1 00:00:17,070 --> 00:00:29,190 We have been a group that has worked together for a for several for, well, year and a half on this installation. 2 00:00:29,190 --> 00:00:35,310 It we first were going to work on this via workshops where we would all get together and 3 00:00:35,310 --> 00:00:42,690 built the the mathematical installation of which I'm telling you in this presentation. 4 00:00:42,690 --> 00:00:48,270 But a first workshop was going to be in mid-March 2020. 5 00:00:48,270 --> 00:00:50,370 And of course, you know what happened then? 6 00:00:50,370 --> 00:01:01,800 The whole United States were all based shut down because of the coronavirus crisis, and we had to reorient ourselves immediately, but we did it. 7 00:01:01,800 --> 00:01:09,270 So it's a story not just of how the installation was made, but the story also of how all these people, none of whom knew everybody else. 8 00:01:09,270 --> 00:01:21,870 Some of them actually met for the first time when we started building the whole thing at the start of July 2021 and how we all hung together, 9 00:01:21,870 --> 00:01:31,170 how we met regularly on Zoom, how we work together, and we joked and invented and created. 10 00:01:31,170 --> 00:01:39,810 We got to know each other quite well. And in fact, a year after the first online workshop, 11 00:01:39,810 --> 00:01:48,060 we celebrated by having a anniversary just start that we shared with everybody where 12 00:01:48,060 --> 00:01:54,960 everybody where we alternated between pictures of everybody and an object or a scene. 13 00:01:54,960 --> 00:02:02,430 Our concept that they stood for. So this is a story of how maths is beautiful, creative, 14 00:02:02,430 --> 00:02:09,810 fun of people getting together to build an installation that did that and of the solace it has 15 00:02:09,810 --> 00:02:19,410 given us to work on this during the whole pandemic and the companionship we got out of it. 16 00:02:19,410 --> 00:02:27,750 OK. How did it all start? Well, I have been fascinated for many years by beautiful mathematical objects. 17 00:02:27,750 --> 00:02:29,670 Examples of these are here. 18 00:02:29,670 --> 00:02:41,310 Just two examples that I grabbed from the Gallery of the Bridges, a maths and art association, works by George Hart and Robert Fattore. 19 00:02:41,310 --> 00:02:46,260 So that's one increased. The other ingredient is great. 20 00:02:46,260 --> 00:03:00,420 The other element that made that that really sparked off the whole initiative is the wonderful skills of Dominic Elman, 21 00:03:00,420 --> 00:03:05,380 a textile artist she made, for instance. 22 00:03:05,380 --> 00:03:11,790 This is one illustration of her art. She made this miniature quilt. 23 00:03:11,790 --> 00:03:19,110 Yes, what you see there at the very bottom is a small coin, penny sized coin. 24 00:03:19,110 --> 00:03:27,150 That's how big this quilt is and all these different pieces and and the whole composition. 25 00:03:27,150 --> 00:03:31,530 So in order to make something like that, I don't even know how it can do that. 26 00:03:31,530 --> 00:03:37,890 But she is that skilled seamstress. But not just does she have the skill, she also has a fantastic imagination. 27 00:03:37,890 --> 00:03:45,720 Another piece of her is a windmill made out of fabric where different pieces 28 00:03:45,720 --> 00:03:54,990 rotate and you see different views of of of the the the the the quilt itself, 29 00:03:54,990 --> 00:03:59,020 as well as two different wings rotate with respect to each other. 30 00:03:59,020 --> 00:04:12,150 I mean, just amazing. But what the first work of her that I saw was this one time to break free and to break free is an astounding artwork. 31 00:04:12,150 --> 00:04:16,710 I mean, the quilt itself is already beautiful, 32 00:04:16,710 --> 00:04:26,130 and you see here it's aspired by some kind of steampunk machine that has the magic to transform the flat 33 00:04:26,130 --> 00:04:35,190 characters on the quilt into 3-D figures that constantly stride into the world in order to conquer it. 34 00:04:35,190 --> 00:04:39,240 And I thought this was a fantastic, imaginative. 35 00:04:39,240 --> 00:04:45,000 I mean, the workmanship was incredible, but the beauty of the whole, 36 00:04:45,000 --> 00:04:52,200 the whole installation and the imagination that went into it and the incredible amount of detail. 37 00:04:52,200 --> 00:04:55,800 So I was flabbergasted by it. So I looked her up. 38 00:04:55,800 --> 00:05:01,470 I didn't know about her and I wondered, I mean, when I saw this, I wondered, 39 00:05:01,470 --> 00:05:06,990 Could we use that idea, that transforming machine that we saw in time to break free? 40 00:05:06,990 --> 00:05:16,520 To maybe illustrate other? And I had this idea of maybe doing a transformation of mathematical ideas mathematical. 41 00:05:16,520 --> 00:05:24,260 Is that come up in one context, like here, I imagined two little critters getting as homework, 42 00:05:24,260 --> 00:05:35,930 finding out whether numbers divided evenly by smaller numbers and how the mouse that got 12 as its assignment found. 43 00:05:35,930 --> 00:05:40,010 Yes, two three four six all work. I mean, five doesn't. 44 00:05:40,010 --> 00:05:48,170 It leaves a remainder of two doesn't divide it. But the poor one that got 13, of course, finds that nothing divided because 13 is primes. 45 00:05:48,170 --> 00:05:53,270 So the discovery of primes leads to the wonder What are these special numbers leads 46 00:05:53,270 --> 00:05:57,800 to discovery of special properties of primes like Frommer's Little Theorem here, 47 00:05:57,800 --> 00:06:05,100 the fact that if you take a prime and you take that prime power of any number at all. 48 00:06:05,100 --> 00:06:16,440 Then when you divide it, that enormous power, because we can be huge when you divide that enormous power by the prime, 49 00:06:16,440 --> 00:06:23,430 you'll get the same remainder as if you had divided aid, sell off the original number like always. 50 00:06:23,430 --> 00:06:35,850 So that's a beautiful observation. It turns out to be at the root of some of the cryptographic machinery that we now use in open key public diplomacy. 51 00:06:35,850 --> 00:06:41,100 So I thought of could one imagine something that abstract? 52 00:06:41,100 --> 00:06:47,210 It showed the abstraction and then it would come out again. And I contacted them. 53 00:06:47,210 --> 00:06:52,260 Given I didn't know I looked up her name, I found that she had a Facebook account, 54 00:06:52,260 --> 00:06:56,880 something I've never done before, and very seldom since I used Messenger. 55 00:06:56,880 --> 00:07:07,470 Just send a message call. And I said, You don't know me, but I was absolutely bowled over by your piece and I'm a mathematician. 56 00:07:07,470 --> 00:07:12,840 There are mathematicians who are beautiful crust press people. 57 00:07:12,840 --> 00:07:18,900 Would you be willing to consider designing a piece and directing a piece where 58 00:07:18,900 --> 00:07:25,110 different people work in order to make this installation that illustrates mathematics? 59 00:07:25,110 --> 00:07:33,630 And to my delight and still surprise, Dominic answered and said, Yes, let's explore this. 60 00:07:33,630 --> 00:07:41,250 Let's talk. And so we started talking. I told her I actually gave her examples of people who are now in our team. 61 00:07:41,250 --> 00:07:46,560 Carolyn Yeakel and her beautiful Tamara walls that illustrate algebraic symmetries. 62 00:07:46,560 --> 00:07:52,950 They had Tamina and her her beautiful crochet with hyperbolic properties. 63 00:07:52,950 --> 00:07:57,480 And Susan Goldstein, who works on both the beading imaging. 64 00:07:57,480 --> 00:08:07,140 And here she has an illustration of the frieze groups. These three examples I had taken from the galleries of the richest association 65 00:08:07,140 --> 00:08:12,390 that organises annual conferences that bring mathematics and arts together, 66 00:08:12,390 --> 00:08:16,530 and Dominique was interested. Yes, let's do it. 67 00:08:16,530 --> 00:08:28,970 So we started exploring. Over the next few months, Dominic and I met weekly, we explained, we talked, 68 00:08:28,970 --> 00:08:39,720 we elaborated the ideas I had and things transformed a lot already from this very vague. 69 00:08:39,720 --> 00:08:48,950 Prep. I had to something more concrete and I would try to realise things. 70 00:08:48,950 --> 00:08:52,730 And so ideas came up, she would make little drawings. 71 00:08:52,730 --> 00:08:57,650 She would show me things. She would dig the examples. 72 00:08:57,650 --> 00:09:06,950 I would be interested already in seeing how she transformed those, those ideas into concrete things. 73 00:09:06,950 --> 00:09:11,840 I would also find that I had. I explained mathematics to her, which was a lot of fun. 74 00:09:11,840 --> 00:09:18,980 So we were fascinated by the Cakir conjecture and beautiful figures that come out of that. 75 00:09:18,980 --> 00:09:23,460 That didn't make it into the final installation. Maybe a future one. 76 00:09:23,460 --> 00:09:29,630 But so after all that I saw Demeke started building in my market here. 77 00:09:29,630 --> 00:09:41,690 You see, as youth use of this market and the market came, the idea was that she would take this market to the joint meetings in Denver, 78 00:09:41,690 --> 00:09:46,580 the annual meeting of the American Society in 2020 in January, 79 00:09:46,580 --> 00:09:55,100 and Dominique and I met there for the first time, even though we had talked a lot online and we do meetings. 80 00:09:55,100 --> 00:10:03,710 We had not met in person and we presented together at a session organised by 81 00:10:03,710 --> 00:10:12,290 the I Made the Mathematical Association of America on art and Mathematics. 82 00:10:12,290 --> 00:10:19,220 And we launched a call. We said, Look, this is a conception that we have. 83 00:10:19,220 --> 00:10:24,140 The piece itself will consist of many different elements. 84 00:10:24,140 --> 00:10:30,050 The idea is that this would not be realised by one person, but that we would have a team. 85 00:10:30,050 --> 00:10:32,600 So if you're interested in coming in, 86 00:10:32,600 --> 00:10:42,590 working on this completely different type of mathematical art and the smaller projects on which you typically work by yourself. 87 00:10:42,590 --> 00:10:54,230 Come and join us. Come to our party, organise party one evening in the hotel and people came and it was absolutely wonderful. 88 00:10:54,230 --> 00:11:03,800 After that, we we at the party, people signed up and we had our first team. 89 00:11:03,800 --> 00:11:08,480 And then after that first evening, people said, Oh, 90 00:11:08,480 --> 00:11:13,880 but I know such and such who would have loved to discuss this idea, but who wasn't at the annual meeting? 91 00:11:13,880 --> 00:11:22,190 So what it does, is it just limited? So we added a we talked to others, we added friends and friends of friends. 92 00:11:22,190 --> 00:11:27,260 And in the end, we ended up with our full team of 24 people. 93 00:11:27,260 --> 00:11:32,450 And as I said, we were going to. 94 00:11:32,450 --> 00:11:36,260 Once we had a team together. I applied for funding. 95 00:11:36,260 --> 00:11:41,180 I found funding for us to have the workshops and then the coronavirus arrived. 96 00:11:41,180 --> 00:11:48,860 So there was no way we could get together at first, but we decided to nevertheless still meet have a workshop online. 97 00:11:48,860 --> 00:11:54,220 And during that workshop, we redesigned we. 98 00:11:54,220 --> 00:12:02,770 Built little components and we put things together, so we had many, many meetings where we explained to each other, 99 00:12:02,770 --> 00:12:13,540 where we learnt about different techniques, where we, we we we sometimes chuckled at, at, at what's going on. 100 00:12:13,540 --> 00:12:20,410 We learnt about lots of different mathematics. We sometimes were sceptical. 101 00:12:20,410 --> 00:12:27,820 We put together different components. Here you see the bay and the bakery. 102 00:12:27,820 --> 00:12:31,150 We illustrated to each other what was going on. 103 00:12:31,150 --> 00:12:39,280 We laughed out loud sometimes, and we gradually the whole thing took shape. 104 00:12:39,280 --> 00:12:44,410 Dominique made plans and presented them to all of us. 105 00:12:44,410 --> 00:12:54,310 She showed different components. Others demonstrated things that had been 3D printed or embroidered. 106 00:12:54,310 --> 00:12:58,730 Here I'm showing off an embroidery on which I'm progressing here. 107 00:12:58,730 --> 00:13:06,190 Carolyn Carolyn Yeakel is showing off an enormous bowl that's going to be a tomorrow bowl. 108 00:13:06,190 --> 00:13:15,670 And here Cathy is showing how the the head of the ceramic head of the baker fits his body. 109 00:13:15,670 --> 00:13:24,610 So we had marvellous conversations. I mean, like, for instance, excerpts that people remember is what motivates the Chipmunks to search for. 110 00:13:24,610 --> 00:13:31,580 Devices might have become chipmunks, and of course, the tortoise has to take her lunch along when she sets out for her walk. 111 00:13:31,580 --> 00:13:38,440 The Lozano's bass and baker Arnold suggests we still should have a deal because he has a date 112 00:13:38,440 --> 00:13:43,570 that has to be covered in nothing for hygienic creatures and the reasons in the bakery. 113 00:13:43,570 --> 00:13:53,350 Let me go away my chickens, said Liz Bailey, talking about little ceramic models that she made and we make many, many puns. 114 00:13:53,350 --> 00:14:00,310 One big fun is that the nautical theme of the bay with the Boat becomes a nautical scene. 115 00:14:00,310 --> 00:14:09,880 And suddenly marine life was full of knots. And here you have a couple of pictures of the nautical scene. 116 00:14:09,880 --> 00:14:18,520 Not the real thing. All these pictures are pictures of the SecondMarket that Dominique made after we had designed everything. 117 00:14:18,520 --> 00:14:24,460 We took about three months to design, and then she made a second Typekit, which is one quarter scale. 118 00:14:24,460 --> 00:14:30,400 And the real thing is only being built right now. So here are pictures of the second date. 119 00:14:30,400 --> 00:14:37,300 Here are the fisher fishermen in an article seen are these beautiful herons. 120 00:14:37,300 --> 00:14:48,850 Here are the Herald. Some the boat boys are 3D printed a pallet dopes, but the fish are incredibly interesting fish. 121 00:14:48,850 --> 00:15:00,130 They are really not because it's a nautical see. And here you have a nutty bench with nutty things as its appendages. 122 00:15:00,130 --> 00:15:10,900 So that was what a nautical scene became. So we have many different scenes within the Muslim Alchemy installation, and this is one of them. 123 00:15:10,900 --> 00:15:21,520 In the installation we tried to use to bring in so many different objects and beautiful mathematical illustrations, 124 00:15:21,520 --> 00:15:27,190 and we used so many different techniques. It was really amazing. 125 00:15:27,190 --> 00:15:31,840 So let me show you just a few. 126 00:15:31,840 --> 00:15:43,570 So we used 3D printing and in several different types of objects, we used beating it again, little sisters. 127 00:15:43,570 --> 00:15:51,280 And in this interesting country, not. We used ceramics for critters of all sizes of all types. 128 00:15:51,280 --> 00:15:59,530 We use crochet crochet for this beautiful octopus, but also crochet for hyperbolic surfaces. 129 00:15:59,530 --> 00:16:03,850 We use embroidery. We use knitting in several objects. 130 00:16:03,850 --> 00:16:10,510 We use laser cutting. Use steel welding. We use needle felting. 131 00:16:10,510 --> 00:16:16,830 Origami painting. Ultimately. 132 00:16:16,830 --> 00:16:20,450 Quilting. Sewing in several components. 133 00:16:20,450 --> 00:16:32,430 A stained glass door, marbles weaving, metal, welding, wire bending woodworking in a lot of different components. 134 00:16:32,430 --> 00:16:39,350 If we could have a technique for which we could, we could that we could use with wooden. 135 00:16:39,350 --> 00:16:45,830 As a result of all those meetings in which we discussed all the things we were going to do and put in, 136 00:16:45,830 --> 00:16:53,450 but we also in which we also explained different mathematical aspects that we wanted to illustrate. 137 00:16:53,450 --> 00:17:05,030 Dominik learnt a lot of mathematics, and she actually told us that she wished that she had known more about all this, 138 00:17:05,030 --> 00:17:12,160 all the things that we were showing her prior to working on honour. 139 00:17:12,160 --> 00:17:19,730 Our artwork and a result of it was that her perception of what mathematics has changed. 140 00:17:19,730 --> 00:17:27,800 And I think it was portrayed very nicely by how she imagined the silhouette of an adult mathematician who's looking 141 00:17:27,800 --> 00:17:37,670 at the whole scene changed itself or her earlier conception was a mathematician kind of puzzled and but very static, 142 00:17:37,670 --> 00:17:49,100 very. And then her later perception was just much more fluid person who reaches out to her. 143 00:17:49,100 --> 00:17:58,520 Her clothes are flowing in the wind. So much more relatable person than I think the first one was. 144 00:17:58,520 --> 00:18:03,770 So I thought this was a fun, a fun, a side effect. 145 00:18:03,770 --> 00:18:11,150 OK. I'd like to illustrate, maybe in order to get you feeling full of what went on in this school construction, 146 00:18:11,150 --> 00:18:15,560 one of the scenes in more detail in I'm going to talk about the bakery now. 147 00:18:15,560 --> 00:18:20,210 We also have a button bakery. We have the nautical see the nautical scene. 148 00:18:20,210 --> 00:18:23,060 We have to garden coral reef. 149 00:18:23,060 --> 00:18:33,020 We have the curio shop, we have the roof terrace, we have the lighthouse, we have tortoises walk, we have Integrale Hill. 150 00:18:33,020 --> 00:18:39,260 So we have many, many, many scenes and I can't explain them all. But look, let me go into into the bakery. 151 00:18:39,260 --> 00:18:46,430 It all really started with the observation that sandy cookie summit cookie cutters are very restful. 152 00:18:46,430 --> 00:18:52,250 Why do I say, well, this is what cookie cutters look like, and this is how you use them? 153 00:18:52,250 --> 00:18:59,280 And when you make cookies, you punch out your cookies, put them on the baking sheet and then you have all this crap dough. 154 00:18:59,280 --> 00:19:08,570 And what do you do with it? You put it together. You need it again, or you assemble it and you roll it out again and then you punch up more cookies. 155 00:19:08,570 --> 00:19:13,550 But every time you do that, you in order to to roll out your dough, 156 00:19:13,550 --> 00:19:20,930 you need to have some flour so that dough absorbs flour every time you do this and your dough becomes less fine. 157 00:19:20,930 --> 00:19:25,590 So why having to do this? All this, all this rolling out all the time? 158 00:19:25,590 --> 00:19:33,830 Well, because you have the scraps around it. So why couldn't people design shapes where you wouldn't have scraps? 159 00:19:33,830 --> 00:19:44,330 So design cookie shapes the tile, the plane? And in fact, as I was telling the others some years ago for Pi Day, I had done exactly that. 160 00:19:44,330 --> 00:19:53,900 I had made by shapes that when you that just follow each other nicely so that you can punch out all your cookies without having. 161 00:19:53,900 --> 00:20:05,480 And I had even made cookies like that, and I made them into different dough so that I could make a startling picture of the cookies in this picture. 162 00:20:05,480 --> 00:20:10,220 People said, Oh, that's a great idea. We should have a bakery with such cookies. 163 00:20:10,220 --> 00:20:19,220 Fine. So timing, of course, reminded Susan Goldstein of wallpaper groups. 164 00:20:19,220 --> 00:20:24,860 So what are wallpaper groups? Well, let me give you a demo. 165 00:20:24,860 --> 00:20:31,220 Let's look at a few wallpaper patterns I borrowed from a website made by Martin McGovern, 166 00:20:31,220 --> 00:20:35,390 and he used real wallpapers to illustrate all the wallpaper groups. 167 00:20:35,390 --> 00:20:42,260 So let me go out of this and try to get you the wallpaper. 168 00:20:42,260 --> 00:20:50,570 OK, so I have a design of an old wallpaper here, and in fact, 169 00:20:50,570 --> 00:21:02,210 I have the wallpaper design itself and a copy and I triggered I I you see that it's an exact copy and it's a wallpaper. 170 00:21:02,210 --> 00:21:09,880 So I can you see that if I move this exact copy around? 171 00:21:09,880 --> 00:21:18,080 You have. It just repeats exactly. 172 00:21:18,080 --> 00:21:30,350 So that's what it means to have a wallpaper. So, wallpaper, it repeats exactly it would also repeat if I just moved this way. 173 00:21:30,350 --> 00:21:39,110 So they're two different directions where if you go by exactly the right amount, the wallpaper is a variant under what you do. 174 00:21:39,110 --> 00:21:50,130 So let's go back to the original. I also. 175 00:21:50,130 --> 00:21:58,770 Have in this case, I have also a cemetery. Let's go over here to the site. 176 00:21:58,770 --> 00:22:10,520 I don't know why I'm not saying this. We start. 177 00:22:10,520 --> 00:22:18,140 OK. I have my copy here over on the site in this extra one that I have here on the site. 178 00:22:18,140 --> 00:22:22,820 I am. I can make a slip on that layer. 179 00:22:22,820 --> 00:22:27,780 I'm going to transform going to flip it horizontally. 180 00:22:27,780 --> 00:22:33,930 And you see it changes a little bit because the wallpaper is not absolutely perfectly printed and so on, 181 00:22:33,930 --> 00:22:37,440 it's an old wallpaper and that's how we actually see something happens. 182 00:22:37,440 --> 00:22:44,010 Otherwise you wouldn't even seen it, but you see the it was the left bird that had this little white on the shoulder. 183 00:22:44,010 --> 00:22:52,980 And now it's on the right because I flip things. Around a vertical axis undo that slip, you see? 184 00:22:52,980 --> 00:22:58,140 OK, but if the wallpaper were perfect, then that would be a symmetry that it would have. 185 00:22:58,140 --> 00:23:03,570 It would be. It's not just something that I can translate into directions and stays the same. 186 00:23:03,570 --> 00:23:08,040 I also have this symmetry. That's one of my wallpapers. 187 00:23:08,040 --> 00:23:20,020 Let's look at a different one. Here again, I have that if I take this copy and I move it around. 188 00:23:20,020 --> 00:23:29,900 To surmount. It's exactly the same. I could also have moved it around by this amount, and it would have looked exactly as. 189 00:23:29,900 --> 00:23:38,600 OK, so let's understand, and I have this translated copy here, and I can do operations with it. 190 00:23:38,600 --> 00:23:45,250 It's obvious that I have a symmetry again. I mean, if I flip this. 191 00:23:45,250 --> 00:23:52,930 Transform it, I flip it horizontally. Then it looks the same. 192 00:23:52,930 --> 00:23:57,040 Well, not quite. It has moved a little bit. 193 00:23:57,040 --> 00:24:09,200 But if I flipped it horizontally around the right axis, like, for instance, here, not for instance, here in the middle. 194 00:24:09,200 --> 00:24:17,500 The one the line that I'm now drawing with my cursor, and it's clear that I would have complete symmetry. 195 00:24:17,500 --> 00:24:27,760 But I have more I also can do vertical flips because if I transform my layer. 196 00:24:27,760 --> 00:24:31,570 I flip it vertically. Oops, again. 197 00:24:31,570 --> 00:24:37,660 Now this one looks completely the same because it happens to be that in the middle of my figure, 198 00:24:37,660 --> 00:24:42,910 that's exactly this horizontal line is exactly an axis of symmetry. 199 00:24:42,910 --> 00:24:50,410 I flip it around there I have I have no no change, but I can do even more. 200 00:24:50,410 --> 00:24:57,580 I mean, let me do a transformation on this extra piece. 201 00:24:57,580 --> 00:25:06,860 Let me put this a centre of rotation here. 202 00:25:06,860 --> 00:25:15,030 Then. Clearly, after every day, 90 degrees, I get the same figure again. 203 00:25:15,030 --> 00:25:22,590 So I have that additional image. In fact, I also have other places where I can rotate. 204 00:25:22,590 --> 00:25:31,410 I can if I put myself around with my rotation axis here. 205 00:25:31,410 --> 00:25:40,350 Then I don't notice this there if I now rotate. 206 00:25:40,350 --> 00:25:49,060 By 90 degrees, I don't get exactly this yet, 90 degrees, I get the same and. 207 00:25:49,060 --> 00:25:57,220 Wow. If I get same after nine degrees, of course, I will get the same after after 180 degrees as well. 208 00:25:57,220 --> 00:26:06,460 So you have rotation by 90 degrees and you have this symmetry axis, so it's different from the wallpaper that we saw earlier. 209 00:26:06,460 --> 00:26:16,560 I have yet another one here. This one, again, I have a property that I can move. 210 00:26:16,560 --> 00:26:22,910 Thanks. And they will look the same like here. 211 00:26:22,910 --> 00:26:30,140 That's one Wolf or I could rule things obliquely, and they look the same. 212 00:26:30,140 --> 00:26:33,300 So I had a horizontal motor did it and oblique move. 213 00:26:33,300 --> 00:26:43,860 So again, I have two directions in which if I move by the right amount, I get exactly the same as before. 214 00:26:43,860 --> 00:26:54,810 Let's look at the the copy. 215 00:26:54,810 --> 00:26:59,940 OK, rotations, well, first of all, there's again, a symmetry. 216 00:26:59,940 --> 00:27:11,310 I mean, if I transform by flipping burgers horizontally, nothing changed. 217 00:27:11,310 --> 00:27:16,130 But. Or something silly? Nothing would change. 218 00:27:16,130 --> 00:27:26,210 But if I flip a house, nothing changed because I was on the wrong layer, so lets me do it transform horizontally. 219 00:27:26,210 --> 00:27:32,360 You see, I had a change. And again, it's because it's a pattern is not printed exactly symmetrically. 220 00:27:32,360 --> 00:27:37,820 The sixth branch here has become the thick branch there because its horizontal strip. 221 00:27:37,820 --> 00:27:42,560 But I it it, it looks very similar. 222 00:27:42,560 --> 00:27:50,790 And if it were printed perfectly, it would look exactly the same. Let's do a flip. 223 00:27:50,790 --> 00:27:57,480 OK. If I do a vertical flip. 224 00:27:57,480 --> 00:28:06,180 Things do change, that's a difference in these, these these little these these this a peace sign tonight is instead of white figures. 225 00:28:06,180 --> 00:28:11,220 So that does not mean I don't have A. Some of that. 226 00:28:11,220 --> 00:28:22,450 What if I start rotating? So let's try to rotate, which says some procedures. 227 00:28:22,450 --> 00:28:35,160 Clearly. This gives again to be cited, not the right, but if I go by 120 degrees instead of 60 degrees, I have the same old cigar again. 228 00:28:35,160 --> 00:28:45,620 So here I have in. The button is invariant by under rotations, by 120 degrees. 229 00:28:45,620 --> 00:28:52,280 OK, so that's different from. 230 00:28:52,280 --> 00:29:00,290 For these three different bets, I have different properties. These are different wallpaper groups. 231 00:29:00,290 --> 00:29:08,870 It turns out that there are in total 17 different wallpaper groups and you'll recognise the ones we saw here, the green. 232 00:29:08,870 --> 00:29:14,630 Why figures we had the one else that we see. 233 00:29:14,630 --> 00:29:26,000 We had this particular figure. And then we had one that had injuries under this one here with the 90 degree rotation. 234 00:29:26,000 --> 00:29:30,200 There's 17 different wallpaper groups and no more. That's it. 235 00:29:30,200 --> 00:29:37,430 I mean, that's something that can be studied, approved mathematically. In particular, there are no groups for rotations. 236 00:29:37,430 --> 00:29:43,280 So we have rotations under 90 degrees, under 120 degrees. 237 00:29:43,280 --> 00:29:48,080 We have others that have rotations under 60 degrees. 238 00:29:48,080 --> 00:29:52,340 And we'll have some that have rotation. So the 180 degrees. 239 00:29:52,340 --> 00:30:02,390 But those are the only possibilities. So we have by or by over two or by over three or five or six, not by over five. 240 00:30:02,390 --> 00:30:08,990 That doesn't give rise to a wallpaper group because there are so incompatibilities there. 241 00:30:08,990 --> 00:30:17,910 OK, so these are 17. So what? What, what? 242 00:30:17,910 --> 00:30:23,040 What Susan said was we could I mean, tiling translation, 243 00:30:23,040 --> 00:30:32,610 so we could illustrate the wallpaper groups that fits with this idea of tiling of repeating and filling the shape, filling the whole space. 244 00:30:32,610 --> 00:30:42,810 So she proposed to knit the wallpaper, a wallpaper for the wallpaper groups that would illustrate this. 245 00:30:42,810 --> 00:30:55,260 Now, if you knit, then you can have reflection and light symmetry very nicely represented, but not 06, 246 00:30:55,260 --> 00:31:02,160 because you see knitting and knitting, which is something that has very nice symmetry particularly. 247 00:31:02,160 --> 00:31:07,500 And it translates nicely, but it doesn't rotate very well. 248 00:31:07,500 --> 00:31:09,360 It doesn't look like itself when you rotate. 249 00:31:09,360 --> 00:31:24,330 So he made a design in which benign wallpaper groups that have just translations and mirroring and glide symmetries are illustrated. 250 00:31:24,330 --> 00:31:30,510 Nine of the 17 there remain eight well. 251 00:31:30,510 --> 00:31:37,280 Three of those eight have 90 degree rotation symmetry in them. 252 00:31:37,280 --> 00:31:44,640 I hear you saw the button and here is actually the knitted finished knitted wallpaper, which is huge. 253 00:31:44,640 --> 00:31:48,900 Twenty by twenty eight inches. And it's all double digit. 254 00:31:48,900 --> 00:31:55,650 I mean, meaning it looks beautiful on both sides. It just to the negative of what you see on the side on the other side. 255 00:31:55,650 --> 00:32:03,120 It was a real work of items. OK, what do we do with the the eight that we haven't gotten here? 256 00:32:03,120 --> 00:32:11,520 Well, three of them have 90 degree rotation. Symmetry and knitting doesn't do that very well, but cross stitching does. 257 00:32:11,520 --> 00:32:16,050 Just cross stitch in by itself is something that is nine degree symmetry. 258 00:32:16,050 --> 00:32:20,400 So we each Susan makes three designs for Cross Stitch. 259 00:32:20,400 --> 00:32:24,760 The strict little maps are little maps that we. 260 00:32:24,760 --> 00:32:33,370 Did Cross Stitch and that will be part of the courier shop and the arteries on top of it then? 261 00:32:33,370 --> 00:32:43,720 OK, that gives us three of the missing eight. There's still five remaining, while those five have rotation of 60 or 120 degrees symmetries. 262 00:32:43,720 --> 00:32:49,450 So those don't lend themselves well to eternity or stitching across stitching. 263 00:32:49,450 --> 00:32:59,200 But they were very, very well with piercing. And so she made a little quilt design that has the remaining five symmetries in them. 264 00:32:59,200 --> 00:33:05,850 And so that will be a little wall hanging in the courier shop. So. 265 00:33:05,850 --> 00:33:09,930 It's a beautiful illustration of how we have all the wallpaper groups, 266 00:33:09,930 --> 00:33:20,580 but it also exploited the different particularities of different ways of crafting and what their strengths and weaknesses are. 267 00:33:20,580 --> 00:33:26,190 You can also deal with Non-Regular Polygon's and in fact, in the 70s, 268 00:33:26,190 --> 00:33:36,000 Marjorie Rice discovered after reading a column by Martin Gardner in the Scientific American unless semantical games where he 269 00:33:36,000 --> 00:33:43,860 thought where he reported on all the Pentagon dealings with irregular particles that have been found that these were all of them. 270 00:33:43,860 --> 00:33:52,170 But no, they weren't. And material response, some stress. And so this is one of them, and we use it in order to do the floor of the bakery. 271 00:33:52,170 --> 00:34:00,510 It's the tile floor bakery, which is realised via a stitch pattern that this varnish to look like that. 272 00:34:00,510 --> 00:34:04,410 But you can tell, not just on the blade. This is a hyperbolic telling. 273 00:34:04,410 --> 00:34:20,430 Everybody knows these figures from Escher and Escher were actually inspired by his his correspondence with Exeter, a janitor to make this. 274 00:34:20,430 --> 00:34:28,560 And they are real to Alex, but of a hyperbolic bling, which means that that as you go further away from the centre of the disk, 275 00:34:28,560 --> 00:34:33,030 this sense changes, but the angles are still all the same. 276 00:34:33,030 --> 00:34:48,630 This is a bullock darling by quadrant, while there are also triangles, quadrangle and triangles that we well, the bakery has a diagonal. 277 00:34:48,630 --> 00:34:52,740 We decided that the Pentagon was a special figure for the bakery. 278 00:34:52,740 --> 00:34:59,790 I mean, we had the pentagonal tiling on the floor, so this would be a hyperbolic pentagonal tiling, 279 00:34:59,790 --> 00:35:07,050 and it was used in this wheel for the display carts of the bakery. 280 00:35:07,050 --> 00:35:20,040 Here you see standing outside the bakery. The real does have has not only this nice styling, it also has a gasket with five circles around a circle. 281 00:35:20,040 --> 00:35:29,670 So this is a big circle. And then you have these the central circle with the five bigger circles of same diameter around it. 282 00:35:29,670 --> 00:35:37,080 And then from then on, you just fill in every little gap by as big a circle as you can. 283 00:35:37,080 --> 00:35:47,760 And if you do that, you'll end up you have this, this, this, this gasket, you end up with beautiful construction, which we combined on this wheel. 284 00:35:47,760 --> 00:35:54,120 The gasket reminded us of a beautiful figure from a book called In Referrals, 285 00:35:54,120 --> 00:36:04,050 which has similar arrangements of circles and which where you then when you do symmetries of circles, 286 00:36:04,050 --> 00:36:11,760 reflections of circles with respect to circles, you actually find that you build implicitly another gasket. 287 00:36:11,760 --> 00:36:15,600 This the yellow lines on this are built by that beautiful book. 288 00:36:15,600 --> 00:36:21,240 I recommend it. You really should look. Should look at it if you're intrigued by this kind of thing. 289 00:36:21,240 --> 00:36:26,430 So we decided we would try to see if we used five circles what we would get, 290 00:36:26,430 --> 00:36:34,920 and this built a beautiful figure that we liked very much off of with all these reflections of circles with 291 00:36:34,920 --> 00:36:46,290 respect to each other that we used in order to make the the cast iron door for the often in the bakery. 292 00:36:46,290 --> 00:36:51,660 So the bakery has all these different geometric constructions in it. 293 00:36:51,660 --> 00:36:59,910 Now in the reflections, I was doing a flexible circles over and over again, so that's iterated maps. 294 00:36:59,910 --> 00:37:07,320 And that actually is how we ended up with our baker was a guide. 295 00:37:07,320 --> 00:37:12,030 How did that come about? Well, there is something that's called a baker's transformation. 296 00:37:12,030 --> 00:37:18,510 She start. So we're not thinking of maps that take, let's say, a rectangle and map it to itself. 297 00:37:18,510 --> 00:37:32,020 One way of doing that is to take the rectangle. And flatten it so that it becomes twice as long, but only half as high got it in two fold this over. 298 00:37:32,020 --> 00:37:44,500 And you have a rectangle of the same original size, so you start from this big cat here at the top and end up with the cap at the bottom. 299 00:37:44,500 --> 00:37:49,030 Actually, the biggest transformation as mathematicians do, it really doesn't have to folding over, 300 00:37:49,030 --> 00:37:56,170 that's what bakers really do, and that's why I wanted to try it this way. Bakers do fold over and so roll out and fold. 301 00:37:56,170 --> 00:38:06,040 Typically a folded three, actually, but well, but the mathematical simplification is to just cut it in two and put one on top of the hour. 302 00:38:06,040 --> 00:38:07,540 Why did I choose to cut? 303 00:38:07,540 --> 00:38:17,080 Well, because the biggest transformation was originally so that a map of Rectangle two itself was originally proposed with a slightly different, 304 00:38:17,080 --> 00:38:22,420 more complicated version that has the same property of mixing things up. 305 00:38:22,420 --> 00:38:28,000 You see, the property here is that when you do this many, many times with your cat, 306 00:38:28,000 --> 00:38:33,640 then instead of having this black figure on white ground, you end up with something that's grey all over. 307 00:38:33,640 --> 00:38:39,520 Things get mixed very much, and that's a property. It's called the mixing property of this map. 308 00:38:39,520 --> 00:38:48,010 So an earlier map that had been proposed and that has a mixing property had been proposed by a mathematician called loudmouth Arnold, 309 00:38:48,010 --> 00:38:49,090 a Russian mathematician. 310 00:38:49,090 --> 00:38:59,740 So he said, let's stretch in one direction and it dramatically compresses in the other direction so that we keep the same area. 311 00:38:59,740 --> 00:39:04,360 Now, if you what means is, do you have left your original rectangle? 312 00:39:04,360 --> 00:39:12,310 But if you now imagine taking just what is sitting into the next rectangle over in red or 313 00:39:12,310 --> 00:39:18,670 the next rectangle to the right of that beige or down a rectangle on top of that in blue? 314 00:39:18,670 --> 00:39:26,140 Then you can just move those pieces back. I mean, the red one, for instance, moves back to the corner here. 315 00:39:26,140 --> 00:39:31,480 The best one to the upper corner. And then what you're missing is exactly that little blue map. 316 00:39:31,480 --> 00:39:41,030 And so because the area was preserved, you have a way of mapping the original cat to this kind of scrambled to get. 317 00:39:41,030 --> 00:39:48,620 And he showed this figure with a cat, and because his name was William Arnold, everybody called Arnold Scott. 318 00:39:48,620 --> 00:39:53,150 You can look at Arnold Scott in Wikipedia and you'll find this figure. 319 00:39:53,150 --> 00:40:02,060 So when we had decided we were going to look at iterated maps, it was obvious that we were going to have a cat and the cat was going to be called. 320 00:40:02,060 --> 00:40:14,270 Arnold R. Baker is a cat and he's cold, and he's assisted by his friend Max, because cat and mouse, of course. 321 00:40:14,270 --> 00:40:25,580 But also Arnold and Moser did work together, and there's a famous construction mathematics that's called Gilmore Girls Arnold Moser. 322 00:40:25,580 --> 00:40:33,380 And so it seemed natural to have Arnold and most in the in the bakery. 323 00:40:33,380 --> 00:40:37,730 Here is the bakery you see that goes all over. It's really beautiful. 324 00:40:37,730 --> 00:40:47,570 It also has here an illustration above the door of a periodic orbit in the Pentagon. 325 00:40:47,570 --> 00:40:58,040 I mean, imagine having a billiard with a pentagonal border and shooting and billiard ball that gets reflected all the time. 326 00:40:58,040 --> 00:41:05,720 Well, if you have a perfect billiard ball that's never slowed down by friction and you choose your angle well, 327 00:41:05,720 --> 00:41:12,200 then it will after many, many turns and up in exactly the same position with same velocity as before. 328 00:41:12,200 --> 00:41:20,580 So it's a real after many turns will repeat its orbit. 329 00:41:20,580 --> 00:41:24,990 Bakery, again, it's a beautiful object. 330 00:41:24,990 --> 00:41:32,280 Now the bakery, of course, is known for a very special cookie bakes kind of biscotti with almonds. 331 00:41:32,280 --> 00:41:40,560 Those are also called Mandelbrot, because Mandelbrot, of course, is a mathematician from Rosetta is linked to iterated maps. 332 00:41:40,560 --> 00:41:51,510 So you see here how one simple idea led to many, many different things in this bakery which are all there, and I even described all of them. 333 00:41:51,510 --> 00:41:59,130 You'll have to come and see that piece when it's finished. The adventure continues. 334 00:41:59,130 --> 00:42:09,140 And we are building it as I speak. And at the end, I will show you some pictures of how we're building it now. 335 00:42:09,140 --> 00:42:17,900 I would like to end by thanking all the other members of the mass-market team. 336 00:42:17,900 --> 00:42:26,180 It's been an exciting, wonderful adventure for me and it continues because we're building right now. 337 00:42:26,180 --> 00:42:32,810 Here you see all their pictures again and their names. 338 00:42:32,810 --> 00:42:36,470 We are. Most of us are mathematicians. The M stands for mathematicians. 339 00:42:36,470 --> 00:42:44,210 I have spelled that out in full than I would have had to go to a smaller song than I want you to see their names. 340 00:42:44,210 --> 00:42:49,640 But we also have an engineer and an architect and a physicist and a manager. 341 00:42:49,640 --> 00:42:57,590 We cover a wide, wide range of bases. It's been a fantastic collaboration. 342 00:42:57,590 --> 00:43:07,550 The collaborative nature of the whole adventure is what made it special and is what made it so exhilarating to all of us. 343 00:43:07,550 --> 00:43:14,500 So thank you, all of you. And as I said, we are building it right now. 344 00:43:14,500 --> 00:43:31,150 So in the final instance of this lecture, we'll share some pictures of the installation as it was taking place on July six, which we are building. 345 00:43:31,150 --> 00:43:40,510 You will be building until nearly the end of July, when the whole thing will be complete for real now in scale one to one. 346 00:43:40,510 --> 00:43:46,670 No longer one two eight one two four one two four one two one. 347 00:43:46,670 --> 00:43:52,590 The big switch, and we'll see it for Real. And it's. 348 00:43:52,590 --> 00:43:57,390 I expect as wonderful as we hope it will. 349 00:43:57,390 --> 00:44:39,442 Thank you.