These changed how I think about higher dimensions

this video is sponsored by curiosity stream home to over 2,500 documentaries and nonfiction titles for curious minds here we have the classic game of snake and you can think of this as a 2d snake living in its own little universe but here's a question what shape is that universe now some of you probably said a square and that's right yeah this is not a trick question because well we can see it's a square with edges and a boundary that the snake cannot escape alright now what about this game of asteroids what would you say is the shape of this universe here some people say a rectangle and others will say a torus or a doughnut which will cause the people who said rectangle to say what but it's true in a way let's rewind because notice there's no boundary this time there's no edge that stops our character after going through one side of the rectangle we appear directly on the opposite side and same goes for the left and the right so how can we mathematically differentiate between a world with boundaries and one that has this wraparound effect well for the boundaries we can just show a normal square no big deal but for asteroids we represent the universe like this the different colored arrows are directly connected together so if I go off the screen where the tip of this blue arrow is I'll come back at the tip of the other blue arrow just like in the asteroids game and the same goes for the red once these are both directly connected now I actually drew the arrows on a sheet of paper on both sides because instead of just saying oh hey this sides came to this one you go up here you come back here what if we were to physically connect them physically connect this side to this one well we were to do that here so it's what the arrows lined up we'd have first get a syllabus right but now I still have to connect these arrows together these I just also connected right and in order to do that I would have to physically you know wrap the paper around itself which I can't really do that would crumble a lot if I tried but if this were made of some stretchy rubber for example and I could do that connection then we'd be left with a torus as you can see here so with that connection that arrows would line up and this is why the universe of the asteroids game can be thought of as a torus because they're topologically the same these are not the same geometrically though geometry is concerned with things that change when you deform and morph an object like angles lengths areas curvatures and so on since the torus is curved and the object on the Left known as a flat torus is not they have different geometries but their topologies are the same because topology doesn't care about smooth deformations it only cares about poking holes or tearing the surface for example and real quick just note when we're discussing a torus we're only referring to the surface because we can only play the game on a flat 2d surface right and the connections that we do don't change that we're still only on the surface itself there's no there's no inside so this remains a 2d manifold it's just a surface but it is embedded in 3d space but now this opens us up to a new way of thinking about higher dimensions we know this represents a torus topologically so what would this be or this or this if you've never seen this and want to think about it on your own they go in order of difficulty this one is the easiest as it's a cylinder it says connect these two sides such that again those arrows line up and we saw before that leaves us with a cylinder but there are no arrows shown on the other two sides so we don't connect them this would be Poppa logically like a video game where you go off the top and come back on the bottom but besides our actual boundaries that you cannot cross now this next one is new for us because it says we have to connect these opposites types together but not like before so we try like before with the cylinder then the arrows will be facing opposite directions that's not allowed they have to be facing the same way after doing the connection so in order to make that possible we'd have to do the same thing but put a half twist in the paper I can't really do that with paper of this size that's okay we can do it with this this is the same thing just longer paper arrows and facing the opposite directions and then if I do the connection here again wait a lot for the problem but to resolve it I can just put a half twist in the paper and now the arrows are facing the same way we do the gluing we're left with a mobius strip something we have seen before on this channel so that fundamental polygon is a mobius strip which is a one-sided 2d manifold that is non-orientable because as i've shown before an object that lives in that world and goes around once we'll come back completely inverted this would be like playing a video game we're going off the top right makes you reappear on the bottom left but again inverted and whenever this is possible whenever you can invert yourself while remaining in that space then you're living and then non-orientable universe and moving on we have this last one here which if you try it on your own with no prior knowledge probably gave you a lot of trouble because what it says to do is connect these two sides just like the cylinder no issues there but now we have to connect this side to this such that these arrows line up I have a copy of that which I tape so it's easier to hold now doing that would be very difficult because if I were to do the same you know wrap around here the arrows would be pointing in the opposite direction which is not allowed but what we can do instead it's take the bottom which I cannot physically do but I'm going to kind of demonstrate by cutting it off and moving it up here so again if I took this I took this bottom and moved it up like that the arrows were facing the opposite direction so that's an issue but we can do instead is cut a hole in this thing and take this and move it the roof that hole and now hook it up through the top and the arrows line up so that's pretty much the best I can do with paper but again just imagine we have a cylinder from before and then I have to take the bottom pull it around again imagine it's a stretchy rubber pull around put it through a hole in the cylinder itself and through the top social if the arrows are then connected and if you were to do that with material that when it tear or crumple then you would get a climb model this is something we've also seen before and you can see a better construction of it here so topologically this is a klein bottle and the only way a client bottle can exist in three dimensions is with self-intersection which is what you see right here but this object can exist without self-intersection if we had access to a fourth spatial dimension that's where you'd find a true Klein bottle where this issue can be resolved note this is still just a 2d manifold fills a surface but this a true Klein bottle is embedded in 4d space now I was talking to John from the YouTube channel epic math plan who has been a huge help on these topology videos seriously he is cleared up and fact-checked a lot for me so definitely check out his channel which is linked below but after releasing one of my older videos he told me to mention this the next time I bring out the Klein bottle in order to help with the issue of higher dimensions when you look at this picture you know hopefully that it's a 2d representation of a 3d cube even though there is self-intersection that exists on this picture which doesn't for the cube like this edge physically intersects this one there self intersection or this edge intersects this one the cube is intersecting itself in a way that doesn't happen here these edges don't cross over one another and the same thing can be said about this we're looking at a 3d representation of a 4d object here we see self intersection I'm just like here but if we had higher dimensional brains I could think about the full a four spatial dimension we would see that be like yeah there's self intersection but clearly that doesn't happen for a real client bottle or a real cube this is just a lower dimensional representation of a higher dimensional object I had never heard that before but I thought was a great way to help people kind of come to terms with the self intersection now if I were to ask you whether this is orientable or not that'd be pretty tough to answer without prior knowledge unless you go back to the fundamental polygon representation where you'll find just like with the mobius strip you will become inverted by going through the top or the bottom so the Klein bottle is or anthem okay if you've noticed so far we've only talked about 2d manifolds and where we're headed is 3d manifolds that could represent our universe and that's where things get really weird but first we need to talk about the product of shapes as in what two circle times a line segment or a client model times a circle and so on well I'll tell you the first one a circle times a line segment is a cylinder and here's why a cylinder can be thought of as either a line segment of circles all these circles stacked linearly or it can be thought of as a circle of line segments it works both ways which is why we can write this cylinder as a product of the two other shapes and you'll notice that those circles are all perpendicular to the line segments but here's a tougher one what's a circle times a circle and by circle I just mean the points that make up the perimeter not the inside a circle when just referring to the perimeter is called s1 in the world of mathematics so I can write the multiplication like this anyways to answer this we have to realize that this product symbol here really represents the Cartesian product of two sets which isn't too hard to understand if we have some set a made up of the numbers 1 2 & 3 and then another set be made up of 4 5 6 then the Cartesian product of a and B is the set of all ordered pairs a comma B where a is an A and B is in B so the Cartesian product of a and B would include 1 comma 4 and 1 comma 5 1 comma 6 2 comma 4 2 comma 5 and so on it's just every point that includes something in here comma something in here what's cool is how similar this is to regular multiplication though if you had to explain what three times four visually looks like you'd say it's four rows of three of something or vice versa and the total number of items would be the actual answer so the visual leaves us with a rectangular region of whatever apples now look at all these ordered pairs in the Cartesian product set because if we were to plot those we'd get in this case a square region it's the same idea though the first set was 1 comma 2 comma 3 so those are the only possible x coordinates or columns then the points are found in the rows or for the y coordinates 4 5 & 6 which was the other set so the Cartesian product left us with a rectangular region of points but this isn't limited to just integers or finite sets if set a was instead every single real number between 0 & 2 then set B was every real number between 4 & 5 the Cartesian product would be will every element in a comma every element in B that means we'd have 0 comma 4 point 2 4 5 comma 4 point 6 7 1 1 comma 4 point 3 and so on everything up to 2 comma 5 or simply put everything inside this rectangular region you can think of this as all of the numbers in B 4 to 5 just a thin sliver stacked on top of each element with an A from 0 to 2 by the way for those who might not know this is why the XY plane is called r2 it's the Cartesian product of the set of all real numbers with itself all real numbers perpendicular to all real numbers anyways now it's time to find out the Cartesian product of two circles what we're gonna do is draw a line segment and say these two ends are connected by I don't know putting two little lines on the ends this would be like a 1 video game we're running off one side brings you back to the other it's just like the arrows from before but arrows would be more confusing here so I'm just gonna use line segments so what we have here is topologically a circle or s1 even though it's not shown the ends are glued together and thus we get this closed loop now we need to find the Cartesian product of this and another circle but remember from before to do this you just take one of your sets like zero two two and then stack the other set four to five perpendicular to it and then sweep across so here we'll stack another circle perpendicular to our first I'll use dots this time to show that these ends are connected so this is also a circle then we sweep across to get a rectangular region now since each of these dots on the top is directly connected to the opposite then we can just say this entire side of our rectangle is connected to that opposite side and also since this end was connected to this one from before then we can say this entire side is connected to the opposite one so we have this rectangle where all opposite sides are directly glued which can be represented by this fundamental polygon which topologically we know is a torus this is the Cartesian product of two circles because a torus is a circle of circles or a different circle of circles so that's the basic idea behind the product of shapes it's amazingly similar to multiplying integers together and with that if you are now asked to find the product of a torus written t2 and a circle well it's not too difficult just take a flat torus and stack s1 or a circle perpendicular to it where those ends are connected then do that throughout the entire torse sweeping those circles across the entire thing until we get a cube this is not a normal cube though because opposite sides of the tourists are glued and all the circles going from the top face to the bottom had their ends glued as well which means that all the opposite faces of this cube are connected leaving us with something called a three torus now if you try to imagine physically connecting the opposite faces or the arrows together as if like before the cube are made of stretchy rubber then you'll have some trouble because it requires a fourth spatial dimension but this is still a 3d manifold and the first one of this video 3d manifolds are we start getting into possible topologies of our universe where yes this three torus is one that has been theorized but that will be the topic of the next video because this took way longer than expected however if you have a strong interest in space physics and cosmology then you can learn a lot more over at curiosity stream this video sponsor what you're seeing here is called Stephen Hawking's favorite places which is probably one of the most visually impressive series I've seen on the site it's three videos narrated by Stephen Hawking would show him going on a journey through space while he discusses some big ideas like black holes parallel universes quantum physics and plenty more this series covers some of the history of the universe and also some of the biggest current ideas in physics and cosmology and they have an entire category of their site dedicated to space and the universe which is my favorite category of documentaries to watch curiosity stream is available on a variety of platforms worldwide and it only comes out to $2.99 per month but if you sign up by using the link below you'll get your first month's membership completely free so no risk and giving it a try and with this you'll have unlimited access to top documentaries that I'm sure many of you will enjoy and with that I'm gonna end that video there thanks as always my supporters on patreon social media links to follow me are down below and I'll see you guys in the next video you

Channel: Zach Star

Views: 491,036

Keywords: zach star, zachstar, higher dimensions, the mathematics of higher dimensions, higher dimensional mathematics, topology, topology for beginners, fundamental polygons, spacetime, space time, universe, global topology, topology of our universe, 3 torus, three torus, torus, mobius strip, klein bottle, 4 dimensions, applied math, shape of the universe, orientable, manifold

Id: lmcT2mP2bfE

Length: 18min 53sec (1133 seconds)

Published: Tue Jun 30 2020