 I'd like to welcome Richard Garriott back to MoMath tonight. He'll be our introducer. He is both a private astronaut and a game designer. So if you have questions for Richard after the talk, I welcome you to stick around. He's a very interesting guy. But with that, let me bring up Richard Garriott. Thank you, Cindy. It's always a great pleasure to be here at MoMath. And tonight, you all are in for a particularly special treat. Our speaker tonight, Andreas Daniel Matt, like myself, has a background in video game development. My first video game played on some very primitive computers even before personal computers was a game called Hunt the Wompus. And I'm curious, does anyone in here remember the game Hunt the Wompus? One, two, three. OK, by the way, there was zero in the earlier session. So I'm glad there's three of you who might remember that this game about hunting down the dreaded Wompus was played in a series of tunnels that were aligned on the edges of a dodecahedron. And of course, the connection between math and video games and geometry especially are very deep, as I'm sure all of you all know. But back in the 1990s, when I was halfway through creating a series of games that I'm best known for called Ultima, Andreas was a gamer of a different sort. He was actually a champion competitive game player over in Europe, a multi-year champion of playing video games. And while he did feel the calling to get into creating video games, his love of mathematics took over and he kind of made himself his own path, shall we say. And it's interesting that when I talk to educators about video games, very commonly people come to me and encourage me to add in educational elements into the games that either I'm creating or that our industry should create. And fairly universally my answer to that is, give up, don't try, because what you'll do is you'll take an already low probability of making a successful game and make it an impossibility to make a successful game. But Andreas has sort of turned this whole issue upside down. He has really created a phenomenal set of both entertaining and informative experiences. Even before I heard his earlier talk, I had the chance, so I was preparing for this introduction to go to his websites and poke around at some of these things that he's created. And it is truly a marvel. I know you're gonna have a, you're in for an incredible interactive journey through the many worlds of geometry. Like all great gaming experiences, our mathematical journey will include bonus and secret levels, boss fights, shortcuts, and power ups. Please, everybody join me in welcoming Andreas Daniel Matt with Super Math Adventureland. Thank you. Yeah, thank you, Richard. Thank you, Cindy. I'm very happy to be here. It's beautiful to be in New York. It's very nice to be in this special museum. And we're going to start on a place that you all know. A geometric shape which looks like a sphere. It's a bit squeezed, a squeezed sphere. And I'm going to kidnap you to a place which is a famous, famous research institute situated in the Black Forest in Germany. Its initials are MFO. And we're going to start with the first task. Video games are always full of tasks. Who dares to pronounce the name of the institute? So it's, yeah. Oberwollfach, exactly. That's also how all mathematicians would call it. They would skip the Mathematicius Forschungs Institute. And it's a research institute where you have about 2,500 of the best mathematicians meeting every year always for about a week, some stay a bit longer. And they do pure fundamental, sometimes applied, but mainly pure fundamental research in mathematics. So they fill the blackboards with all types of equations and formulas and ideas and concepts. And as it happens in video games, you have the bad, evil monsters. This time it's a monster called Lapsseptic. It's an algebraic surface, an evil algebraic surface. And one day it kidnapped all the mathematicians. Some of them are also developing video games and visual tools for video gaming. And as you see, it's all empty. And I was there once at night at 3 a.m., 14 minutes and some seconds. And I found that the sculpture, you see that interesting sculpture? It's the so-called boy surface. So it's a sculpture which has a triple intersection point in the center. And if you go there at this particular time and you touch it, then you would switch words and you would enter a different word. So here you have the same surface, but in form of a soap bubble, which is a mathematical soap bubble because it would never exist in this form because you don't have intersecting soap bubbles. But in the mathematical world you can do it and I'm happy to introduce you to a different world. So geometry, adventure, land is one of the many lands we have in the abstract mathematical world. And I really like to look at mathematics in terms of continents and countries, little villages, cities, forests, mountains. And the beautiful thing is that most of it is still not discovered. Our planet is kind of discovered. You still have the oceans and the space, some things. But here a lot is still undiscovered. And I think the new discoveries will be in these abstract scientific words and mathematics is full of it. I'm going to show you, this is just one continent. It's the geometry continent. And I'm going to show you a few of its countries or regions. So one will be the land of algebraic geometry. We heard about that evil septic. So it comes from that land. Then we would have up north the land of differential geometry. Further right there's a land which is a bit older, an ancient land of fractal geometry. Maybe you know this is already very old in the 80s of the last century it started. And then you have the land of patterns and symmetries and tessellations. The land of polytopes. That's also a land that we're going to discover. It's not too far from the land of algebraic geometry. And then there are some new lands coming up. Oh no, there's still an old one, knots. You know, knots. But this one is a very new one. It's called tropical geometry. Let's start with one land. So I'm going to pick the first one which is the land of algebraic geometry. So we're going to have different levels. The first section of the talk is single player. So I'm going to play. But then we're going to start with multiplayer later on. I'm going to give you the sign. And we will start with a small motivational tutorial. So the first level, let's get motivated. So of course, as a good mathematician the big motivation is always a nice equation. So this equation has a lot of whatever, parenthesis, numbers. If you look at it, you would spot a few x's and y's and sets and a few squares and cubes. And what is interesting in this equation is nothing else in a spoon. It's the equation of a spoon. So all points in space that fulfill this equation I'm going to show you just give you the spoon. So one nicer form to show the equation is just to show the spoon. But in terms of compression it's probably better to just have the equation because it gives you all the infinite many points of that spoon. So how does it work? In geometry one of the basics is first we have to orient ourselves. So what we do usually goes back to the 18th century where French mathematician Descartes made the basis for this Cartesian system so you would fix one point, which is the zero point. And then from that point you can orient in two dimensions you would have two directions, x-directions and a y-direction and you can identify all points, two numbers. For example, this is one point, it's the point one zero so it's one step of foot or meter, whatever to the right, to the x-direction, and zero steps to up and down. So this is the point one zero. This would be the point zero steps to left and right and two steps upwards. This was the point zero two. You can imagine you can also go negative if you go to the left or if you go down so this would be the point minus one and two. So this one would be the point about four or five to the right and then one or two downstairs. So I think I put four minus two. So you can do the same thing in three dimensions so it's not only left, right, up, down, so left, right, forth and back and up, down. So we have x and y on the plane, on the bottom and then we can go up and down using the set axis. So if you have a point here, for example, we go two to the right, two to the back and three up. So with this system we can identify all points in space. So the one four five four should be a plus one four five four. I changed it, I don't know what happened. So where is this point? You start here, you go 450 feet to this direction, 750 feet back and then you go 1,454 feet upwards. Any idea? It's the top of the Empire State Building. So this is exactly the point on top here. So any point on the planet. The interesting thing, of course, is that on the sphere, on the sphere it's not plain, so actually it's a little bit different but I think you've got the idea. In the ideal three space we can identify all points. So now we're ready for the first level. So we're not just playing around with points, we of course we play around with curves and then with surfaces. Surfaces is the real interesting part. So looking at curves. What we do now is not only to have points, we kind of relate the x's and the y's somehow together to paint lines and curves and parabolas. So one thing would be to paint all points where the y number equals the x number. So here below you see I would always put an equation. In this sense this equation would say y minus x equals 0. So this is the same to say y equals x. So you would get all points where you have y equals x which is the point 0, 0, 1, 1, 2, 2, minus a half, minus a half and all these points together would give you a straight line 45 degrees up to infinity down to minus infinity. So we have to cut it here somehow and you would see the line. Now what is interesting, I can just play around with that relation between the x's and the y's. I could say okay let's have all y's equal to x squared. I will get this equation. Then I can say okay I'll multiply it with a number. A number that I can change here. So a number between 0 and 1. So I can make that parabola squeeze and not so squeeze. I can say okay instead of minus I make it a plus. Okay it goes downstairs. Instead of square I make it a cube. Okay it will look like this. I can also multiply another y for example. I'll get something completely else and so on. So the idea is to play with equations. To really play with equations. I have an equation which paints my surface or curve. So the next level now is to really jump into the fantasy surfaces. So we add in order to have a surface in space we have to add a dimension. Here as you see we only have y and x. So what we can do is to add a set coordinate. But before that I just wanted to show you one equation that you might know. It's x squared plus y squared equals to 1. Do you know that one? That's the circle. So that's all points that are in that circle. If I now look at that circle in three dimensions. So adding a set which is not there. If it's not there it means it can have any value. So if I turn that circle it's nothing else than a tube. You see the set value going back and forth. Again up to infinity down to minus infinity you would have a cylinder or tube. And of course I can play with the set. I just added here and I would for example add set square and I would have a nice ball, a sphere. And I can play around. I can say okay I'll open up the sphere. Here I make it a cube for example. Or again here I make it a cube. And one interesting thing is with these equations I can separate them. I can say I have one equation and I multiply it with a second equation. Let's say times set. So that's instead of planet earth we have Saturn. And the interesting thing is this is algebra downstairs. Which means you have a product. Something times something equals to zero. So you know what happens if you multiply two things. So if one of them is zero everything is zero. And the interesting part here is you multiply in the equation and you would add in the image. Because every image is independently turning the equation zero. So I can just multiply here by y. I get another sphere. Another plane. I multiply by x. And I would get another plane. So it's very, very interesting what's happening when you multiply and you add in the image. And this is now equal to zero. But I can also make them not equal to zero but equal to some small value. So for example here I have a small value b. Which I can change. I can make it bigger and smaller. They're not independent anymore. But they're connected to a small value. So they would kind of melt together or kind of pull each other apart. So what is interesting is that you can take any equation and you can just play with it. Here you have an equation. And then you just multiply something and you would look at it and get a nice image. And as you've seen with the spoon all the points you see here are just the points that fulfill this equation. This is the first bonus level. So the bonus level is they always come along with some gadgets. So the first bonus level is called the heart bonus. So we've read a lot of postcards. We're going to just give through. Choose the one you liked. And of course we're going to look at the equation of the heart. I don't know if you ever saw it. But it's rather a bit of a longer equation here. Not too long. And this is the equation that gives you all the points on that beautiful heart. So what once happened, if I change that equation, I just played around, you see that last set cube here on the equation if I make it a set square. What do you think would happen? Difficult question. No mathematicians in the world would know that because you have to compute all the points in space and put them in that equation. But with this software we can just try. So I'll make it a square. And the funny thing is that the heart gets a nice slip. You see that? And we have a proverb in German which says that my heart just fell into my trousers when you get a bit nervous. Embarrassed. So I think in English it falls into your boots or some it falls further down. I don't know exactly the proverb. But in Germany it gets stuck in the trousers in the slip. So here we have that equation for heart with slip. It's interesting that it exists. And what is also funny, we can look into the heart. You see these are surfaces which are not infinitely big. You can call them compact. They're not extending like the ones we had before. And we can cut them. I just zoom in a little bit and then I would just cut the heart on the corners so I can look inside the slip. And if you look inside, it's really just a mathematical surface. An infinitely thin surface in space. All right. Now we have the option to go to the alien versus virus detour to show you some very, very special constructions. Or if you're up to, we can also go directly to the singularity monster which is always open questions of mathematics. Let's briefly look at the alien versus virus. So the idea is how can we create surfaces that look like this. Very thin kind of smiley virus structures or something like this. So it's always difficult to question how can I create an image that looks like this because I have to work with equations. And there's one simple trick if you want to show an intersection curve. What you can do here, we have two cylinders. If you look at the equation, I mean you see them here, the cylinders, they come also in the equation. So this is the first cylinder. So it's x squared plus y squared minus 1. And here you have the other cylinder, x and z. So it's just another direction. You can also here if I make it a y instead and I would just change the direction. Now, can you imagine how the intersection curve of these cylinders would look like? The curve where they meet here in the center. Do you have any idea how it could look like? So we can also try to display it. So what we have to now change the equation in a way that it shows both equations. So there's one mathematical trick that you can do. You just square the first equation and then you would square the second equation and you would add them. I don't know if you remember, but if you square something, it's bigger equal than zero. It cannot be negative. If you take the sum of two squares and you put them equal to zero, both of them have to be zero. So now what we see now is both of them together. And both of them together is just the intersection curve of these two. And with this trick, we can create now alien surfaces. So what we would do, we just take any surface, here for example, the hummingbird surface, and we would intersect it with any other surface. I don't know, let's try this one for example. So what I have to do is I take the first one, I square it, and then I add the second one. Oh, this looks like spaceship room. So I take the second one. And then I would, in order to show the intersection curve, add a very small value and then I get these very thin structures here. And I can still change the equations and would get other different types of aliens. All right, I was talking about the monsters of each of the lands. At the end, you would always have a big boss fight. And here it is all about resolving open questions in mathematics. And the open question is related to singularities. Sometimes you would also call them catastrophes. So what is a singularity? I give you a very simple example. Look at this so-called double cone. And you can imagine that instead of living on a sphere, we live on a double cone. It's a very different equation. It has a minus here and a minus somewhere else, and then it would be the sphere, you see? And I just change here one plus and one minus and I would be back on a double cone. So it's not that different. And imagine you live on that planet and you have your little house up here and your very best friend lives down there. It's not a problem. You have to take a little bit of a detour and take a T. But what happens if there's a little of a wind or something happening on that cosmic equation system that changes a parameter of that equation? If you look down here, the A now is put on 0.5. So now I take that A and I put it on, let's say, 0.49. I just change it a little bit. You would see the planet, the whole geometry, would change drastically around this singularity. And that's exactly what happens with singularities. They're very sensitive towards little changes in the equation. So now, of course, it's easy. I can visit my very good friend just going through the hole, which is nice. But imagine the wind comes from the other direction and changes it to 0.51, not even one, just a little bit to the other direction. Now the two worlds are separated and I would never see my very best friend again, which is not what I want to have. Catastrophe again. If you look at this thin, like a peak, or this single point, you would always have problems. And actually, this happens a lot, for example, in robotics. If you have robot arms going around surfaces, they would always get stuck in the singular points. So what do mathematicians do, of course? They solve problems. So there's one side of mathematicians and they try to resolve the singularities. And actually, it's very interesting. There's a proof by a Japanese mathematician, from the 60s, which is still one of the biggest proofs in this whole area, algebraic geometry, which says that in principle you can resolve the singularities. You can find for each of these surfaces, you know? For example, I take this one, the lemon surface. It has two singular points. You can spot them easily, one here, one there. So the proof says, given any surface, it doesn't matter, this one, I can find another one in a higher dimension. Let's say this is in three dimensions. I can find one in four dimensions, five, six, seven, any higher dimension. And that higher dimensional surface is completely smooth. There's no singularity. And its projection down to a lower dimension is exactly what is my lemon. I'll give you one example. I have to do it here from the backside. So let me see. I'll try to make a singularity. You see that one? The peak here from top, that's a peak in two dimensions. If I show you the peak, it's not a peak. It's completely smooth. You see, there's no corner on this one. It's just how you put the curve in three dimensions and how to make it in two dimensions in order to have the peak. So you can resolve this one. You can go up and at the end, I do all my calculations here, my robot arm here, there's no problem. But, you know, the proof is very nice. It says there exists one, but the proof doesn't say how to find it. And that's, for example, still an open question about how to find for each of them the smooth surfaces. Now, what is interesting is that you have the mathematicians who try to resolve the problem, and then you have the other mathematicians who try to better understand the problem. So this is like, if I cannot find my enemy, I make my enemy my best friend. So what do they do? They create singularities. So they become evil themselves and they say, okay, now I'm going to make the worst surfaces of the world. I take, you know, any equation, which has a lot, a lot of singular points. I was talking about the lab septic, the really bad one, and I'm going to show you one first, which is the Bart Sechstich. Usually these equations, they have the name of the inventor, of the creator of the mathematician. So here it's Wolf-Bart. And then it's called Sechstich because this one is a very long equation, and the highest, like the degree of that equation is 6, which means the highest exponent for one part of the equation is 6. There cannot be to the power of 7. It's not allowed. But everything else is allowed. The equation can be very long. And now the question is how many singularities can you create with an equation of degree 6, with a Sechstich? How many? And this, Mr. Bart, found that this surface here has 65 equations. And it's interesting that he constructed one, and he also made a mathematical proof that says it cannot be more than 65. So this is a world record-safe surface. We call it this way. So you cannot beat it anymore. But if you go to the so-called septics, so now you allow to the power of 7, degree 7. So we have this one, found by Oliver Lapps, a German mathematician, and this one has 99 singularities. 99 singular points. And there's a mathematical proof that says more than 104 are not possible. So now it becomes very interesting. Either you find a better proof, or if you construct one that has 100 singularities, just by changing here the equation, you can just play around, and you find one that has more singular points, it would immediately become a famous mathematician, and the surface would be called after you. I don't know what, yeah. So I can show you here the table. The D is always the degree, like the highest exponent of the equation, and below you see the one that was found already, like here 99, and below is the proof. So you would see until 6, there is one available with 65, and we know that 65 is the maximum, but here we just don't know anymore. And that's an open question in algebraic geometry. So I think we are ready for a bonus level. Bonus level just gives us impressions of very, very nice surfaces. These are all constructed again just through equations. So we can have a cup of coffee, a cup of tea. We can also go to the toilet. Very important equation. Here we have today that one. Beautiful artistic equations. Here we have a butterfly, or a Buddha, and here is one. It's very interesting. There's one hobby mathematician, and he found that one, claiming that he broke the record of the 99 singularity, but at the end he just miscalculated. So it was another version of the 99 singularity image. You can also eat singular surfaces. You can also wear them if you want. So there's a fashion designer in Slovenia. He put that on very nice models. He made a whole collection out of singular surfaces. All right. I think we are up, or we are ready for a new continent. And as you see here, the little red star, it's a detour, and the detour goes from algebraic geometry now to the country of polytopes on the lower left side. And I'm going to show you how. So if you remember, we had the equation of the sphere, and now the question is, I don't know if you know what is a polytope, but we're going to see it, but how can I connect algebraic geometry to polytopes, so to polyhedra? And one thing is, if I just change the equation of the sphere, and instead of two squares, I just make it to the power four. You would see the square look a little bit more like a cube. So I can make it higher, make it to the power of eight, and it looks even more cubic. So I can even, you know, let's say I make it to the power of 14, and it gets more like a cube. So the higher I go, as long as the numbers stay even, I would become more and more like a cube. As long as, as soon as I put an odd number, for example, 13, it would immediately break up. And why? Maybe you have an explanation for that, we can show you here. That's not very difficult. It breaks up because if you have an even number, negative numbers, an even exponent, negative numbers would behave the same as positive numbers. So you have a kind of closure. You know, minus two squared is the same as two squared. And that's why it looks more round or not. Now, polytopes are different type of mathematics. And the good thing is with the world of polytopes that we have the multiplayer enabled. So now we can play a few of these games all together. And the only thing is with them, with the world of polytopes, that there is an entry test. We cannot just enter and look at the different levels. We first have to all together which is called match the net. So you can take your iPads out. And if you if you go in Safari on one of the browsers and you put match the net dot de, maybe it's still there. And very important, put your iPad horizontally. You know, if it doesn't change the orientation, maybe you have to pull that little switch on the side. But it's very important to keep it horizontally. And then you would enter a game that looks like this. Match the net. So here you have the website match the net dot de. And as you are all video gamers I would not explain the game. The idea is just to make the high score. And I give you 10 minutes or 5 minutes. So just to motivate you, the one who has the highest score would get a very nicely 3D printed shape geometry which comes out of the algebraic geometry world but it has straight lines on it. So it kind of connects to the polytope squeed geometry world. So just one tip. So you can change the difficulty level here by just clicking somewhere between easy and difficult. And you can change the number of polytopes. The less the easier, the more difficult it is. And if you want to change difficulty, I think you have to reload the first page. Because sometimes because of cookies it doesn't it kind of doesn't change the difficulty. And I would not explain anything else. You just try to play and break a high score. Make a very high score. And I would help you if you have issues or problems. So if you have any scores already, just tell us. Maybe I'll help you out a little bit. You probably figured it out. But you have to match the planar nets of the polyhedra. You have an object on top. And this object would unfold in something below. And now you have to look what matches to that one. For example, this one looks like it would match here. And if you found the match, you click on submit. Then you would see, okay, this was right. This was wrong. And you can go to the next one. And you have five rounds. And after five rounds, you would get the sum of all the points you made until then. And we're aiming for something like, I don't know, 60, 70, 80 points. 40? Okay, so the first high score here is 40. Somebody beats that. 10. No, that's so it's 10 points. That's good. With the simplest one, you cannot get too many points. So you have to take three. You can restart. But it's good. All three correct. Go to the next one. Now you get the next challenge. 60. So the next high score is 60 points here. First row on the left. Again, if you want to change difficulty, maybe you refresh the page. Because sometimes it would not take the... So you go and restart and then you just refresh. And you can change a new level of difficulty. So any new high score? 70? 77. Wow. That's very good. So I just tell you, the maximum number possibly is 145 for the high score. But it's really difficult to get. Anybody beating 77? Ah, look at this. How much do you have? 80. Wow. That's cool. 80 points here. But I'm already getting the trophy out. And approaching the 80 points person. You still have a few seconds left to break that. I think I'll give it to the first person with the 80s. But we give you a special applause for the second 80s. So thank you. Thank you too. Now you can keep the iPads out. We have a few levels that we can play together. So the only thing you have to do is to put the link that I give you. It's always momas.imaginary.org slash a special key. So the first special key is just one. And if you put that in your browser you have that momas.imaginary.org that would give you a simple version of what we just did. And now let's do it calmly. So this was the game and now let's look at it. So what is a polyhedron? So it's an object like this that has facets. You see the different facets here, you have pentagons as facets. So the definition is that you have polygons. Polygons are just whatever triangles or squares or pentagons or hexagons. Any number of vertices that give you facets and if you put these facets together you would get an object. Now the definition of a polyhedron is that within this object if you go from one point to another point any point to any other point you can do that in a straight line without leaving the object. So you can imagine here I go from this point to another one and with a straight line and it works. And the object itself doesn't have any intersections. So there are five the very famous five platonic solids that you might have heard of. It's these five. And these are the only objects that are built with facets that are all equi, have the same length. The same type of facets with the same length. So you have here the cube is the most famous one probably. There are six squares that build up that cube. And all the other four platonic solids are made out of triangles. All triangles are the same. So you can only build these five. And you have seen the nets. Now what is interesting coming back to the open questions in mathematics it's still not clear for any type of polyhedron how to get to that net. I mean you have seen the game we calculate the nets. But that's still an open question. It's not very easy to do that. And for many polyhedron you would have a lot of different types of nets. For example the cube you can imagine you can rearrange the net and you would have 11 different types of nets for the cube. For the tetrahedron for the tetrahedron it's easy there are only two different type of nets that you can make. But for the icosahedron like this figure here this one you can actually have 43,380 different nets that recreate that unique object more than 43,000 different nets so of course it's tricky and if we go to the next to the next level the task we go to polyhedron morpher instead of 1 put into your browser 7 it's mo-mass.imaginary.org slash 7 can you do that so the next number is 7 and you would get to this program mo-mass.imaginary.org slash 7 so what we can do now we choose one of the 5 platonic solids 3,345 for example here the tetrahedron and we can apply one of 3 operations you can see here you have one operation maybe the easiest one is to cut the corners it says move points on edge but it's actually just to cut the corners so you can see here you can cut the corners I can shrink the facets I just make them smaller or I can take the center point of each of the facets and pull it out so now we have the next task who can create a soccer ball a soccer ball which looks like this using that formula morpher polyhedron morpher program so you have to choose one of the 5 platonic solids and then try to apply operations in order to get that classic soccer ball which has pentagons and hexagons so anybody you have one here you have one so what base object did you use which one of the 5 to do it I'll switch back to the program if you take the icosahedron that's the solution and you just cut the corners you see until you get a nice soccer ball it's interesting that many people would start with the todicahedron here but you would not get to that soccer ball because if you cut it you would always have to triangle so you have to start with a triangle and you cut the corners here with this one and you shrink the facets here and this one ah yeah ok very clever of course very good but is it the same type of soccer ball or is it a different one let me just check this one has 5 hexagons and just look at the other one and the pentagon in the center perfect next level if you're up to I just wanted to show you that polyhedra they're connected these are the classical ones you saw the 5 in the center the 5 platonic solids if you start cutting them you would get the so called acchimedian solids so the one we were looking at was the truncated icosahedron soccer ball but then you can continue applying operations and you would get the very famous catalan solids and then you have the chonson solids and all types of solids and it's really interesting to study one into each other and there are a lot of properties to study very simple objects but a lot to study mathematically good if you look at the monster of this level it's on the far left down side the 120 cell I can already tell you that this is a monster that lives in the fourth dimension not in the third dimension so we're going to build up now on a few levels which are not multiplayer but we're going back to the multiplayer again so just introducing dimensions so let me show you this example how can you imagine the fourth dimension let's start with the simplest thing which is just a point the point lives in zero dimension there's nowhere to go it's just one single point because I switch from zero dimension to one dimension and I have a line you know I can switch to two dimensions I would have a square you see the movement so I just have the line I kind of extend the line I would get a square then I take that square and I make a cube of course always have in mind that what we see here on the screen is two dimensions so it's a projection of the cube but we can imagine a real cube so now what happens if I switch from three to four dimensions I can do that I would just do the same you know I take the basic structure and I just drag it into one more direction and what is very interesting if you observe what happens that we all know what is a point what is a line, what is a square what is a cube but we don't know what is a hypercube and we don't have a feeling because we don't live in a four dimensional world but that's the beauty of mathematics that it's just a number you know it's like dimension N in this meaning it's four or five or whatever and you would just apply the same kind of definitions to each of them if you look at the different objects you would see that the facets they are always one dimension lower for example the facet of a square here is a line if you go to a cube the facet of a cube is a square and similarly the facet of a hypercube like the four dimensional cube is a cube so here we would have cubes as facets now if you go to the next level I don't know if somebody of you has seen the film Flatland or read the book Flatland so it's a very interesting story that geometric objects the heroes of that film and book they live in two dimensions like if you can imagine an ant that cannot look upwards or downwards it can just look left and right and go forth and back imagine that you want to explain this ant who lives on that plane in the two dimensions in Flatland the blue world is Flatland and we are the gods of the third dimension we want to show the poor two-dimensional Flatlanders the ants living there we want to show them how does a cube look like so what we can do we just take the cube and we send it through their two-dimensional plane and you see what the thing what they see what they can observe is that square here of course they would look at the square from one side if the square is transparent they would look through it they can walk around it but they can really perceive that square you see so what I can tell them okay look this is a cube just look at it this is a cube but what I can do I can just I can also turn that cube and I can say okay now this is also a cube look it looks like this do you get it little ant I can also turn it the other way around and say okay look now I turn it again this is how the cube looks like you see so we can do the same you know let's say we are the little ants now living in the third dimension and there is the god of the fourth dimension and he or she wants to show us the fourth dimensional hypercube so what that god does he or she just sends that polytope through the three space and shows us the intersection with the three space with our space so we do the same now I'll show you so the fourth is four dimensional objects sliced with our three dimensional space so I can do the same so this is what I see now we can see three dimensional objects because we live in the three space so I send the hypercube now through this is how the hypercube looks like but I can also turn the hypercube and I can send it through again you see turn it I have one direction more to turn it not only three this looks like maybe familiar here it looks like a cube depending on how I turn it so you can imagine the flat lenders they don't have an easy job to understand three dimensional objects by just looking at the sections so what we can do is we can apply here too if you look at this pyramid the tetrahedron it's a three dimensional object but we show it on the screen so we do a projection of it you can see that's a projection from three space to two space and now I'm doing the same projection from the fourth dimension to the third dimension so this is the hypercube you can see that's the four dimensional cube and it's very interesting it can turn it in many directions I can turn it this way I can turn it that way I can turn it this way many more possibilities I have six degrees of freedom to turn that hypercube and now we are going to get to the monster you see in three dimensions we have five platonic solids that's the bodies which are built together by the same the same facets the same facets with the same length like the cube but in the fourth dimension I have not only five, I have six and the one we didn't see before was this the 120 cell so we can have a look at it it's interesting that this 120 cell I think it has totally cohedron as facets I can turn it in all ways I can mark one facet here you can see the green one it's just a facet of this 120 cell and that facet is a dodecahedron try to turn it you can see alright I think we are going back to so this was the hyperspace the four-dimensional platonic solids I think let's go to get a detour via the monster in order not to get dizzy because there is a very nice underground world you see the pipes they all connect downstairs to the underground world there is a different abstract mathematically connecting all the different planets below the ocean and now I'm going to show you one connector between many of these worlds it's a different type of geometry called dynamic geometry actually it's used to do all types of geometries in an interactive way it's its own field what we are doing now is all dynamic geometry because we put something and we can change it directly so that's software used to interactively display geometry with this dynamic geometry we can connect the different continents and now we are going down in the underground and on the way I'm going to show you one application of dynamic geometry a robot simulation I'm just going to start it presentation of a robot who cleans a staircase you see that little robot it's going up and down and it has some small whatever cleaning pads on its legs and it goes down and then you have a nice lift that brings the robot up again and what you want to have is to have in all the skyscrapers of New York for the staircases you would have robots cleaning them and what you want to have is a robot who is really fast, really clean really reliable, it never gets stuck and it's suitable for all types of staircases so that's what you want to construct now you can do a simple geometrical simulation what is interesting here with the dynamic geometry I can just change the staircase I think I bent the legs of the robot oh yeah still moving but it's a bit slow you know I was moving the soil below its feet but it's still crawling up on the lift so you see this one is not the ideal one because you just change the staircase a little bit and it gets very slow it doesn't even hop on the lift and I'll help a little bit out so now what we can do of course we can have a look at it we can change the staircase we can have the robot which is really nice I just make here one super leg and one big body and let's see what the oh no that's the wrong one so you can see what this one with the super leg is doing it's too big okay now it's coming down okay so this one is maybe very clean but it's also a bit slow oh it moves so now we would have a model this is a very simple model of a problem the problem is construct the best robot possible so you would define it and then you would simulate and on the way you would have a lot of mathematical proofs and then you would for example be able to say that there is an optimal robot it's a climber robot that's another type of robot so there's an optimal robot for this staircase for example which is the fastest but then you would have I don't know other robots which are super generalized so they can work in all the skyscrapers but they're probably not the fastest for each of them but on average they're the fastest for all and so you define certain criteria and then you would build the best robot mathematically build it and then at the end you can visualize it using geometry or not so this is a good example I'll just show you one more let's make one robot okay so that's a very interesting field in machine learning where you work with robots where you have algorithms that actually find an optimal strategy for any type of problem there's a very nice deep learning algorithms that can now play all the Atari games and beat human beings by playing better and they learn themselves the only thing is what they get is just some they have the pixels of the game and they get the high score and they want to maximize the high score so they can learn on their own and it's the same algorithm it gets stuck so it's also not the best one alright so via the robot simulation now we enter another world which is the world of fractal geometry so what is interesting in this world that I told you it's ancient from the 80s and it was the first time that mathematics went into the big public in terms of images you probably have seen it already this fractal image was very very fashionable suddenly the beauty of mathematics was discussed lay people could look at mathematics and attached to the theory of chaos was also very fashionable they have things that just this chaotic behavior which is not deterministic and interesting so we're going to have again multiplayer enabled a look at a few examples of fractal geometry so I'm going to start with one application it's the secret code here number 39 so it's again mo-math.imaginary.org slash 39 and it's an application called light billiard so it should look like this so you have a light beam sent into one direction you see and you can change the position of the beam like of the torch light and the position of the direction and now what happens it gets reflected here you have circular mirrors and it's very simple entry angle equals exit angle so the very simple rule so you send it and then it will just get reflected now what happens if I put that light beam inside a circular mirror you can create very very beautiful patterns depending on the angle you would get special stars and what is very interesting is now getting to the chaos point if I just distort it a little bit this mirror I just put this circle a little bit inside here and then it would get immediately chaotic image and this is really like changing the angle a little bit you would get a very very different image and this is one one of the definitions of generating chaotic behavior you have an iteration so you have a light beam which gets reflected and again produces a light beam that gets reflected and gets reflected and so on and so forth so you iterate something and then once you change a little bit of that whole system you would get something which you cannot predict you can play around with this and I'll show you another very interesting example which is also easy to rebuild as a physical exhibit as a physical system it's a so-called double pendulum so if you enter momas.imaginer.org as the next example you would have a simulation of a double pendulum so how does a double pendulum work you have a fixed point in the center you have one long pendulum and on the end you attach another one so now what happens if you let this loose you would get immediately a chaotic behavior you cannot really easily predict the position of this pendulum in 10 minutes for example so you can turn on here the trace of the small pendulum and what you can do this is especially important for Richard our space astronaut you can put gravity on zero that's extremely important so you can look at the pendulum you don't have to fly to space you can look at the behavior of that pendulum you have no gravity or you put gravity the other way around on the top below what you can also do is to change the positions of the pendulum so I can make one longer than the other and have a look at it how it's going to behave of course this one you can easily construct and you can look at it physically how it would look like it's really amazing you turn it and the big task is can you turn it twice the same way no it's really very sensitive towards the initial positions I'm going to show you now just to relax a little bit another bonus level this is latest technology visualization in fractals which uses 3D, three dimensional fractals and that's just a movie with films that I'm going to show you I've been away like a rainbow been washed out by stone hash to paint your time paint your time I've stopped there and I'll show again the Mandelbrot set so this was the famous image and what is interesting coming back to the dimensions these fractals can have non-interture dimensions so the dimension can be somewhere between 1 and 2 for example 1.2 or 1. that makes it really interesting and again here you have an effect that if you zoom into that Mandelbrot set you can iteratively zoom to an infinite end so it would never stop you would always see details and details but the time is almost over and I just as it is for a good computer game look at the credits for this game and then I have a small bonus at the end so what we were looking at was one program, the Surfer program which is on algebraic surfaces that we developed at Imaginary they matched the net it's online, developed in collaboration with us and the Technical University Berlin and the PolyMake project which is a research library for polytopes all the other applets from the robot to the polytopes they are done by Jürgen Richtergebert a professor in Munich using Cinderella and CineJS CineJavascript they are all free to use you see the authors of the images I want to mention Thorsten Stier who made that 3D Mandelbrot film and you can find everything on the Imaginary.org website but we are going to put these special links also online on the MOMAS website and also on the website of the talk at the Imaginary website you can have the links then at home so before ending I just wanted to show you an independent video game that uses these 4th dimension slices into 3 dimensions so I just switched to a small explanation where you can see this is particularly interesting because of the known connections between high dimensional space and certain crystal structures I can move a little bit in the 4th dimension and the scene will look slightly different again slightly different there are technically an infinite number of unique slices one could take here is another more complex example the surface of this 40 shaped called the 120 cell is made out of 120 dodeckahedra in this case I cut a hole inside each dodeckahedra to make them hollow while you could ignore all this when playing the game to me it feels even more beautiful when you know more about what is happening I wanted to share some of the things you may not realize when you finally get to play Miyagakure so the game is not out yet it's called Miyagakure it's interesting that it just plays with the slices of the 4th dimension alright we can go to questions I would have one extra app if you want to play but I don't know I think we have a few minutes left for questions so thank you very much so if you have a question raise your hand I'll bring the microphone to you we were seeing a 2 dimensional projections of a 4 dimensional hypercube is there a way to see a 3 dimensional projection like a hologram or something so the question was we have seen a projection of a 3 dimensional of a 4 dimensional object into 2D can we do that into 3D and we were just discussing before with Richard that now with new technologies like virtual reality of course you can explore other projections into 3D space that's a completely new world that has to be explored 3D printing there's a lot of 3D printing going on in terms of projections other questions let's give him a hand thanks for coming