Klein Bottles - Numberphile

Key Takeaways

math aid about numbers if you think math is about numbers you probably think that shakespeare is all about words you probably think that dancing is all about shoes you probably think that music is all about notes math ain't about numbers math is about logic it's about beauty it's about connections it's about how you get from one place to another and for me the cool thing about math is a part of it called topology i'm not even going to tell you about topology because what i'm interested in is a little section of topology called non-orientable manifolds more specifically i'm interested in klein bottles this is a klein bottle wait a second wait a second we haven't even touched upon a mobius loop how can i talk about a klein bottle without talking about a mobius loop i take a piece of paper as everybody knows if i have this piece of paper it has two sides four edges turn around this way it becomes a cylinder give it a half twist and a piece of tape [Music] becomes a mobius loop well the cool thing about a mobius loop then there's about 50 or 60 cool things about a mobius loop is that it's got one side it's got one edge take two mobius loops one left-handed one the other right-handed one and i try to connect them hey this one has one edge this one has one edge what happens if i take two mobius loops and sew their edges together well i know from from experience that if i have four edges here and four edges here and i connect them oh i'll get something that's fewer edges so if i apply that to a mobius loop i've got one edge here one edge here sewn together i should get something that has oh i'd lose this edge i'd lose that edge i ought to have something with no edges cool if i take two mobius loops hook them together i'll get something without any edges okay i'm gonna try it hot damn far out here's a mobius loop here's another mobius loop the cool thing is that if i take two mobius loops and try to sew them together i end up with a shape that requires four spatial dimensions to exist and that shape is called a klein bottle named after felix klein felix klein this guy in 1882 came up with the absolutely nifty idea that if you can take a cylinder turn it around and orient one end of it and loop it through so his idea was take a cylinder bring it through itself and have the end of the cylinder welded to the base that this would have the cool property of having only one side well what do i mean by one side if i have an ant crawling around here ant or caterpillar crawls around oh clearly it can cover this whole thing it can then slide continuously over there through what through this tube and get to the other side of this piece of glass we never cross an edge to get to the inside absolutely nifty in an ordinary let me get an ordinary bottle ordinary bottle well if i take if i leave the cap on the bottle clearly it has two sides here's an outside and the inside of the bottle is where i'm touching right now an ant walking around here couldn't get to the other side if i take the top off watch this an aunt walking along here can get to the other side but he has to cross a peculiar location called an edge if i make this bottle arbitrarily thin this edge becomes sharp and the ant cuts cuts herself when she tries to walk from the outside to the inside in other words an ordinary bottle has two sides an outside that i'm touching and an inside that my finger is touching right now and they're separated by the lip of the bottle and okay we all know this a klein bottle meanwhile watch this you can walk around here just like walk around here but the other side of my finger i can get to without ever crossing a sharp edge by carefully walking along never crossing an edge going through this tube and getting to the place that's just on the other side of my finger without ever crossing an edge in other words this is a bottle without an edge cliff i'm comfortable with that except you just told me that this wasn't possible in three dimensions if we lived ah if we lived in a universe with four spatial dimensions four space this tube wouldn't have to intersect itself this weird place where there's self-intersection right there that would not self-intersect and an ant walking along this way we go and go right through this piece of glass here and keep walking to the inside and an ant walking around this way would go right through this and keep walking this way a cliff that hole that you've created there so the tube can get through is like the floor this is worse than the floor it's this place right here where the where the manifold where the glass intersects itself it's a figment of living in a three-dimensional universe if if if we lived in four space we could go right through on the other hand that guy felix klein showed that in three dimensions you have to have this intersection in a quality universe that has four spatial dimensions klein bottles would be terrific here unfortunately my klein bottles have to have this self intersection when i make klein bottle hats right here's a here's a hat that is the same thing as that glass klein bottle instead of welding it together i've knitted this hat so that the peak of the hat comes through itself so that the hat itself can the loop of the hat can be pulled through itself and turn it inside out but i'd rather not do this because it takes about five minutes to do and i'm stretching things that shouldn't be stretched but ultimately i ought to be able to pull this sort of inside out sort of it's continuously deformable back to itself cool and along with a matching mobius scarf ah i'm i'd like to say i'm ready for a california winter look look look look look friend of mine robert lang made a paper origami klein bottle i mean absolutely sweet one sheet of paper it's an origami klein bottle one of the neat things about mathematical topology is that you don't care what size something is i can expand this contract it i can make this i can stretch it and it'll still have the same mathematical properties same number of sides same number of holes same number of handles instead of making a big one i could make it smaller i could make an even smaller klein bottle i mean i can make a klein bottle even smaller to a mathematician these are all the same thing and of course i can go in the other direction as well i can make them big so i had to put a thousand klein bottles someplace so i put them under my house and the wheels you know their little motors drive the wheels around [Music]