgotten about it or something because i can't imagine it's that difficult a problem but it's_ it hasn't been solved. <P :04> uh so in three-dimensions when you strip away, and you make the cells soft instead of hard you get a foam (in) space, and the corresponding problem in three-dimensions is also unsolved, you want to put together volumes, equal volumes, in such a way that you minimize the surface, of the cells. <P :06> uh, this is how far we are on solving the problem, this uh was done by a few researchers a few years ago, they solved the problem when you have, two boxes or two cells two pieces of foam. uh this is called the double bubble problem, <SS LAUGH> and they showed that this is the best way to break up, space into two equal volumes in a surface-minimizing way the interesting thing about the solution is that it was also a computer solution and it relied on interval arithmetic which is one of the methods that i used a great deal, in the proof of Kepler Conjecture so there are certain points of contact in the method of, proof... so, the three-dimensional problem has a conjectural answer, which is, uh it was obtained_ this was obtained by Lord Kelvin. um he said take these things, and pack, them together so you tile with these pieces to see that these things tile if i hold it this way you see that i get an oc- octagon. everybody see the octagon? so, octagons well they almost, tile space but you get these squares that are left out, but if you notice, that this octagon has a square right here in front and that plugs the hole, so that you can build up layers of these things to fill space... so here's the second layer, plugging, the hole. and Kelvin said, that he thought that this arrangement, was, the solution to the problem. that if you want to, pack equal volumes in a surface-minimizing way, you should use this, and to prove his point, he made a set of bedsprings, uh illustrating the arrangement uh you really have to look carefully so this was actually constructed by Lord Kelvin. it's in a museum, now, but if you look very carefully you can see that's one of the squares for instance, that corresponds to a square, in this, diagram. so uh people believed for a long time that this was the best way, to pack cells together... uh Herman Bile for instance in his book on symmetry says that he was inclined to believe that this was the best possible, arrangement... uh but, two physicists shocked the mathematicians a few years ago. when they constructed a counterexample to Kelvin's, conjecture. uh the two physicists or a fellow named Weir and what they, did with their ke- so this is a picture of it and, uh here's a model of it. they used two different types of cells, that have the same volume and, it's, it takes a while to see but you can fit these things together i put, for each yellow box, i put a whole bunch of these things around it and they fit together around the yellow box and then i can continue out and fill space, with these things. and, they have the same, volume but you get a smaller surface when you do it with this construction, than with this. so uh this really took, mathematicians by the surprise, but now nobody has any idea what the correct answer should be because they don't dare conjecture that they've found the best possible answer they just know that what Kelvin thought was good is not so good... so this is the Kelvin problem, um, i've_ i don't know if you've noticed i've been putting stars by these problems saying how good a problem i think they are and this is a ten star problem, <SS LAUGH> to show that the best the surface-minimizing way, uh to break space into cells of equal, volume. you want to determine the best way, to do that. uh i should say that, Kelvin called this thing a, tetrakaidecahedron, and everybody has condemned Kelvin ever since for that term, uh, Coxeter calls it an outrageous, thing to call this <SS LAUGH> and perhaps that's_ maybe it was doomed to... uh die according to a counterexample... so, i want to conclude with a, few words about Hilbert and his problems. Hilbert in nineteen hundred, produced a list of problems, where, uh that have been extremely influential in twentieth century mathematics. and his idea was that these should be problems that are easy to state problems that anybody could understand that you really aren't explaining your theory until you can explain it to somebody, in the street. so, that was nineteen hundred now we're coming on to a new century, and everybody has a list of problems that they want to make, for the new, Hilbert problems i'm sure most of the mathematicians in this audience have their favorite problems that they'd like to see on the list for the twenty-first century. um but i'm giving the lecture today so i'm gonna propose mine, <SS LAUGH> and that is the Kelvin problem i think it's uh, a really great, problem, it's very simple to state you're just packing equal volumes in a surface-minimizing way, it's easy to understand, and so if uh, Kepler's problem which was included on the eighteenth of Hilbert's problem is to be replaced by another problem it'd be my, preference to see, Kelvin's problem, there. uh so this is just what Hilbert said as part of his eighteenth, problem. the eighteenth problem is a long problem uh with several parts to it... uh so at the end i'd just like to give credit to a couple of people, on the right is Sam Ferguson a graduate student who helped me a great deal with this project his thesis solved an important step, in the solution of this problem and he helped me a great deal with the computer work, as well, and on the left i'm there in the middle and then on the left is Sean McLaughlin the undergraduate who proved um, this what's called the dodecahedral conjecture, that this shape gives the smallest possible volume, for a box in a, sphere packing. um so i'll just close, with a, statement that, uh sphere packer Neil Sloane made at the international congress last year, <READING>in short many beautiful packings have been discovered, but there are few proofs that any of them are optimal and that gives a challenge to all of us to prove some theorems uh, corresponding to all these conjectures.</READING> um and another person involved in sphere packing Laslo Fejes Toth, uh he spoke about the sphere packing problem, but then he spoke about another problem called the covering problem which is very closely related to this Kelvin problem. and he said <READING>still worse are the prospects of solving the co- covering problem, and yet we have a hope that the human heart is strong, in will to strive to seek to find and not, to yield.</READING> so i think i've explained, all my models now, so with that i'll quit. thank you very much.
S2: um if anyone has any questions i'm sure that Professor Hales would be happy to entertain a few brief questions, and um, don't forget there's also the reception, right through those doors to the right, afterwards, thanks, for coming everyone
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