John Conway Distinguished Lecture - The Symmetries of Things

this is a first for this room English pleasure to introduce our speaker today mr. Conway I think the national treasure and nobody I don't think we need to introduce him so I try to say a few words about just for maybe the people here that you may be students so John Horne in Liverpool England and actually his father taught chemistry the deals two of them and I would guess that John and I have forgotten which / they were I can find out by the way later the Beatles both to school about this so there are too many awards that Professor Conway received I'm going just to say fellow of the Royal Society boy a prize and across their weak ties many others work in many areas of mathematics and physics computer games here in geometry geometric topology look theory number theory algebra algorithmics and critical theories and he is the gentleman and professor of mathematics and physics not of physics Nevada mathematics a lot of physics yeah but I am distinguished John for Norman distinguished professor but well on the on this visiting card thing it says a dist professor and served my friends or my enemies whoever they are say what does it mean disturbed distracted distraught ooh never so his jump from them and distinguished professor of mathematics I think if I would like to name one mathematician that using the same mathematical welcome to that type of mathematical research like John Konami I don't think I can have another name other than John he worked from so many areas making so many fundamental contributions so I thought about a number because general likes numbers so I said what number could be associated with John so John phenomena in the 50s said that mathematics is too broad now for one person to know all of it no more than one quarter can be done by a person he was referring of course to him I guess is that John is jazz number it's probably similar I'm not sure can do it evaluated without further ado yeah well okay that's too much really but pretended yes so where are we so the title of this lecture is some kind of attempt to drum up enthusiastic it's the same as the title of the book namely the symmetries of things and what do I mean by things you will ask well I'm thinking of quite homely things like well let's say chess is this a nice chair and now this has a very simple kind of symmetry oh yes what do I mean by symmetry that's probably a good thing to say first well I mean some kind of way I'll take it well let's say a geometrical congruence that puts the chair or whatever object we're talking about back in the same place that you start it well there are two for this chair one of them is the identity you just leave it where it was and the other is to reflect in this vertical plane here okay so that's the kind of symmetry that's most common we all have an approximation to that symmetry you know we have not only a left arm but a right arm and so on which is roughly the same as its mirror image not exactly you'll notice if if you see yourself you know you most often see yourself looking into a mirror so you see the reflected self not the real self and if you have occasion to see what your real self is like you get a little bit of a shock because it's not the picture you normally say okay well symmetry has long being of interest to all sorts of people if you're building for instance a bridge a really big expensive bridge you know then they usually have fourfold symmetry there's a reflecting plane in one direction and a reflecting plane in the other and if you sort of cut the bridge along those planes you get four quarters and each quarter is congruent to all the other quarters and that saves money if you want to save money anyway because you only have to do a quarter of the design work really you know it took a quarter of the calculations and so on and in solving mathematical problems symmetry often you know enables you to save effort in some way because you can say well without loss of generality we discussed the left-hand side roughly speaking or whatever it is so similarly it's a very practical subject that's not really the reason I'm interested in is it's a very beautiful subject you know in general my typical way of thinking of a highly symmetric object is to think of some kind of gem which has many facets but you know a beautiful sort of rose shape or something in okay in chemistry in geology particularly symmetry arises in the ways in which things crystallize and this was noticed a long time ago you know if you have alum crystals you can tend to pick out pretty close to regular octahedra nice shape but then there's another way when you think of crystal symmetry when geologists or mathematicians think of crystal symmetry they actually think not of the symmetry of the crystal very much they used to think of a symmetry of the crystal as a physical object here but you imagine this crystal continuing you know you magnify it up and then there's a regular repetition of the atoms involved and so on and you imagine that continued forever to infinity and then there's an infinite symmetry group the symmetry group of all the translations and rotations and reflections that fix that crystal pattern historically the finite symmetry groups were enumerated first and then roughly in 1890 the possible groups for of this infinite kind I just mentioned for crystals William weighted there are about 200 of them they were dead anyway so by free teams Federoff good the great version I say teams two of the team's only contained one member oh here the terms contain teams contain the only one member wasn't my Federoff the great russian crystallographer who turns out to have done it first in every way but then his his work didn't reach the West so to speak and until later the other people assurance whose German and whose notation is generally adopted nowadays and less well known was equally interesting figure Peter Barlow and English crystallographer and the author of a book of mathematical tables that were still in use when I was a kid II as well a day numerated these things they each made a few mistakes it seems to be a rule that if you enumerate at least 100 things you know or something like this then you'll miss one out or you'll put one in twice or something they did it all about the same time 1890 and in a few years from then you know they communicated with each other and the inconsistencies were ironed out from the situation has stayed the same ever since then and it's a staple of physics chemistry crystallography and so on okay now the ways they did those things there's a theorem that the type of symmetry group were having always Candice infinite kind always has a lattice of translations called graph a lattice in the literature and so the way they did it was take every possible shape of lattice and discuss what symmetries you could add to this this lattice is invariant under water for while that one is invariant of some under some other rotations of affections so and that lasted for essentially 100 years no perhaps 80 or 90 years and then the correct way of doing it was found and it was found by two people really independently mary macbeth it was at the time at birmingham university in england and William Thurston who was one of my colleagues in Princeton not at the time you know I later joined Princeton and met Thurston and I remember very much you know what one of the sessions made the 2 dimensional analog of this kind of assertion is something you may know that there are 17 symmetry types of repeating plane patterns these are often called the wallpaper groups I think that's almost criminal to call them the wallpaper groups because it leads people to expect to find them on wallpaper and you can't find them on wallpaper you can find about four of them on wallpaper but if you go to posh hotels you can often find mosaics in the lobby or even in the bathroom and these tile and you can find them outside too and the tiled patterns you know of these mosaics will exhibit all 17 different types so I prefer to call them the plain crystallographic group so I'm sorry that's rather technical or if I want to be non-technical I talk about repeating patterns or something don't say wallpaper please because you know it turns the students on and then disappoints them you know you set us an exercise try and find examples of these seventeen groups and they comb the streets of say Providence and they don't find any you know okay but that I mentioned that the initial way of doing it lasted for about 90 years and then the correct way of doing it was found well then perhaps I shouldn't say correct the best way of doing it I mean the old way was correct it was just rather long-winded and turns out to have been you know not a good way to approach the problem and that is the way of discussing what geometrical operations are in the group you know which which operations of rotational translation or whatever fix your object and Thurston was a topologist he still is and he did it a different way which is to consider the quotient of the space that you're acting on by the group and that's called an orbital and I think the etymology of orbifold is orbit - manifold but it's not quite a manifold so well let's try and find the symmetry group of this chair well somebody once said I believe but if anybody can find the quotation I'm referring to I'd love to see who it was that said this that even quite ungainly pieces of furniture can to become spherical if you wrap them in enough brown paper this is one one of those great things and it's too so to speak so what we notice what I notice about this chair is that it's finite fine that's a good thing it's it's not a repeating pattern through all of space okay fine so we can think of wrapping it in brown paper we'd better take it outside this room because it gets a bit crowded in here and you know we want a lot of brown paper so well actually let's not use brown paper let's use emperor's new paper emperor's new paper is so wonderful and Finley and everything that it's just completely transparent you can't even see it's there but it is the you know and so here by magic I've wrapped it in some emperor's new paper and okay here we are and let's make this fear well the bigger let's call it the celestial sphere that's a pretty good idea so um where are we yes and so I'll simplify the shape of the chair if you don't mind because I'm not a fantastically good artist okay so there's the chair and now we take some MCAS new paper I suppose a celestial blue is the right color and oh is every symmetry of this finite so the physical object fixes its center of gravity so I don't work quite where the center of gravity is let's say the you know um so now this is if it fixes the center of gravity it automatically fixes any sphere whose center is the center of gravity so let's have this sphere you know probably I'm not doing it to wealth doesn't look terribly psychical okay so here's the the chair wrapped in the Imperial celestial sphere right and what's the reason why I wanted the celestial sphere well the celestial sphere is a rather simple kind of space I mean the surface of the sphere otherwise I would have said the celestial ball the celestial sphere is a two dimensional manifold okay and moreover it has all sorts of homogeneity properties and so on the chair is rather ungainly object okay well now let's I'm going to write in red something that is sort of highly relevant here it is if you had the mirror that reflects this chair into itself it's where it would cut the chair but it's rather simpler to say where it would cut the sphere well I hope I'm not you know drawing that so badly that you can't see what I intend it would cut the sphere in this circle which apart from the fact that I've drawn it vertical we might call an equator and so on and right now what's the orbitals well what I mean by the old befouled is this remember I suggested in etymology for the word orbifold that etymology is orbit manifold so the points of the orbitals are all bits of the group okay well this group only has two elements so here's a point on the surface of the sphere let's call it acts on X here is a point the point that X is reflected into by this symmetry I'll call that let me call it Y for simplicity and then X comma Y is a point of the orbital the set consisting of x and y but Z here okay Z happens to be in an orbit of size 1 ok fine and now what we're going to do as mathematicians we can't necessarily do in real life which is to identify X with y and z with itself and you know every point with its mirror image across this this mirror and then whoops if we the way to do that so to speak let's remove the chair itself by magic it's not there anymore and then let's come along and sort of punch the sphere so that the right-hand half of it sort of goes inward and collapses onto the left-hand half so in a sense the orbifold is now a hemisphere ok I'm fine now Thurston's wonderful idea which actually was had before him by macbeth and nobody outside of other small coterie of mathematicians knew about Macbeth was that better Thurston by the way took this idea and made it his own and if that sounds like a description of theft well that's actually an impression I want to convey slightly but not too much I don't yeah I don't sort of mean that he knew he was thieving it but he really made it his own and all sorts of things about orbitals and their relation to symmetry and so that Macbeth did not Macbeth only did a rather special case on the other hand that special case includes all the ideas so what you know Thurston's idea is that the topology of the orbitals actually determines the group that's an astonishing thing because the group is a metrical object you know it's got symmetries which is it the members of the group are geometrical congruent which means their distance preserving things and yet what's topology topology is the study of those things that are invariant you know you with distance isn't invariant and the topological transformations or continuous Maps but it's the stands amazing observation that geometrical groups are in fact determined by their orbitals and it's part of his revolutionary program for proving things in topology the the so-called metallization conjecture which Thurston made and which eventually has been used to solve the Poincare a conjecture by perma but so it really says topological problems reduced to geometrical ones I mean certain topological problems which is rather strange anyway I am fine well let me take another piece of furniture this time since it's got things on it I don't propose to disturb it there's this table here now this has a rectangular shape at the top of it what's the easiest way for me to come across that one well god I'm a trying to to do it we did it oh is this the black that works so again now I'll take the slip shape to be a slightly different shape to the particular example I was looking at whoops a rectangular table and now um let's draw the celestial sphere around it OOP well this is a bit like the Emperor's New Clothes you can't see them terribly well but now there's a reflection let me draw the worthy reflection hits the table so to speak now we get to reflection lines and right angles to each other and whoops on the sphere boom I have gone outside the city yeah must stay inside the celestial sphere memo anyway here it is okay well I'm not going to bother to draw the orbifold consists of it's rather like a piece of orange peel here you all understand it that is a right angle if you cut an origin to quarters in the sort of obvious way and then then eat the orangey part the good bit and then the peel will look rather like this fine and now I think I want to start describing the notation perhaps I should say when I first met Thurston I said to him something you know I said to him I have a rather nice notation for the 17 plain crystallographic groups let me show it to you Thurston was always rather irritating if you wanted to show him anything he said no no no let me show you my ideas first and I said oh you know what gives you the right to have first go you know I said it first proposed all this discussion he said well anyway you know do it so I said okay well look how about are you taking five minutes and then I take ten and then you take ten and then I take ten and some will you accept that he said yes but I won't need more than five minutes so he took his five minutes and I'm not normally a modest person but after Thurston's five minutes I didn't bother to take my own 10 minutes or any time because I realized his ideas were much better than mine and and that was well surprise to me but it shouldn't have been because he's a very famous topologist and you know he's made his life's work in this way and it was obvious that this was the right way to do things and then I became a propagandist for Thurston and I'm still acting as a propagandist for Thurston in this way okay and so what we've got to do is describe a notation and a way of naming these orbitals now that's what I provided in the and I said that you know I will find a beautiful notation which will make your ideas very clear and I did exactly that so let me say what the notation is in this case where the I'm going to gradually introduce the various features you look at the orbifold and you'll notice things about it and then I'll tell you the notation for each feature it turns out there are only four features only four types of feature perhaps I should say and once you've got them you've done a lot and the the notation for this orbital which you'll remember as a hemisphere is star so this is star star indicates a group of order 2 the precisely 2 operations here but what star really names is not the group operations but their effect on the orbital the way in which they change it from being a sphere which is the simplest sort of thing it could possibly be to something more sort of complicated and one of the things you can do to an or befouled or to alexei to a manifold is cut a hole in it and star really means i've cut a hole in it so this hemisphere can be regarded as what's left of a sphere when you cut out a hemispherical hole the other hemisphere so star is the name of world when we talking tough topology star is the name for a disc because the hemisphere is a disc topologically okay it has metrical information as well it's not only a topological manifold but it turns out that that the topology is the most important thing that's all you need really which is a remarkable discovery because as I say what we'll talk about our geometrical congruence --is which are defined to be the things that preserve distance okay so that's star and now this one up here oh by the way usually you well let's say this this one is star because after all this is a disk but remember that star really names the hole in the disk and then to this is read 2 2 and let me redraw the thing emphasizing just the topology there's a right angle right angle which is the angle is actually PI over 2 up here and PI over 2 down here and then there's a hole and this is the group I call star 2 - okay so that is the symmetry group of a rectangular table let's suppose instead that the table had been square now if the table were square I'll just sketch it very quickly down here then there's more symmetry namely as well as reflecting it in these two lines and let me just sort of continue them around there to suggest the sphere without having to draw you understand me but there's one of those two lines it's also symmetric on the reflection in these lines and so I am not controlling where the lines go exactly okay and the orbifold is a rather thinner version of the thing I've described which is this and the this angle is now PI over 4 and so the table is star 4 4 okay what other things can happen so far these have really been kaleidoscopic groups if I fix two mirrors facing each other at an angle of PI over 4 and I then put a random object in between the two mirrors I will see something whose symmetry group is Starfall 4 there can be more complicated kaleidoscopes than that so let me um where did the arrays ago I got it okay a cube has really rather a lot of inflections it's going to be hard for me to remember where I'm putting me and there's a line there okay one of the coordinate planes of the queue there's one here there's one there but then there's one that does this diagonal and goes all the way around and so on there's one that does this well I'm not going to bother to draw them all I suppose I should because it should be a bit more patriotic if this cube has a Union Jack on each in on each face okay but now the orbifold has angles right now was are they PI over four up here that's that one PI over two here and that there and then PI over three here now don't get confused I drew this these this network of lines on the surface of the cube but it's really not on the surface of the cube it's on the celestial sphere and so the angle here is not 45 degrees okay because six of them make up a whole revolution at 60 degrees it's not PI over four it's PI over three there's another way to think of this we can ask how many mirrors go through the point and therefore mirrors going through this point in the face of the cube you see it there are three mirrors going through the vertex and there are two mirrors going through a point on the edge so this has kaleidoscopic points of orders two three and four and okay and so the let me draw a joist again and not be too worried about getting the angles right because as long as I label this four three two for the star there remember only the topology is important really and so this is star four three two is the symmetry group of the cube okay um but now suppose that we didn't take all the symmetries of a cube but only say the rotational symmetries well oh god it's gone again no it hasn't if we take the rotational symmetries a curious thing happens the best way of indicating this although it's view of fairly recent history it's not too nice is to draw a swastika on each face of the cube oh sure it's a great pity that I sort of always feel a bit guilty when I do this but I really shouldn't I mean the swastika is a nice geometrical pattern there's known in ancient India and so on okay so now what the object we're thinking of is this object and it doesn't have any reflection symmetries at all these are all as it were right and it's wast occurs actually there's another name one tends to think of swastika as a German word but it's not they call this a hacking quoits but it's really an Indian name but this object has been has had many many names over history and one name for it which I go like is the tetra gamma because it's made of four gammas but anyway what happens if you do that well let me just take a piece of paper first and I was like I'll take a piece of paper like this and just draw a swastika on it and then wonder what that does to a job if oh can I steal a piece of paper from you and oh you will you even drawn the swastika well is it okay forget to sir I was taught this trick I was taught this trick by a an elementary school teacher how to tear paper up and get reasonably straight lines but anyway here it is um now how can I identify this up what I've got to do is take this point here say and identify it with this point and then also with this point you know and so on well what I do is I turn my way to the center somehow does very much matter and then I turn it into a sort of cone and just fitting this X on top of that X so this is what happens when you're symmetry group contains a generation now a generation is a rotation well let say a gyration point is ago is a point of the figure which through which there's a rotation that takes it to itself but that point is not on any reflection line okay not you notice I emphasized I emphasize that word because experience teaches me that people forget that condition so I get a rather annoyed with them and it's more efficient to get annoyed with you before you make the mistake rather than after you know you learn quicker um okay that's called a cone point and this is a cone point of order four and the angle is two I over for because before I did it there was two pies worth of space a whole revolution of space around this point and now there's only two pi divided by four ninety degrees I'm okay so this is the orbifold of the original square with the swastika on it under this rotation of order four it's rather harder to see it within more complicated cases but there's something else I wanted to say about it the concept of an over fold can be made to carry some metrical information and in particular the angles the total amount of space around the point is quite important this point is really only a quarter of a point because a full manly sized point you know has 360 degrees around it but after this identification then we have ninety degrees so it's only a quarter of a point that will be quite important later so what's the orbifold of the cube under its symmetry groups well it turns out that the the only important thing really is what the singularities ten points the singularities are and there's a singularity here this is a four-fold cone point there's one here that's a free focal point and then there's one here that's a two focal point if you locally rotate the things through one half of evolutional one-third of a revolution or one quarter of a revolution about these respective points it comes back on to where it was and a revolution remember is 2pi we work in terms of units of revolution so this group is four three two and you notice I've changed color there's a color convention in this notation red is involved with reflection blue well they're true motions so to speak so okay that's this group well now it's a wonderful thing let us try to work out what the Euler characteristics of these various things are now you possibly remember the definition of Euler characteristic V minus E Plus F you draw a map on your object and then you can't V as the number of verses is the number of edges F is the number of faces and V minus E Plus F is an invariant of the topology now what we have here is an extension of the notion of oiler characteristic so let's draw a map on this orbifold here which has one face which is this two-sided thing okay and well let let me say something that's wrong it's going to have two edges and two vertices that's the wrong thing but no that's not true because this edge is only half an edge the Euclid was wrong when he sort of defined us you know lines of things that have no thickness and so it lines have a thickness let's say we can let me take this green here here's a line okay and here are some vertices at the end of the line and so on and now and the green thing is going to be part of my map on the on the surface and now I'm going to start considering what that becomes in the orbifold well you know I'll make it a bit I'll suppose that at the top here there were several edges coming in so what happens now is the mirror line goes along the edge of the map and the edge was this quite thick green line and it splits it into two and and then there's some other reflection line of the map maybe I don't quite know which map I'm talking about this is only an example and it goes these red lines go through the vertices and split it into six you know this vertex is split into six so the part that comes out in the orbifold is only half an edge here the right-hand half and the this the part of this vertex that comes out in the orbifold is only one-sixth of a vertex and so on so now let's try and work out the Euler characteristic of this thing well there's V in this case is 2/4 okay because we only got a quarter of a vertex in there a quarter of vertex down here and then E is two halves because I have a half of an edge there and half an edge there and F however is one so the Euler characteristic so this is of course another way of saying two quarters as a half two halves is one plus one so the characteristic is this thing which is a half there are the characteristic of that orbifold is a half this is appropriate because the Euler characteristic the orbifold is a sort of divided manifold and so the older characteristic is performed by doing the corresponding divisions on the vertices edges and faces let's work out what the Euler characteristic where did I draw it of staff or fee to is I don't seem to have got it anymore must have rubbed it out let's work it out again um so staff or fee - why let me just remind you roughly what it looks like there's a 4 and a 3 and a 2 and I put star there to indicate there was a hole well what's here is only 1/8 of a vertex oh sorry I must write in a different color remember this was what came out of the vertex at the center of a face of the cube and it was on eight triangles so only 1/8 of it is around on in on time this is 1/4 and this is 1/6 so V is 1/8 plus 1/4 plus 1/6 which is some number of twenty-fourths 324 + 6 + 4 24 C and 10 is 13 so the number of vertices in this map is 13 24 sorry 30 24th what's the number of edges well the number of edges is three halves because there's half an edge here half an edge here and half an edge here we only have the inside halves of the edges and the number of faces is one so the Euler characteristic is 13 - 36 + 24 24 which is 13 minus 12 which is 124 okay and that's 248 and I usually measure these things in monetary terms ever since I came to America all money is green and it's measured in dollars and what's the significance of that well this is the way you work out from the orbifold symbol what the order of the group is how many symmetries are namely look what what is the orbifold the orbifold is 148 of the sphere we cut the surface of the sphere into 48 triangles and remember 8 for each face of the cube and there are 6 faces of a cube that makes 48 so the orbifold is 148 of the it's not a sub manifold it's a quotient manifold but still you know it counts as one forty eighth of the fifth what's the Euler characteristic of the sphere you ought to know V minus E Plus F is two that's all this original theorem so so when when I work at these other characteristics if they come out to be fractions I always renormalize them so that the numerator is two dollars because I think of this in properly American terms an amount of money and but there's another way of getting to it which is this you can work out for each feature of the orbifold what it does to the characteristic and it turns out it's like paying for a menu and I want you to think think of it like this your mother when you got in the morning gives you your daily allowance of two dollars she says now be good don't spend too much and so on you know don't be profligate and then you buy various things and you have some change sure and let's discuss what happens if we buy for instance the symmetry group of a cube well it turns out that here's what everything costs yeah we've got it this is the menu star costs whoops $1 I'll explain why later a red number after a star well here it is to costs half a dollar three costs two-thirds of a dollar for the costs three courses of a dollar and so on you can guess what the general rule is costs $1 minus one end of a dollar that's what the red numbers cost the red numbers are children the blue numbers are adults and if I got this right now I've got this wrong curses okay well I'll correct it it was the blue numbers that cost those things and the red numbers are the children and they cost half as much namely oh so he namely quarter of a dollar one-third of a dollar and a fee eighths of a dollar and so on an in general a blue n costs and over N minus one over n dollars and a red n costs and minus one over two n dollars okay let me try to explain why I don't think I'll write anything about this explanation on the board but I'll just say it and I've got to keep the colors the caps on the right colors I'll say I'll say why these things cost what they do well the first thing is if you punch a hole in in the manifold the significant that you do this significant thing you do is remove one face okay well no that's not all you do though you also have the edges and vertices around that face but there's just as many edges around that face as vertices so that doesn't do anything because V and E or equal to V minus e cancels so when you punch a hole in the thing you reduce the euler characteristic by one because it was a face okay that's when you punch a hole what happens when you introduce a corner no I won't make it a corner I'll make it a cone point well do you remember when I had this object before I made it a cone point this was a man-sized point a full point but after I made as a coma point of order four it was only a quarter of a point so what have I done I've subtracted three quarters of a point that's why a comb point of order for just say a blue for cost three quarters okay and what happens what what why does a red number cost what it does well if before I introduce let's say this corner there wasn't a corner there and and all the points along this edge were half points because only half of the angle is there after I've introduced it I made this a quarter point okay or in general I made it one to n of a point if there were n Millers meeting at it and so that explains why the red digits cost half they were originally half points and they later became two n thermal points and that's exactly half of what the blue things did so when we took the symmetry group of the cube in the form which one was it this group I started off with two dollars or so so let me try and explain that's what my mother gave me when I set out in the morning I'm now going to do the calculation I think I'll do it in black rather than green and the cost of this is three quarters plus two thirds plus a half that's written on the tariff over there okay and so the change is to minus 3/4 plus 2/3 plus 1/2 and if you work it out that turns out to be 240 eighths of a dollar my mother is rather irritating and she always says it was a little homily you know let me see the change and I held up the change in my hand and she says you started off with two dollars and you've come back with only 148 for that she always says it's in the form of what my original capital has been divided by that's the number of symmetries down there of the cube well I won't um yeah when did I start foolish wasn't it yes no wonder they I mean I should be stopping them say yes well say yeah well I'll tell you roughly what happens so take a finite symmetry group acting in three-dimensional space fixing some piece of furniture say and therefore acting on the celestial sphere if that symmetry group has order n in other words if there are n symmetries then the Euler characteristic will be two dollars divided by n okay and you understand why because the orbifold was the sphere whose other characteristic is $2 divided by n fine and you'll notice that that number is positive if you take two dollars and divide it by a positive integer you get a positive number and so the Euler characteristic of a finite group of motions of three dimensional space is a positive number that's a wonderful thing and we can work it out by this recipe okay there's a cost for each symbol which is given on the carry over there and so whatever the waste of fitting symbols together that can be the notations for finite groups in three-dimensional space ah ways of buying something as a menu with this as a restaurant with this menu that leave a positive amount of change okay that's a truly wonderful thing now I'm going to tell a lie now in the interests of simplicity well I'll tell it a bit later actually I won't tell you when I'm telling it that's a good idea now suppose you take an infinite symmetry group in other words a wallpaper pattern on the plane and work out its orbifold well what you get is by a sort of limiting argument $2.00 divided by infinity which is zero so the symbol for a wallpaper pattern is necessarily an item on the menu away of buying something at this restaurant here that costs exactly two dollars so you get zero change okay it turns out that there are precisely 17 ways of doing this if you go to that restaurant and want to spend exactly two dollars you can buy 17 meals there's not a wonderful theorem that's the reason why there are 17 of these pain crystal crystal graphic groups now okay what happens if you only want to defy to produce a finite group well you've got to produce less than two dollars you go home to mommy and have some change and again there's correspondence between those things what happens if you want to spend more than two dollars you profligate little child well then it turns out you're buying a scoop of symmetries of the hyperbolic plane so you can get the enumerations of these three types of groups just by asking what symbols there are that cost the appropriate amounts okay I successfully told the lie what was I going to say this is the new way this is the way people should have enumerated these groups and that's you know that's the message really that I'm sure there's something else that I should have said yeah yes it is the gaspin a theorem says that the the integral sigh they bought the the integrated curvature over the air over the manifold is 2pi in the case of a sphere but over the orbifold it would be 2pi divided by n where n is the number of symmetries or if if the orbifold is obtained from a torus that's a thing of characteristic zero the integrated curvature would be zero yes that's the so to speak the analytic aspect of this stuff very good question I didn't pay you to ask that today any questions this just happens that's what it is I mean it could have been another number and then the theorem would have said the number of the plane crystallographic groups is that other number let me tell you what they are I mean it's it's quite easy now I think I'll just go unencumbered or where is it now is it still attached to me in some funny ways okay so okay so let me name them you see if you if you have just blue numbers it's really quite easy to work out what goes on if there are three blue numbers well let me just tell you what the answers are first six three two I'm doing the case of plane crystallographic groups 4 4 2 3 3 3 2 2 2 2 and then I'll add something which is if you or what maybe I won't do that if you have just rotations in your group then the sum of the reciprocals of these numbers has to a just K of them has to be just 1 because the sum of 1 minus the reciprocal 1 - of course it was 1 minus 1/2 plus 1 minus so plus 1 minus of course plus 1 my son is to okay now whenever I have to equal blue numbers oh no no I won't do that I'll I'll put a star 6 P 2 star 4 4 2 star 3 3 2 star 2 2 2 2 since a red number costs our Lulu blue number and the star costs $1 these also cost exactly 2 dollars and now I can replace 2 equal copies of a red number whoops by a blue number because a blue number cost twice as much as a red number that's sorry one of the advantages the only advantageous advantage of having had a stroke is that I had to learn to write left-handed and so now I can do either where are we fighting remember which color I wanted to write here of this was that's a fever and this can be replace by edge groups of blue 2 star to two or another I can replace these two blue tooth by these two red tooth by a blue tooth so I've verified that all of these things cost exactly 2 dollars ok 1 2 3 4 5 6 7 8 9 10 11 12 that's the way it goes basically I mean there's some other things that I didn't mention for instance you could you could have two stars that cost two dollars but if a star gets to the end of a symbol you can replace it by a cross what across is is a cross cap on the manifold that's another way of changing the manifold it changes the topology now and then this star is at the end of the symbols only followed by crosses so these are a few more then I could have just left the orbital be a Taurus which is a sphere with a ring on it and so on anyway they're they're just 17 ways you can do it I mean there's no magic to this number 17 you know there's a combinatorial problem being solved it just happens to have 17 solutions you can verify it if you like well except that I didn't quite tell you all of the symbols this circle it's Taurus oh oh my oh to to cross this star got to the end of the simp symbol and when it gets to the end of the symbol and sorry I thought I was still some way off the end when it gets to the end of the symbol you can replace a fire cross let me discuss what happens the orbifold if you ignore the singularities on it you just get a manifold and we know the complete classification theorem for two manifolds they're all obtainable from a sphere by adding handles and that's what a circle gives for you I seem to have rubbed out the circle arrows or cross caps and so the first thing to do is decide on the overalls apology of the manifold and then you can add singularities which are to punch a hole that's star or to convert something into a cone point or a kaleidoscopic point those are the only features you can have by the classification of two manifolds essentially and now we have to see which of those two manifolds have a total cost of $2 which mentions no snow change and then we've got one of the plane crystallographic groups that was a rather hurried explanation and a rather incomplete one but that's what you do this is connected incidentally with Thurston's wonderful work on three manifolds but there the information transfers in the other direction you know why how does this work well we have had for a century now a complete classification of two manifolds and essentially this enumeration of groups is parasitic on that because you can completely enumerate two manifolds you can completely enumerate two orbitals which are essentially the same thing of symmetry groups but the real power of Thurston's ideas lay in the three-dimensional case and they're the information transfer was in the other direction we don't have a complete enumeration of three manifolds we still don't its but Thurston realized that maybe you could use these ideas to get information about three manifolds and that was his metallization conjecture which has now actually been proved and you know the simplest thing al the simplest thing you get in that direction is the thing now known as pellman's theorem Balt the Poincare conjecture but the next simplest thing also handled by Perelman was a proving Thurston's conjecture about the nature of free manifolds they're all cut a ball into pieces and each piece has a metrical structure and then you glue the pieces together in certain ways and there are seven types of eight is seven or eight different types of piece I forget which and so in this case we didn't have the classification of manifolds to start with but we did use something about it from the geometrical situation you

Channel: Istrail Laboratory

Views: 32,762

Keywords: CCMB, John Conway, The Symmetries of Things, University, College

Id: 8z6T-7ovA5Q

Length: 72min 13sec (4333 seconds)

Published: Thu Aug 23 2012