Higher dimensions get really weird... | The Mathematics of What We Can and Cannot Imagine

this video was sponsored by novo resume if you have a paper and pencil nearby go ahead and put four dots anywhere on the paper and connect them to make a quadrilateral it doesn't matter how weird the shape is and I'll just put a few examples here to illustrate that point now put a dot at the midpoint of all of those lime segments you just drew and connect those to make another quadrilateral turns out no matter what connecting those midpoints will always yield a parallelogram or shape or opposite sides are parallel to each other the reasoning behind this is a theorem that says if you connect the midpoints of two sides of a triangle that line segment will be parallel to the third side I could then just form another triangle which shares that third side and then connect those midpoints you see here so again as a result that theorem I mentioned earlier these two line segments you see here are parallel to the third side which means they're also parallel to each other then if I connect the other two corners I can make the same argument which gives us two new parallel line segments leaving us with a parallelogram what we just solved was math in two dimensions something that could be done on a flat piece of paper we'll soon get to three dimensions though which we can of course grasp since the space we live in is three-dimensional but once we move to higher dimensions you'll see the intuition is mostly gone leaving us to rely on mathematical analysis which can still be done without that intuition but to set up a lot of what's coming I'm first gonna ask how many different types of three-dimensional dice are possible to make now I'm using the word dice pretty loosely here because what I really mean is three dimensional polyhedra that meets certain criteria the first one is that every face must be a regular polygon and the same regular polygon so for example this cube or a die we're familiar with is an example of this as every face is a square with the same sides and same angles the second criteria is that add the same number of faces must meet at each vertex for each corner for a cube that would be three and then the third is that shape must be convex so it can't look like this or anything so including the cube how many 3-dimensional objects fulfill all these requirements well I've got a few here that do like a tetrahedron which has four faces that are all equilateral triangles we of course have the cube here's the octahedron that's also made of equal our triangle faces X four phases in me at a corner here not three there's a dodecahedron that has twelve phases which are all regular Pentagon's then we've also got the icosahedron which has 20 phases but how many more are possible I mean in two dimensions you can have infinitely many regular polygons or shapes with equal side lengths and angles but when it comes to putting those on the faces of a 3-dimensional polyhedron what is possible well this is it these are the five platonic solids the only shapes that meet the criteria we discussed earlier that's definitely not obvious for example I found it weird that we can't have a die with hexagons as faces okay what we can like this one here but as you can see there Pentagon's as well so it doesn't meet that requirement having the same regular polygons on each face now this is the point where I was gonna do a proof as to why there are only five platonic solids but I decided to a dedicated video just for that so if you want to see you can click here or down in the description however we will do a little more analysis with these shapes take a look at the vertices edges and faces of each solid there's something these all have in common and it's not obvious at first but if we subtract the edges from the vertices and then add the number of faces in each case the value comes out to two this relationship of vertices minus edges plus faces is known as Euler's characteristic yes another one of his and it turns out for any convex polyhedron Euler's characteristic is two in fact even for a sphere with regards to the surface Oilers characteristic is still - that's because the sphere can be morphed into any convex polyhedron aka the shapes are homeomorphic and Euler's characteristic is preserved under homeomorphisms but things get even weirder when you realize for example that Oilers characteristic for a four dimensional Klein bottle is 0 which we're going to get to soon now in a previous video I showed you what a mobius strip was and this is a very famous shape in the field of topology where you take a flat length of paper and put a half twist in it then connect the ends then I also showed how when you cut it right down the middle and go all the way around you're still left with one piece but that one piece is not a mobius strip as it has more than a half twist in it then we also saw how if we start our cut about 1/3 of the way in rather than and go around twice we end up with two pieces that are linked together but there are plenty more interesting results that we can get like what if now I decided to put one two three half twists in this strip and then cut it down the middle well let's see what happens and here we have holed up I'm gonna need a flat surface for this the bends in the paper are making it not behave well so I just kind of press everything down now what we have here if you can tell is a knot you can see if we start somewhere and follow the paper all the way around we get to the original spot and the loop just keeps going so it is one connected piece knot rings like before this is actually a specific type of knot known as the trefoil knot here let me try rearranging it so it looks more like the graphic now this isn't a knot like in your shoelaces where they have loose ends because in this case there's nothing I can do to unknot the paper without tearing it but if I could tear it or move the knot through itself in just one location like here it would become totally unknotted otherwise known as the UH knot to demonstrate moving it through itself I will just cut it in one location and retake the pieces on the other side of that crossing and now if you just rearrange the paper a little you'll see we no longer have a knot since I was able to do that in one cut one is the unknotting number of our trifle knot that's not too difficult to see but it's not as obvious that for example this stevedore knot with six crossings rather than three also has a nun knotting number of one well the cinquefoil knot with five crossings has a nun knotting number of two we can see several other unknotting numbers here as well but it turns out that determining those numbers is actually not so easy to do mathematically we know it for several of these simpler knots before example we haven't determined it for this knot here when you move into higher dimensions you'll see there's a lot of unsolved problems there actually but let's go back to that mobius strip and answer another question what would I have to cut in half in order to get to moe strips or what shape would I get if I could sew these together well the answer is that Klein bottle I mentioned earlier which is a shape that only exists in four dimensions that means if you really did connect to mobius strips like numberphile has shown before you only get a 3d representation of the Klein bottle which is the best we can do to make a real Klein bottle you need access to a fourth spatial dimension if you're wondering why it's because right here it appears like the client bottles intersecting itself and in three dimensions it is but a true Klein bottle does not self intersect and that's only possible if this is embedded in a higher dimension however the Klein bottle is still a two dimensional manifold meaning close up or locally it looks like a two dimensional plane since this really is just a surface see this is just like the surface of the earth of course we know the global structure of Earth but to something small like us standing on that surface it does look flat and two-dimensional because of this the surface of the earth is also a two dimensional manifold as is the Klein bottle a torus and plenty more locally they appear to be two-dimensional but they all curve or are embedded in higher dimensions then when you get to relativity space-time and the curvature of the universe you move into those three and four dimensional manifolds but that's beyond this video let's now get into how to think about a four spatial dimension though and to do that we should first imagine a two-dimensional object living in flatland that can only see along this completely flat surface if a three-dimensional sphere entered that world all the flat creature would see is the two-dimensional cross-section or a circle if the sphere said hey I'm a three dimensional object and the flatlander said prove it move in this third dimension you speak of all they'd see is that cross-section gets smaller and the same goes for us I mean if there were some two-dimensional world living on this piece of paper I could see everything that's going on and there's nothing they could do to hide for me then if I wanted to at any point I could enter their world but all they would see is the two-dimensional cross-section of my fingers in this case and this is exactly how it be for a four-dimensional object visiting us they could see our three-dimensional world and there's nothing we could do to hide from this they can then enter at any point and we would see a three-dimensional cross-section of that object so if it was a four dimensional sphere for example we would see a 3d sphere just appear out of nowhere it would get bigger and bigger and then smaller and smaller and eventually disappear see our brains aren't really made to understand how that object is moving through that higher dimension but as you can see we can still draw comparisons and do the mathematical analysis even if we don't have the geometric intuition to back it up for example we can answer the question how many four-dimensional dice are possible and again what I mean by that is how many four-dimensional platonic solids are possible that meet those same criteria we discussed earlier we saw before how only a few regular polygons could be put on the faces of those platonic solids but can we connect these shapes to each other in the fourth dimension to create 4d dice the answer is yes in fact every platonic solid in three dimensions has a four-dimensional counterpart like if you join tetrahedrons together at three to an edge you get the hyper tetrahedron there's also the hypercube hyper octahedron hyper dodecahedron and the hyper icosahedron as well but there is a six platonic solid that emerges in the fourth dimension and it's the octa cube or hyper diamond the analog of this constructed in three dimensions is the rhombic dodecahedron something which is not a platonic solid do the irregular faces but in four dimensions amazingly the necessary criteria are all met and you'd think it gets even crazier as we move up but guess what happens in five dimensions there are now only three platonic solids we move to the sixth dimension they're still three and as we continue to go up we unfortunately only find three platonic solids now moving on here's another example of something we know very well for up to four dimensions but afterwards we know very little and this has to do with kissing numbers or the maximum number of units spheres that can be arranged to barely touch or kiss another common unit sphere for example in two dimensions the kissing number is six that's because if you have a single circle at most six other circles can be touching it in this way this wasn't too hard to find but in three dimensions it already got difficult as it took mathematicians a while to figure out that in fact 12 is the kissing number and after much more effort we have determined that in four dimensions the kissing number is 24 however five dimensions we don't know the kissing number we just know as of this video that's between forty and forty four and four six dimensions and seven dimensions we also only have a range of numbers it could be but funny enough we have figured this out for eight dimensions as the kissing number is 240 and even stranger we have determined that in twenty four dimensions the kissing number is a hundred ninety six thousand five hundred and sixty it turns out something weird does happen in eight and twenty four dimensions when it comes to spheres but I'll get to that soon the last thing I want to mention about four dimensions though is that there are no knots like the ones we're familiar with in the fourth dimension well other than the trivial knot which isn't really knotted the reason is because a knot can be thought of as a one-dimensional string embedded in three-dimensional space as it needs that third dimension to exist you couldn't make a knot on flat paper without self-intersection but remember how I said I could unknot something by having it essentially passed through itself which I demonstrated by cutting it well in the fourth dimension a knot would be capable of passing through itself through that extra dimension meaning it's not really a knot it'd just be a loop in disguise and in mathematics this is just the trivial not nothing that's special however in four dimensions it would be possible to have knots made out of two-dimensional sheets though no clue what that would look like but we need that difference of two dimensions in order to create a true knot so using that same logic we could technically not a three dimensional object but we'd have to move it through five dimensions to do so and with that let's now transition to something else interesting that happens in the fifth dimension which goes back to the video I made about adding up all the volumes of even dimensional units fears which comes out to e to the PI for anyone who hasn't seen it many of you guys want more of an explanation and the biggest thing to know is that since the series converges that must mean the volume of these spheres gets smaller as the dimension goes up for a one-dimensional quote sphere or really just a line segment with radius one the length is of course to a two-dimensional sphere or circle with a radius one has an area of pie or 3.14 a three-dimensional sphere with radius one has a volume of 4/3 pi or four point one eight nine then from the equation for the volume of an n-dimensional sphere we can calculate that a four-dimensional sphere of radius one has a volume of about four point nine three five and a five dimensional unit sphere has a volume of five point two six four notice that the number keeps going up with every dimension but once we get to six dimensions a sphere with radius one has a volume of five point one six eight it's finally lower than the previous dimension for seven dimensions it goes down again and continues to do so if we look at the plot for the volume of a unit sphere versus the dimension we're in we see that the fifth dimension is the peak and it eventually does approach zero but geometrically why is that the case well I'll give the best explanation I could find in regards to two dimensions let's say we perfectly rest a circle inside a square which has side lengths of 1 well it turns out the circle takes up 78.5% of that squares area when we move to three dimensions and put a sphere inside a cube also a side length of one that sphere takes up only 52.3% of the volume adding the extra dimension decrease the ratio of those volumes if we go to four dimensions and embed a hyper sphere inside a hypercube the sphere only takes up 31% of that volume but notice that the cubes volume is not changing as you move up in dimensions for two dimensions the area of the square is one squared or one in three dimensions the volume is one cubed or one and in four dimensions it'd be one to the fourth or of course one yes we are comparing completely different units so it's not the best physical comparison but the sphere is taking up less and less of something that always has a volume of one thus the spheres volume will go to zero another thing that notice is the diagonals of the cube in two dimensions the diagonal has a length of root two well the circle has a diameter of one just a little less in three dimensions though the diagonal of the cube goes to root three well the spheres diameter again stays at one in any dimension n the diagonal of the cube has a length of the square root of n as in those corners get further and further apart without bound as we move up in dimensions the sphere on the other hand is maintaining the same diameter and thus taking up less and less of the space as the dimension goes up okay now I know before I showed how the fifth dimension strangely has that peak volume but actually this was nothing special and that's because we are specifically using it's fierce if we look at spheres with the radius of three the peak would happen at a different dimension but since spheres with a radius of one are just so common I decided to highlight it now let's look at sphere packing for centuries mathematicians have been curious about the optimal way to fill space using identical spheres for example in two dimensions the question was how can we arrange pennies on a table such that they are as dense as possible and it turns out the best arrangement takes up about 91% of the space for spheres it was much harder but we found the optimal arrangement takes up about 74% of the space now for most dimensions higher than three we know very little about the best way to arrange spheres however as we go from dimensions 3 to 4 to 5 and so on the gaps between the spheres start to increase from the 3d case but for some reason in dimension 8 there are new gaps that are just large enough to hold spheres that can be perfectly locked into place resulting in that optimal arrangement which takes up about 25% of the 8 dimensional space so we do know the math in eight dimensions and guess what the math of optimal sphere packing was determined in 24 dimensions as well just like the kissing number problem by the way for anyone saying what's the point of this it turns out sphere packing and the hypercube graph have applications and information and coding theory when it comes to correcting errors and data that show up in noisy communication channels for example let's say if I want to transmit the word yes to someone I'll send them a one and if I want to transmit an O I'll send a zero now this is simple enough but errors could come up when it comes to long distance communication so instead I'll send three 1s for a yes and three zeros for a no that way veneer does occur like in a message that says 1 0 1 you can still say that's more likely to be a yes since two of the bits are one well we can think of each possible message as a point in 3d space located on the corners of a cube we'll put a sphere centered at the one-one-one message and one at the 0 0 0 message if the radius of those spheres matches the length of the cube then whatever messages they cover will be interpreted as the message at the sphere center like here the 1 0 1 message would be interpreted as the 1 1 1 or yes since that sphere includes both of those corners but if I want to include more words than just yes or no I need to pack more spheres into the space yes this example is pretty simplistic but openly it shows how packing steers tightly into space does have direct applications and as we add more bits we do the analysis on higher dimensional hypercube graphs meaning we're dealing with higher dimensional sphere packing now let's see something weird in 9 & 10 dimensions imagine the square that is cut into four smaller squares and then draw a unit circle in each one of those as shown then draw another circle in the center that is tangent to the four others you'll notice that the center circle is pretty small relative to the entire square and it's far from the edges we'll do the same thing in three dimensions which would involve a cube being cut into eight pieces and those each being filled with a unit sphere then we'd pack a sphere in the very middle such that it barely touches the other eight here you also notice that the sphere is much smaller than the cube but as the dimensions go up that central sphere gets larger and larger and in nine dimensions that central sphere actually touches the edge of the nine dimensional cube then in ten dimensions it breaks through the sides of the cube even though the cube well looks like it should be the boundary of the inner sphere holding it inside however when the sphere breaks through the cube it still has a smaller volume than the cube that is until dimension 262 after 262 dimensions the hyper sphere becomes larger than the hypercube volume-wise okay I don't officially quote me on that as I found it in various forum posts rather than an official source but once I came across that information I really couldn't exclude it now let's look at an example that's kind of similar to the sphere packing we saw earlier let's say we have a bunch of oranges or spheres that we need to wrap using saran wrap or whatever the question is what's the best way to arrange these such that you use the least amount of that saran wrap now I only have three oranges here so you don't have many options but in general the question is should you align the oranges in a line or should you clump them just like we saw the sphere packing well at the moment both of those answers are right because it does depend on how many oranges we have when you only have a couple like four or five it is best to just line them up however for some reason when you get to 50 seven oranges it is now better to clump them and the same goes for any number of spheres greater than 57 however what's the best way to arrange them in four dimensions well it turns out again you should arrange them such as their centers are on the same line until you get to somewhere between 50 thousand and a hundred thousand four-dimensional oranges because we have determined that somewhere in that range is when you should finally start to clump them in order to minimize the amount of wrapping paper used and after that we have no idea what the optimal packing method is in terms of higher dimensions until we get to dimension 42 in 42 dimensions and everything after we have determined that it is best to again align the spheres such that all their centers are on the same line and there is no breaking point for this so that it's always the best case no matter how many very high dimensional oranges that you have so just in case you guys need to wrap some forty two dimensional fruit you now know the best way to minimize the cost of that gift wrap now if this kind of math interests you there are quite a few areas that you can look into further things such as hyperbolic geometry differential geometry or abstract algebra I'll deal with complex surfaces or higher dimensional symmetries and so on but the main branch you want to look into is probably topology this is a branch of math dealing with properties of space that are preserved under continuous deformations so like homeomorphisms which we saw earlier are an important topic in this field now this is a course that most people outside math and physics majors will never see but it is usually offered an undergrad along with some of those other courses I mentioned so if you're not in those majors you could always try to tailor your electives to these higher-level math courses or you get a math minor even if these don't directly apply your field of study it's still something you could put on your resume and if your student or professional needing to make the perfect resume you can do so over at Nova resume the sponsor of today's video novo resume is a free online resume and cover letter making platform that will prepare you for your next 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they have the same capabilities for cover letters as well so you'll have all the necessary documents when looking for that next job and if you need more help they've got a career focus blog that'll provide insight on how to land a job as a recent graduate how do video interviews how to tailor your resume for specific industries and much more so if you need to create your next resume head on over to novo resume com using the link in the description and make yours now completely for free and with that I'm going to end that video there if you guys enjoyed be sure to like and comment follow me on Twitter and bring them in for a Facebook group for updates on everything I'll see you all in the next video

Channel: MajorPrep

Views: 1,032,733

Rating: 4.9010444 out of 5

Keywords: majorprep, major prep, higher dimensions, four dimensions, five dimensions, what is the fourth dimensions, what happens in higher dimensions, are there higher dimensions, sphere packing, visualizing higher dimensions, manifolds, spacetime, curvature, math, mathematics, physics, applied math

Id: dr2sIoD7eeU

Length: 23min 16sec (1396 seconds)

Published: Fri Jun 07 2019

Reddit Comments

[u/HeatherAine]

This is beyond fascinating. When he got to the section on how volume peaks in the 5th dimension, I began thinking about the Pleroma and how the shrinking volume after that appeared to be moving toward a Singularity.

Really enjoyed this video, thanks for posting.

[u/strydar1]

Very interesting. 4d craft in 3d space could explain some UFO behaviour.