Subtitles section Play video Print subtitles A 2-manifold is a thin piece of surface, like this piece of paper, or this piece of thin plastic. But a 2-manifold must not have any branches. A book that has many pages, but has a spine where they all come together, that would not qualify as a 2-manifold. But even a weird thing like that would qualify as a 2-manifold. 2-manifold may have borders such as this MĂ¶bius band. Or it may have holes, like this particular Klein bottle. Or it may be completely closed, like this surface of an icosahedron, or the shell of an egg. Now for topologists, all objects are made of infinitely stretchable rubber. And so the geometry really doesn't matter. And if I deform this MĂ¶bius band, you know, it stays a MĂ¶bius band, and the topologist would still classify this in exactly the same way as the original shape. From this point of view, all of these 2-manifolds can be classified by just three integer numbers. The first one is the number of borders. This particular MĂ¶bius band has just a single border. I can run my finger around the border here, and if I go around the loop twice, I'm coming back to where I started. Brady: "So this is like the edge." This is the edge of the thin surface. This particular plastic piece has two borders, you know, one is a circle here, and the other one is this particular edge here. Brady: "So this is the part where if an ant was crawling on it, it would cut itself in half." That's right. And if we look at this piece of paper, this one has three borders. One is the rectangular frame on the outside, and then we have, you know, a circular border here, and sort of kidney shaped border over on this particular case. So this is the first number. The second number is the sidedness of a surface. So this piece of paper is clearly two-sided. As a matter of fact, It has a blue side, and it has a white side, and you cannot get from one side to the other without crawling over this very sharp edge where an ant would cut itself to death. The MĂ¶bius band, on the other hand, has a single side. If you start painting and spreading your paint without ever crossing one of the edges you would come around, and you would notice that by the time you're done, you have actually painted both sides of this particular surface. So that's a single-sided surface with a single edge. On the other hand, the Klein bottle here is also single-sided surface, but, as you can see, it has no edges. However, we cannot really create a completely closed Klein bottle in three-dimensional space, without having either some opening, some punctures, or a self-intersection line. And you can see here, here is one of those self-intersection lines. Now if I don't like these self-intersection lines, I could cut a slightly larger hole into the wall of the thick arm, so that the thin arm then can be passing through. So in this model, rather than living with the self-intersection, I have cut a large enough opening into the green body so I can bring out the thin arm without creating a self intersections. But for the price of creating an opening, a so-called puncture, which has its own border. So this would still be a Klein bottle, but now it is a Klein bottle with one puncture. There's an important message here. And that means that simple holes that you cut in a surface do not change the type of surface for a topologist. Even though this little thing has a whole lot of little holes, it would still be considered a Klein bottle, but with many many many many many punctures. Now from that point of view, this little cylinder is really just a sphere with two punctures. And this becomes a little bit more obvious if you try to cap off these openings. So we put a cap on here. Now it would be a sphere with only one puncture, and by the time I'm adding the second cap, it becomes more obvious that this is topologically just a sphere. There is a third number that is important for the classification of all 2-manifolds, and that has to do with the connectivity of the surface. It's called the genus of a surface. A sphere, or equivalent topological shapes, would be of genus zero. A donut would be of genus one. A two-hole donut would be of genus 2. And here is a more complicated object, and the surface of that would also be a 2-manifold of genus 2. The genus is clearly related to how many holes you have in your donut, or how many handles that there are. A more precise definition would be, how many closed loop lines can you draw on that surface, that if you were to cut along them, the surface would still hang together? If I take my piece of paper, cut a circle loop into it, then the inner part of that hole falls out and is clearly no longer connected. On the other hand, if I take my simple torus, and I cut along this red line then I simply get some kind of, you know, tubing, but it still all hangs together. So this is of genus one, because I draw one such line. If I had a two hole torus, I could draw two such lines, and I can cut along those, and after both those cuts the surface still all hangs together. So that's a surface of genus 2. If I take my MĂ¶bius band, and I'm cutting along that red line, I maintain that this MĂ¶bius band will not fall apart. Now, what do I get? I get a loop twice the size. It has a twist of 360 degrees in it, but it is still in one piece. And since I could do one such cut, that means that MĂ¶bius band had a genus of one. Try to cut it again in the middle, then it actually would fall apart. There's only one way of cutting the MĂ¶bius band along a line and then we're done. to see whether it's not perhaps genus two. Here, ah, now I've two individual pieces, and I cannot get from one piece to the other, so the surface is no longer connected. So clearly MĂ¶bius band is only genus one. The Klein bottle is actually of genus two, and I've tried to indicate that by drawing two such lines which I could cut along. The red line on the one side, and the green line on the other side, and if I did that, the surface would still be hanging together. So, the Klein bottle is a single-sided surface with no borders, and it is of genus 2. Now I would like to make better Klein Bottles of a higher genus. Now how could we do that? I would like to build these super Klein bottles in some modular way. The first possible way of making a module is what I'm showing here. I'm essentially taking the top half of the classical Klein bottle that has the important mouth. The way it is, this is still a two-sided surface. To make that clear, I essentially painted the inside here in silver, so the inside of this green toroidal body is silvery, and then it is being brought out here in this silvery stem, which by itself, on the inside, is still green. I want to use two of these modules to make my super Klein bottle, and I'm contemplating on essentially bringing together into a ring underneath these curved connectors. But I can see if I try to put them together like that, then silver meets silver in the upper branch, and green meets green in the lower branch. And so I never really stepped from green to silver, and so that cannot possibly be a single-sided surface. It's a perfectly good, you know, two-sided surface. Ah! But what if I turn this thing around? So maybe I want to connect it like that. Let me actually try to do that. Some assembly required. So now I have created this contraption. And it looks good, you know, I'm getting from green to silver over this branch But then wait. If I go through lower branch, I get, once more, a green silver transition. And that means I have an even number of changes between the two surfaces, and so I'm not really getting a net single-sided surface. So again, it's two-sided. That's disappointing. Maybe we can put three of those into a loop. So here you can see a loop of three of those Klein bottles. We have three transitions. Clearly, it's an odd number of transitions between the surfaces, so the whole thing is single sided, but if we analyze it in details, we find that the genus is still only two. So there's really nothing new over the ordinary Klein bottles. So maybe we should make something more complicated. What do we need to do in order to get a super Klein bottle of higher genus? You know, nothing seems to really work. Well, in order to get this done right, we need to do what mathematicians call a connected sum of two or more Klein bottles. You start with two ordinary Klein bottles, and then you cut the puncture in each one of them, you connect those two openings with an umbilical cord. And now if we do that, we actually get a single-sided surface of genus four. Here is my simple Klein bottle. But now I have added this branch with a puncture, and here is another Klein bottle with a puncture, and if I put the two together, now I have what I call the first super bottle, which is a surface of genus four. The crucial thing was to create this extra branch here that allows me to have this component coupled into another component, and this is a modular component that now allows me to make higher genus Klein bottles or what I call super bottles. So truly, now, this is a single-sided surface of genus for with two punctures. Because that's what I need if I want to see this thing in three-dimensional space. Brady: "But in four dimensions this would work" In four dimensions I could do it without those self-intersections. Here is a more compact version of one of those genus four single-sided surfaces. There are clearly many possible ways of creating such a branching Klein bottle module, and I wanted to be this as general as possible, to make many different sculptures, and so I have chosen to use this particular version, which has the three arms come out in three mutually orthogonal directions. By doing that, I can then put eight of those around the corners of a cube frame. But even this special cube corner component can be done in quite a few different ways. This is one of my first modules. Then I have made another module, and here the third module, and they all have one thing in common. The ends of the arms come out in three mutually perpendicular directions, and they all have the same distance from the end of the arm to the center of the three intersection lines. So they all can replace one another, and I can put them in arbitrary ways around the corners of a cube to make a modular super bottle based on a cube frame. They're quite different, but they're so similar in style I have made sure all of them rely essentially on one toroidal body somehow that makes them belong to the same family. But in some instances, you can see here, I'm branching out the thick part, where in this case I'm feeding out the thinner tube, and then the thinner tube is the one that branches in two. And in the third module, the thinner tube branches in two, but the actual branching occurs on the inside of the module. Each one of these different modules can go in different places, and each one of them can be rotated in three different positions. A few additional curved elements. I can make different shapes. For instance, a three-sided prism. This is a version where I have a fairly regular prism by adding three pieces of curved branches that each one turn through 30 degrees. Here is a different version where rather than having three inserted pieces, I have two pieces of 45 degrees, to make this, you know, somewhat less regular triangle. And then I just copy two of these triangles behind one another, and I get something that looks a three-sided prism. Another option is curve scatter through 39 degrees, and then in this, I can use four of my modules, and six of those bent pieces, and I get something that emulates a tetrahedral frame, a super bottle of genus six. Whereas, this here was a super bottle of genus eight. Of course, once I had all these parts lying around, I was just playing, and here are a few more fancy shapes that are much less regular. So, the inventiveness now has no limits, you know, with enough parts lying around. If we introduce a new Klein bottle module that actually has a four-way branch, Then one four-way branch gives us the same increase in connectivity that we would get from two three-way branches. With only six of those four-way branches, I can make a super bottle of genus 14. Brady: "Wow, that's genus fourteen!" I also did something a little bit special in one of these branches,