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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 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,