Bridges 2014 talk: The quaternion group as a symmetry group

Transcript

okay well uh I guess we'll get started is this working yes no is this working now it's working okay good um so uh so this is Joint work with viart who sadly isn't here um and uh well so first let me let me uh introduce or or remind you about uh symmetry because what I'm talking about is symmetry groups um so symmetry of a of an object is a motion that leaves the object looking the same so this very simple uh design here um if you rotate it in one of four different angles or you leave it the same then uh nothing happens so it doesn't doesn't uh change the look of it and so uh you get a group by um well so you can add together symmetries you can combine them by rotating by different angles um and so the group is the set of objects do I say this yes uh the group is the set of symmetries uh of an object in this case you can rotate by uh 2 pi over 5 uh 2 * 2i 5 and so on and there are five elements and you can combine them together to to to find others um so here's another example uh presumably everybody knows uh the Symmetry group of this so we again have rotations um and we also have Reflections so there are four uh rotations you can do including the do nothing identity element um and there were four Reflections for a total of eight elements and the Symmetry group is D4 perhaps sometimes called d8 depending depending on your um uh your notation the the dihedral group um and so uh so these show up a lot in in artwork of course this uh uh but but groups are studied independently of uh symmetries of uh physical objects so you know groups have uh been classified the finite groups are now possibly entirely classified um so you can ask which of these groups appear as sculptures or as objects in nature so right so which groups can be represented as the group of symmetries of some real world object and so so this is in a sense this is a little bit of a fuzzy question um uh there are many sort of issues here well well first of all again what do I mean by symmetry I specifically mean there's some motion of some space which takes the object onto itself again that's what I mean by a symmetry some isometry of the space if you like um but you know there's there's lots of issues with this question uh real world physical objects are made of atoms uh is there actually any truly symmetrical object where all of the atoms are actually in exactly the same place after you rotate them probably not anyway here's a here's a second question um which groups have actually been represented as as the group of symmetries of some real world object so just think about uh rotation groups you rotate by some angle your your group is the cyclic group of order n there is there's some n which hasn't been done just because there's infinitely many NS and there's only fin many things in the world uh and the third question which groups have been represented as the group of symmetries of some physical object here at Bridges 2014 so um I spent the last few days going around trying to answer that question uh in a sense so first of all um some of the bilateral symmetry just reflection uh the group is C2 uh perhaps if there's time at the end we can go back and and we can go through these and we can ask spot the Symmetry group where was this does anybody recognize this one yes Cedric villan BR it's it's this here's the problem with you know is this a real world symmetry well so the legs are not quite symmetrical but if we can imagine that well it's it's uh it's sort of symmetrical I mean you can imagine that perhaps the artist was doing this just just for an effect but we can we can think of it as as being straightened out and having um uh actually symmetrical legs of course the the museum itself um there were many many cyclic uh uh symmetries uh throughout Bridges um the best in show winner here um this was another one from this morning uh this from the the dance uh math show anybody know where this was from the drum Dr the drum it was the drum they they also appear on the Subways um so yeah so so many so these are here's another issue uh so does this count as rotation of order two do you count the colors or not um if you if you don't count the colors then then it does but if you do count the colors then it has no symmetry at all so these are the symmetries up to cyclic symmetries up to order four here are the higher ones so Carlo had an symmetry group order five another one uh this I think is Mike maor's 66 78 this one was a little tricky to get anybody recognize it it's a big yellow thing and you if you look down from above then then that's what you see uh nine this one I think I counted as 17 and this one is 19 yes 19 yes okay well so there's the question who were you didn't know you were participating in this competition but uh but you were what is the largest symmetry group represented at Bridges 2014 element say again in terms of total elements [Music] yes well well first of all I have to say the largest finite because you know there are circles everywhere so but that doesn't count um well it's my talk so I get to make some uh judgments here um you could well this is what I think is the largest symmetry group this was Mike ners um if you if you set the mark ners piece in the in the show with the the runes if you set the number to one and the number of dots to 360 then there are 360 dots around here probably can't see this so the Symmetry group is obviously representing d360 however you may say that the uh syet group of 120 cellers more well we'll we'll get on to that one uh the dihedral groups so dihedral group of order two here's a subtle one is this dihedral order four or order two the coloring is a little tricky um 3 three um three again this is four four five um where where was this dinner Hall the ceiling of the dinner Hall this was a photograph on the the uh one of the the Stalls um dihedral group of order 16 I think doesn't beat Mike n's 360 though uh which symmetry group wins Bridges 2014 which appeared the most what's your guess yeah boring boring boring well so I mean it's probably bilateral I mean people's faces but more interesting than that the dedal reflection group is everywhere here um they're all over the place oh actually I made a mistake this one is rotation if you look at the words up here that you you just see there's a word here and a word here that's actually rotational symmetry but okay so so there were at least well I only took photos of 10 of them but I only have so much time um the RO doal rotation group also made a strong showing there are many examples of that um yet another do dihedral reflection group and perhaps you should say that this is really illustrating the four-dimensional uh uh symmetry group with uh 14,400 elements um here's another do heal rotation group from earlier in this session I'm afraid I didn't get any photographs from uh forer's uh lecture because I just didn't have time but there were many many symmetries there as well um the other polyhedral groups uh this one's interesting I think you know if we imagine this is continuing on after it embeds in here then this is uh the sort of cubic reflection group um and well there are many others um the wallpaper groups not so many sharings here it must be said there's a few of these around um and then some other interesting miscellaneous these these are not quite dihedral because um you have uh reflections of this and also another way to do the sort of the dihedral flip and these are somehow um I mean I take these to mean that you can translate upwards or you can rotate and reflect and so on so these are don't quite fit into the other categories and then there's various ones which are sort of showing hyperb things and this is another sort of interesting point um these symmetries you know clearly you're supposed to think of all of these heptagons as being the same even though those are not ukian Transformations right so we allow apparently um not just sort of ukian translations as being part of our um part of our symmetries and uh you'll have to ask Yoshi what is going on with this one I think it's hyperbolic but there's some complications and some Branch points and I'm sorry it's you you say it's not whatever you see whatever you see well but but I mean can I think of extending this onwards but but I this is this is yours and I'm thinking of extending this onwards I mean what's the difference anyway look we can talk later um and then right so are these these These are showing uh four-dimensional symmetry groups I think probably we have to say that that that's true um so maybe this should have won after all but anyway moving on the real question I want to ask are there any glaring gaps are there any small uh groups that should have a physical representation as a symmetric object but don't um and uh as far as I know the Guan group so this this is an eight element group that I'll describe doesn't has never had a physical representation but it's only got eight elements I mean why not so to get at this uh uh to describe the catonian group um uh I'm going to uh get to it by looking at these uh very simple monkey blocks this is V heart's invention of as a way to think about these things so I've got a block it has uh it's just a cube which has patterns on the sides so there are two monkey faces two monkey pores and two monkey tails and these all have handedness and so we fold this up into a cube like this and uh and this is what you get so so the the two faces are opposite uh each other as are the two paes and the two tails something to notice is that um all of the uh the faces um the hand faces the tail faces and the face faces um have handedness so um and they're all opposite each other so this is a left monkey porw and this is a right monkey porw the tongue is sticking out to the left and to the right and the the question mark tail twists in different directions and so this Cube itself has no symmetry whatsoever it looks like it might do but there's just no there's just nothing there and so your question is how can you fit these together so that the faces match and so the first thing that you might think of is just stacking say the pores together so here um because I've got a left monkey porw on one side and a right monkey PO on the other side um if I sort of swap the order if if I take a copy and stack it next to the other one then they match together and you can make an infinite line of these uh monkey blocks and so the first question what is the what are the symmetries of this infinite line of blocks anybody what is the symmetry of this this infinite line of box say again um well so I can translate that's certainly true this guy is the same as this guy and it's going to repeat after every four and so I can just translate four blocks along oh oh Craig is making the right hand gestures 0° screw rot 90° screw rotation left-handed I believe you would not imagine how much of a pain it was to figure out the of the screw rotation so yeah so what is going on is is you go from this to this and and the the cube screws uh left-handed uh rotation as it goes um and that's so so you get the the integers um I can also think of uh sort of well since it repeats after four I can I can imagine just wrapping it around and uh having only four blocks which are which are twisted around like this uh and then then the symmetries are uh 90° screw rotation but it's going around in a circle so so you screw around um you move so the these arrows here as supposed to unintentional puns of a of a sort isn't that what we're all doing all the time anyway um so so you you rotate the this ring of blocks around and you also Twist by 90° as well and so this has a CYCC group of order four um and you can also stack them uh vertically so so it turns out that you can't sort of take this infinite row of blocks and stack just to make a wall um it doesn't work um you can see see so so there's a line of three blocks through here and then you put one up here and then there's no block that can fit in in this Gap here um but I'm sure some of you have already realized given the shape that I've got on the screen here that what you can do is you can clo you know if you could close up these two faces so that three cubes fit around one Edge uh then then it all works and everything matches up and you get a a hyper Cube so uh um so we have to move to four dimensions um and eight of these monkey blocks glue together to make the cells of a hyper Cube um so this is a complicated uh uh image so this is the the the net of the hyper Cube exploded showing all of the different left hand screw motions through those rings of four blocks so I've got um let's say let's let's see there's a there's a ring of four uh blocks going along here through the tailes and then there's another sort of dual ring of four blocks going through here again through the tails and so you can act on this uh configuration of Cubes by doing both of those tail uh screw rotations at the same time and that's a symmetry of the object um and it's the same thing with the K and uh and The The J so um so what is the group of this object there are eight uh there are eight symmetries um well so how do you know there are eight symmetries well uh so one Cube on its own has no symmetries itself if you're going to have a symmetry of the whole object you have to move a cube you can't just have one sit there and do something and that Cube can move to any of the other seven cubes and so there are a total of eight different uh uh symmetries and you you can just work it out they correspond to the eight elements of the Quan group so this is uh has eight elements is one IG J K minus one - i- j- K and the these are sort of it's an extension of the complex numbers to four dimensions um so one is the do nothing uh uh element I and K these these three screw motions minus IUS J and minus K are the the going in reverse screw motions um and minus one sends every Cube to its antipodal opposite and these satisfy um the relations that you expect from the cians if you're not familiar with the cians there you go that's what they do um and so so these these really show you uh what is going on with the Quon group um so how do we make a sculpture which has this symmetry um so each monkey block itself has no symmetry um and so we want to uh or rather it has only the do nothing symmetry being Panic um so if we want to make a sculpture that has this Q8 symmetry we have to put a design which has no symmetry into each cube of the hyper Cube so so if you put something which had more than no symmetry or only the the uh trivial symmetry then your eventual object would have too much symmetry so you want exactly Q8 you have to have no symmetry in here um so obviously it should be a monkey because monkeys are the canonical choice for objects that have no symmetry once you move the arms around a little bit um okay why is it a monkey um so you put copies of it into the other cubes you need whatever the design is to connect into this through to the six neighboring cubes of one cube and a monkey has a tail and four Limbs and a head head so it's the only choice um it's sort of tricky if you if you're going to have something that has no symmetry what do you do I mean I guess you do something figurative because there's no restrictions otherwise so why not um so how do how do you draw four-dimensional pictures in uh threedimensional space I talked about this uh two years ago at Talon um just a a brief recap so this is this is a picture of stereographic project from the two-dimensional sphere to the two dimensional plane um so there's a light up here which is projecting uh rays of light down onto the the plane and so a line goes from the light source at the North Pole of the sphere it hits the sphere once and it hits the plane once and the map just is what is the point on the sphere and that maps to what is the point on the plane um so how do you get uh so if you're going to do this uh from uh two-dimensional sphere sorry this is the two dimensional sphere it may look threedimensional but it's a surface so we would call it a two- dimensional sphere um so how do you get uh a hyper Cube well in this case we're going to do a cube how we going to map a uh a cube onto the um uh the two-dimensional plane so first we radially projected onto the the sphere um so this is a sort of hard picture to see but there's a cube here sitting inside of a sort of rounded Cube and you can see the Shadows um coming from the light this is radial projection onto onto the sphere um and then we do stereographic projection again from this sort of beach ball rounded Cube onto the plane um and if you do the same thing one dimension up um projecting the hyper Cube radially onto the um three sphere the the sphere in four dimensional space and then stereographically projecting it's exactly the same mathematics into three space then this is what you get and uh and if you do this with the monkey design then this is what you get um more fun than a hyper cube of monkeys um I should mention will sagum my brother designed the monkey because he knows how to design things that aren't math um and I think that is pretty much my talk is there anything else I I wanted to say uh that's it thanks very much [Applause] are there any questions yeah when you mentioned the symmetries of the um querian and you said one to another one it's determined is there an intuition for why you can't choose the map of the others uh let's go back maybe you can repeat your question because I'm not sure I followed you you said there's eight symmetries because once you put one Cube somewhere that determines everything oh yeah well so um I guess the question is is is there any choice in how you glue the cubes together if you've determined the shape the the position of one of them um right I mean uh gosh right I mean I think the point is that that uh on once you've decided where one of them is going to go then all of the gluings to determine where the other ones you've already done so I mean you would have to be ungluing then regluing things to get a different um object maybe maybe we'll try and figure it out afterwards uh yes yeah very nice I was wondering about this monkey I mean certainly he wasn't hurt because this is all in in in in3 but your your brother did he design the monkey in in the first space no no he so he he designed the monkey um in this Cube so so I wrote a little program that he he could just a command line script that he could use to see what happened to the monkey as he was tweaking things and it would give him back the the the distorted stereographically projects version it's sort of tricky you have you know how how are the the hand and the tail going to wrap around each other you have to sort of uh keep designing it and going back and forth and seeing what the effect of doing some sort of change here in the ukian cube hasn't in the original in the the final uh picture so it was it was a little bit tricky but he's kind of obsessive like I am so yeah how satisfactory do you find this solution maybe as follow question using similar techniques would you not be able to just create objects for any s right so so well so there was some Distortion in the eventual objects right I mean so if we get to here so this object itself has no symmetries in R3 if you just allow ukian uh motions but in the same way I think as as I was very careful to say that uh Vladimir really is showing hyperbolic translations but those aren't ukian Transformations either but we interpret them as as uh as being symmetries of that object so right I mean that there's some there's some hand waving here I mean this isn't a par this isn't a very precise question but yeah I mean I think I think it's reasonable to say that this this shows enough to to allow the the viewer to infer what what is going on uh and particularly with the animation I think that helps as well um as for the question can you now do everything um uh it's not at all obvious to me how you would do that and so so I'm specifically saying you know a symmetry is um a motion uh right it's an isometry of some geometric space so how do you do that in general and how do you represent it effectively in three dimensions you know if you want to take the monster group 168,000 something dimensional object is there any way to produce some thing which is go we need only need to produce visualization uh well to hold on so okay so there's an issue here how much time do we have plenty um so there's an issue here so you could say why don't we just draw the kaph um you can you can draw the kaph of everything I agree um is and so yes that is a way of visually representing the group You could argue so is just writing down the presentation that is a way to visually represent the group um I want to see sort of visual motion that I can imagine doing which acts on the object and leaves it looking the same um and and I well so I mean it's an aesthetic Choice uh I guess we can argue about whether or not uh this counts or not but hello uh another question I was a bit surprised that you start with the CU and faces of the cube you know already that the group acts on you mean B construction of faces is one you do for the DU the DU is again the you I'm not sure why quite FID yes and you yes that together so the graph that you get when you go through the faces oh is the 16 C this space of the Hy also happens to be a CU that is you have the cube and then you know then you have to do mirac stuff which is I believe the sem product maybe so well so the Dual of the hyper cube is the 16 cell which is a I'm not saying the I'm saying you start your construction it's be nicer to see you know right right start the construction oh in the Dual then you would have an image on each of the corners right you'd have a it would be rather violent to the monkey it would be cut up and there'd be an eighth of a monkey here and an eighth of a monkey there that's right so you imagine the centers of the cubes and the Heart of the monkey I see well designing the monkey itself is harder to do if you if you cut it up into eight pieces I mean s right there were already difficulties with you know the hand had to be cut off and them put on the other side and then Stitch back together again and and yeah I mean already the design is is kind of tricky even with the the connections that there are any other questions all right well thank you very much again [Applause]