We have been a group that has worked together for a for several for, well, year and a half on this installation. It we first were going to work on this via workshops where we would all get together and built the the mathematical installation of which I'm telling you in this presentation. But a first workshop was going to be in mid-March 2020. And of course, you know what happened then?
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. 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.
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, how we met regularly on Zoom, how we work together, and we joked and invented and created. We got to know each other quite well. And in fact, a year after the first online workshop, we celebrated by having a anniversary just start that we shared with everybody where everybody where we alternated between pictures of everybody and an object or a scene.
Our concept that they stood for. So this is a story of how maths is beautiful, creative, fun of people getting together to build an installation that did that and of the solace it has given us to work on this during the whole pandemic and the companionship we got out of it. OK. How did it all start? Well, I have been fascinated for many years by beautiful mathematical objects. Examples of these are here.
Just two examples that I grabbed from the Gallery of the Bridges, a maths and art association, works by George Hart and Robert Fattore. So that's one increased. The other ingredient is great. The other element that made that that really sparked off the whole initiative is the wonderful skills of Dominic Elman, a textile artist she made, for instance. This is one illustration of her art. She made this miniature quilt. Yes, what you see there at the very bottom is a small coin, penny sized coin.
That's how big this quilt is and all these different pieces and and the whole composition. So in order to make something like that, I don't even know how it can do that. But she is that skilled seamstress. But not just does she have the skill, she also has a fantastic imagination.
Another piece of her is a windmill made out of fabric where different pieces rotate and you see different views of of of the the the the the quilt itself, as well as two different wings rotate with respect to each other. 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.
I mean, the quilt itself is already beautiful, and you see here it's aspired by some kind of steampunk machine that has the magic to transform the flat characters on the quilt into 3-D figures that constantly stride into the world in order to conquer it. And I thought this was a fantastic, imaginative. I mean, the workmanship was incredible, but the beauty of the whole, the whole installation and the imagination that went into it and the incredible amount of detail.
So I was flabbergasted by it. So I looked her up. I didn't know about her and I wondered, I mean, when I saw this, I wondered, Could we use that idea, that transforming machine that we saw in time to break free? To maybe illustrate other? And I had this idea of maybe doing a transformation of mathematical ideas mathematical.
Is that come up in one context, like here, I imagined two little critters getting as homework, finding out whether numbers divided evenly by smaller numbers and how the mouse that got 12 as its assignment found. Yes, two three four six all work. I mean, five doesn't. 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.
So the discovery of primes leads to the wonder What are these special numbers leads to discovery of special properties of primes like Frommer's Little Theorem here, the fact that if you take a prime and you take that prime power of any number at all. Then when you divide it, that enormous power, because we can be huge when you divide that enormous power by the prime, you'll get the same remainder as if you had divided aid, sell off the original number like always.
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. So I thought of could one imagine something that abstract? It showed the abstraction and then it would come out again. And I contacted them. Given I didn't know I looked up her name, I found that she had a Facebook account, something I've never done before, and very seldom since I used Messenger.
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. There are mathematicians who are beautiful crust press people. Would you be willing to consider designing a piece and directing a piece where different people work in order to make this installation that illustrates mathematics? And to my delight and still surprise, Dominic answered and said, Yes, let's explore this.
Let's talk. And so we started talking. I told her I actually gave her examples of people who are now in our team. Carolyn Yeakel and her beautiful Tamara walls that illustrate algebraic symmetries. They had Tamina and her her beautiful crochet with hyperbolic properties. And Susan Goldstein, who works on both the beading imaging.
And here she has an illustration of the frieze groups. These three examples I had taken from the galleries of the richest association that organises annual conferences that bring mathematics and arts together, and Dominique was interested. Yes, let's do it. So we started exploring. Over the next few months, Dominic and I met weekly, we explained, we talked, we elaborated the ideas I had and things transformed a lot already from this very vague.
Prep. I had to something more concrete and I would try to realise things. And so ideas came up, she would make little drawings. She would show me things. She would dig the examples. I would be interested already in seeing how she transformed those, those ideas into concrete things. I would also find that I had. I explained mathematics to her, which was a lot of fun. So we were fascinated by the Cakir conjecture and beautiful figures that come out of that.
That didn't make it into the final installation. Maybe a future one. But so after all that I saw Demeke started building in my market here. 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, the annual meeting of the American Society in 2020 in January, and Dominique and I met there for the first time, even though we had talked a lot online and we do meetings.
We had not met in person and we presented together at a session organised by the I Made the Mathematical Association of America on art and Mathematics. And we launched a call. We said, Look, this is a conception that we have. The piece itself will consist of many different elements. The idea is that this would not be realised by one person, but that we would have a team.
So if you're interested in coming in, working on this completely different type of mathematical art and the smaller projects on which you typically work by yourself. Come and join us. Come to our party, organise party one evening in the hotel and people came and it was absolutely wonderful. After that, we we at the party, people signed up and we had our first team.
And then after that first evening, people said, Oh, but I know such and such who would have loved to discuss this idea, but who wasn't at the annual meeting? So what it does, is it just limited? So we added a we talked to others, we added friends and friends of friends. And in the end, we ended up with our full team of 24 people. And as I said, we were going to. Once we had a team together. I applied for funding. I found funding for us to have the workshops and then the coronavirus arrived.
So there was no way we could get together at first, but we decided to nevertheless still meet have a workshop online. And during that workshop, we redesigned we. Built little components and we put things together, so we had many, many meetings where we explained to each other, where we learnt about different techniques, where we, we we we sometimes chuckled at, at, at what's going on. We learnt about lots of different mathematics. We sometimes were sceptical.
We put together different components. Here you see the bay and the bakery. We illustrated to each other what was going on. We laughed out loud sometimes, and we gradually the whole thing took shape. Dominique made plans and presented them to all of us. She showed different components. Others demonstrated things that had been 3D printed or embroidered. Here I'm showing off an embroidery on which I'm progressing here.
Carolyn Carolyn Yeakel is showing off an enormous bowl that's going to be a tomorrow bowl. And here Cathy is showing how the the head of the ceramic head of the baker fits his body. So we had marvellous conversations. I mean, like, for instance, excerpts that people remember is what motivates the Chipmunks to search for. Devices might have become chipmunks, and of course, the tortoise has to take her lunch along when she sets out for her walk.
The Lozano's bass and baker Arnold suggests we still should have a deal because he has a date that has to be covered in nothing for hygienic creatures and the reasons in the bakery. Let me go away my chickens, said Liz Bailey, talking about little ceramic models that she made and we make many, many puns. One big fun is that the nautical theme of the bay with the Boat becomes a nautical scene.
And suddenly marine life was full of knots. And here you have a couple of pictures of the nautical scene. Not the real thing. All these pictures are pictures of the SecondMarket that Dominique made after we had designed everything. We took about three months to design, and then she made a second Typekit, which is one quarter scale. And the real thing is only being built right now. So here are pictures of the second date.
Here are the fisher fishermen in an article seen are these beautiful herons. Here are the Herald. Some the boat boys are 3D printed a pallet dopes, but the fish are incredibly interesting fish. They are really not because it's a nautical see. And here you have a nutty bench with nutty things as its appendages. 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.
In the installation we tried to use to bring in so many different objects and beautiful mathematical illustrations, and we used so many different techniques. It was really amazing. So let me show you just a few. So we used 3D printing and in several different types of objects, we used beating it again, little sisters. And in this interesting country, not. We used ceramics for critters of all sizes of all types.
We use crochet crochet for this beautiful octopus, but also crochet for hyperbolic surfaces. We use embroidery. We use knitting in several objects. We use laser cutting. Use steel welding. We use needle felting. Origami painting. Ultimately. Quilting. Sewing in several components. A stained glass door, marbles weaving, metal, welding, wire bending woodworking in a lot of different components. If we could have a technique for which we could, we could that we could use with wooden.
As a result of all those meetings in which we discussed all the things we were going to do and put in, but we also in which we also explained different mathematical aspects that we wanted to illustrate. Dominik learnt a lot of mathematics, and she actually told us that she wished that she had known more about all this, all the things that we were showing her prior to working on honour. Our artwork and a result of it was that her perception of what mathematics has changed.
And I think it was portrayed very nicely by how she imagined the silhouette of an adult mathematician who's looking at the whole scene changed itself or her earlier conception was a mathematician kind of puzzled and but very static, very. And then her later perception was just much more fluid person who reaches out to her. Her clothes are flowing in the wind. So much more relatable person than I think the first one was. So I thought this was a fun, a fun, a side effect.
OK. I'd like to illustrate, maybe in order to get you feeling full of what went on in this school construction, one of the scenes in more detail in I'm going to talk about the bakery now. We also have a button bakery. We have the nautical see the nautical scene. We have to garden coral reef. We have the curio shop, we have the roof terrace, we have the lighthouse, we have tortoises walk, we have Integrale Hill.
So we have many, many, many scenes and I can't explain them all. But look, let me go into into the bakery. It all really started with the observation that sandy cookie summit cookie cutters are very restful. Why do I say, well, this is what cookie cutters look like, and this is how you use them? And when you make cookies, you punch out your cookies, put them on the baking sheet and then you have all this crap dough.
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. But every time you do that, you in order to to roll out your dough, you need to have some flour so that dough absorbs flour every time you do this and your dough becomes less fine. So why having to do this? All this, all this rolling out all the time?
Well, because you have the scraps around it. So why couldn't people design shapes where you wouldn't have scraps? 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. 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.
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. People said, Oh, that's a great idea. We should have a bakery with such cookies. Fine. So timing, of course, reminded Susan Goldstein of wallpaper groups. So what are wallpaper groups? Well, let me give you a demo.
Let's look at a few wallpaper patterns I borrowed from a website made by Martin McGovern, and he used real wallpapers to illustrate all the wallpaper groups. So let me go out of this and try to get you the wallpaper. OK, so I have a design of an old wallpaper here, and in fact, 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. So I can you see that if I move this exact copy around? You have. It just repeats exactly.
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. So they're two different directions where if you go by exactly the right amount, the wallpaper is a variant under what you do. So let's go back to the original. I also. Have in this case, I have also a cemetery. Let's go over here to the site. I don't know why I'm not saying this. We start.
OK. I have my copy here over on the site in this extra one that I have here on the site. I am. I can make a slip on that layer. I'm going to transform going to flip it horizontally. And you see it changes a little bit because the wallpaper is not absolutely perfectly printed and so on, it's an old wallpaper and that's how we actually see something happens. Otherwise you wouldn't even seen it, but you see the it was the left bird that had this little white on the shoulder.
And now it's on the right because I flip things. Around a vertical axis undo that slip, you see? OK, but if the wallpaper were perfect, then that would be a symmetry that it would have. It would be. It's not just something that I can translate into directions and stays the same. I also have this symmetry. That's one of my wallpapers. Let's look at a different one. Here again, I have that if I take this copy and I move it around.
To surmount. It's exactly the same. I could also have moved it around by this amount, and it would have looked exactly as. OK, so let's understand, and I have this translated copy here, and I can do operations with it. It's obvious that I have a symmetry again. I mean, if I flip this. Transform it, I flip it horizontally. Then it looks the same. Well, not quite. It has moved a little bit.
But if I flipped it horizontally around the right axis, like, for instance, here, not for instance, here in the middle. The one the line that I'm now drawing with my cursor, and it's clear that I would have complete symmetry. But I have more I also can do vertical flips because if I transform my layer. I flip it vertically. Oops, again.
Now this one looks completely the same because it happens to be that in the middle of my figure, that's exactly this horizontal line is exactly an axis of symmetry. I flip it around there I have I have no no change, but I can do even more. I mean, let me do a transformation on this extra piece. Let me put this a centre of rotation here. Then. Clearly, after every day, 90 degrees, I get the same figure again. So I have that additional image. In fact, I also have other places where I can rotate.
I can if I put myself around with my rotation axis here. Then I don't notice this there if I now rotate. By 90 degrees, I don't get exactly this yet, 90 degrees, I get the same and. Wow. If I get same after nine degrees, of course, I will get the same after after 180 degrees as well. So you have rotation by 90 degrees and you have this symmetry axis, so it's different from the wallpaper that we saw earlier. I have yet another one here. This one, again, I have a property that I can move.
Thanks. And they will look the same like here. That's one Wolf or I could rule things obliquely, and they look the same. So I had a horizontal motor did it and oblique move. So again, I have two directions in which if I move by the right amount, I get exactly the same as before. Let's look at the the copy. OK, rotations, well, first of all, there's again, a symmetry. I mean, if I transform by flipping burgers horizontally, nothing changed. But. Or something silly? Nothing would change.
But if I flip a house, nothing changed because I was on the wrong layer, so lets me do it transform horizontally. You see, I had a change. And again, it's because it's a pattern is not printed exactly symmetrically. The sixth branch here has become the thick branch there because its horizontal strip. But I it it, it looks very similar. And if it were printed perfectly, it would look exactly the same. Let's do a flip. OK. If I do a vertical flip.
Things do change, that's a difference in these, these these little these these this a peace sign tonight is instead of white figures. So that does not mean I don't have A. Some of that. What if I start rotating? So let's try to rotate, which says some procedures. 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. So here I have in. The button is invariant by under rotations, by 120 degrees.
OK, so that's different from. For these three different bets, I have different properties. These are different wallpaper groups. It turns out that there are in total 17 different wallpaper groups and you'll recognise the ones we saw here, the green. Why figures we had the one else that we see. We had this particular figure. And then we had one that had injuries under this one here with the 90 degree rotation. There's 17 different wallpaper groups and no more. That's it.
I mean, that's something that can be studied, approved mathematically. In particular, there are no groups for rotations. So we have rotations under 90 degrees, under 120 degrees. We have others that have rotations under 60 degrees. And we'll have some that have rotation. So the 180 degrees. 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. That doesn't give rise to a wallpaper group because there are so incompatibilities there.
OK, so these are 17. So what? What, what? What Susan said was we could I mean, tiling translation, so we could illustrate the wallpaper groups that fits with this idea of tiling of repeating and filling the shape, filling the whole space. So she proposed to knit the wallpaper, a wallpaper for the wallpaper groups that would illustrate this.
Now, if you knit, then you can have reflection and light symmetry very nicely represented, but not 06, because you see knitting and knitting, which is something that has very nice symmetry particularly. And it translates nicely, but it doesn't rotate very well. It doesn't look like itself when you rotate. So he made a design in which benign wallpaper groups that have just translations and mirroring and glide symmetries are illustrated. Nine of the 17 there remain eight well.
Three of those eight have 90 degree rotation symmetry in them. I hear you saw the button and here is actually the knitted finished knitted wallpaper, which is huge. Twenty by twenty eight inches. And it's all double digit. 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. It was a real work of items. OK, what do we do with the the eight that we haven't gotten here?
Well, three of them have 90 degree rotation. Symmetry and knitting doesn't do that very well, but cross stitching does. Just cross stitch in by itself is something that is nine degree symmetry. So we each Susan makes three designs for Cross Stitch. The strict little maps are little maps that we. Did Cross Stitch and that will be part of the courier shop and the arteries on top of it then?
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. So those don't lend themselves well to eternity or stitching across stitching. But they were very, very well with piercing. And so she made a little quilt design that has the remaining five symmetries in them.
And so that will be a little wall hanging in the courier shop. So. It's a beautiful illustration of how we have all the wallpaper groups, but it also exploited the different particularities of different ways of crafting and what their strengths and weaknesses are.
You can also deal with Non-Regular Polygon's and in fact, in the 70s, Marjorie Rice discovered after reading a column by Martin Gardner in the Scientific American unless semantical games where he thought where he reported on all the Pentagon dealings with irregular particles that have been found that these were all of them. 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.
It's the tile floor bakery, which is realised via a stitch pattern that this varnish to look like that. But you can tell, not just on the blade. This is a hyperbolic telling. Everybody knows these figures from Escher and Escher were actually inspired by his his correspondence with Exeter, a janitor to make this. 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, this sense changes, but the angles are still all the same.
This is a bullock darling by quadrant, while there are also triangles, quadrangle and triangles that we well, the bakery has a diagonal. We decided that the Pentagon was a special figure for the bakery. I mean, we had the pentagonal tiling on the floor, so this would be a hyperbolic pentagonal tiling, and it was used in this wheel for the display carts of the bakery.
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. So this is a big circle. And then you have these the central circle with the five bigger circles of same diameter around it. And then from then on, you just fill in every little gap by as big a circle as you can.
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. The gasket reminded us of a beautiful figure from a book called In Referrals, which has similar arrangements of circles and which where you then when you do symmetries of circles, reflections of circles with respect to circles, you actually find that you build implicitly another gasket. This the yellow lines on this are built by that beautiful book.
I recommend it. You really should look. Should look at it if you're intrigued by this kind of thing. So we decided we would try to see if we used five circles what we would get, and this built a beautiful figure that we liked very much off of with all these reflections of circles with respect to each other that we used in order to make the the cast iron door for the often in the bakery. So the bakery has all these different geometric constructions in it.
Now in the reflections, I was doing a flexible circles over and over again, so that's iterated maps. And that actually is how we ended up with our baker was a guide. How did that come about? Well, there is something that's called a baker's transformation. She start. So we're not thinking of maps that take, let's say, a rectangle and map it to itself.
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. 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. Actually, the biggest transformation as mathematicians do, it really doesn't have to folding over,
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. 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. Why did I choose to cut?
Well, because the biggest transformation was originally so that a map of Rectangle two itself was originally proposed with a slightly different, more complicated version that has the same property of mixing things up. You see, the property here is that when you do this many, many times with your cat, then instead of having this black figure on white ground, you end up with something that's grey all over.
Things get mixed very much, and that's a property. It's called the mixing property of this map. So an earlier map that had been proposed and that has a mixing property had been proposed by a mathematician called loudmouth Arnold, a Russian mathematician. So he said, let's stretch in one direction and it dramatically compresses in the other direction so that we keep the same area. Now, if you what means is, do you have left your original rectangle?
But if you now imagine taking just what is sitting into the next rectangle over in red or the next rectangle to the right of that beige or down a rectangle on top of that in blue? Then you can just move those pieces back. I mean, the red one, for instance, moves back to the corner here. The best one to the upper corner. And then what you're missing is exactly that little blue map. And so because the area was preserved, you have a way of mapping the original cat to this kind of scrambled to get.
And he showed this figure with a cat, and because his name was William Arnold, everybody called Arnold Scott. You can look at Arnold Scott in Wikipedia and you'll find this figure. 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. Arnold R. Baker is a cat and he's cold, and he's assisted by his friend Max, because cat and mouse, of course.
But also Arnold and Moser did work together, and there's a famous construction mathematics that's called Gilmore Girls Arnold Moser. And so it seemed natural to have Arnold and most in the in the bakery. Here is the bakery you see that goes all over. It's really beautiful. It also has here an illustration above the door of a periodic orbit in the Pentagon. I mean, imagine having a billiard with a pentagonal border and shooting and billiard ball that gets reflected all the time.
Well, if you have a perfect billiard ball that's never slowed down by friction and you choose your angle well, then it will after many, many turns and up in exactly the same position with same velocity as before. So it's a real after many turns will repeat its orbit. Bakery, again, it's a beautiful object. Now the bakery, of course, is known for a very special cookie bakes kind of biscotti with almonds.
Those are also called Mandelbrot, because Mandelbrot, of course, is a mathematician from Rosetta is linked to iterated maps. 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. You'll have to come and see that piece when it's finished. The adventure continues. 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.
I would like to end by thanking all the other members of the mass-market team. It's been an exciting, wonderful adventure for me and it continues because we're building right now. Here you see all their pictures again and their names. We are. Most of us are mathematicians. The M stands for mathematicians. 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. But we also have an engineer and an architect and a physicist and a manager.
We cover a wide, wide range of bases. It's been a fantastic collaboration. The collaborative nature of the whole adventure is what made it special and is what made it so exhilarating to all of us. So thank you, all of you. And as I said, we are building it right now. 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.
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. No longer one two eight one two four one two four one two one. The big switch, and we'll see it for Real. And it's. I expect as wonderful as we hope it will. Thank you.