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  • 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 thisbius 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 thisbius band,

  • you know, it stays a Möbius 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 particularbius 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. Thebius 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 mybius band,

  • and I'm cutting along that red line,

  • I maintain that thisbius 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 thatbius 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 thebius 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 clearlybius 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,