The Iron Man hyperspace formula really works (hypercube visualising, Euler's n-D polyhedron formula)

[Music] welcome to another mathologer video let me start by showing you something very cute x plus two cubed cute okay okay here's a kitten definitely cute but bear with me x plus two cubed is also cute let's expand 6 12 8 numbers ring a bell no how about now aha a cube has six 2d faces 12 on the edges and eight dimensionless zero dimensional vertices damn no zero inside well actually there is x to the power of zero is equal to one and when it comes to solid cubes let's not forget the 3d interior of the cube very nice isn't it i think you're on to something or is it all just a strange coincidence and i'm just leading you on what do you think of course there's something there lots and lots and lots the mathematical superheroes euler and schlafly iron man you heard right hyperspace one of the most beautiful formulas in maths the shape of the universe all intricately connected to this simple observation lots of amazing insights to look forward to so let's go [Music] okay if there's anything here then all those exponents should stand for dimensions so it's a reasonable guess that this three here somehow has to do with the fact that we're dealing with a three-dimensional cube okay so let's check let's try a different dimension what's the 2d counter part of a cube what's a 2d cube well square of course expanded that x plus 2 squared on the left gives this yep that works one brown 2d solid square four aqua edges and four red vertices just to be sure let's check another dimension what's a 1d cube that doesn't sound very cubey anymore does it but if a 1d cube is anything it's got to be this just a skinny little line segment and yep those numbers pan out again but why stop here what about a zero dimensional cube and mathematicians wonder why people think they're weird anyway there's nothing in zero dimensions except for a single point so a zero d cube got to be that point yep that works too very nice zero one two three what's next four of course right four what's a four-dimensional cube well whatever it is we now have our trusty formula to guide us according to our formula a 40 cube should consist of 16 vertices 32 edges 24 faces and so on now using the binomial formula which you may remember from school or my last video we can write down the general formula so just put 6 or 66 or 666 whatever you like and so whatever an n dimensional cube might be if there's a god the number of m-dimensional bits in an n-dimensional cube should be the general coefficient in this expression that one here very pretty formula don't you think now just in case you're a bit rusty on your school mats the bracketed expression is called a binomial coefficient it's the number of ways in which you can choose m objects given n objects to choose from in terms of factorials the binomial coefficient looks like this it's all coming back now i sure hope so before going on let's look at a simple case that we know and do a quick check that this formula really works and how it works always a good idea so what's the number of 1d bits that's edges in a boring old 3d cube that's 12 of course and what does our formula say well we're interested in the 1d bit so m is equal to 1 in a 3d cube so n is equal to 3. okay now engage algebra autopilot [Music] twelve edges works okay all in order we have our higher dimensional cube formula but does it really tell us anything new are there really higher dimensions to throw cubes into yes and yes there really are higher mathematical dimensions hyperspace and there really are 45 d60 etc hypercubes and yes by expanding x plus 2 to the power of n you really do get the correct numbers of bits and pieces in these hypercubes and of course all will be explained now chances are that on your youtube journeys or maybe even your library journeys for your older guys you've already encountered ghostly shadows of these hypercubes for example i'm sure many of you will be familiar with this iconic 3d shadow of a 4d cube this shadow from the fourth dimension already made an appearance in my video on how to solve the 4d rubik's cube there you actually interact with the 4d rubik's cube via its 3d shadow you should definitely give this a go a lot of fun this mysterious shadow also made two appearances in the marvel blockbuster movie iron man 2 in the scene where tony stark flips through his father's notebook there's one of those and the second time on the next page now the super symmetric diagram here is another shadow of a 4d cube there's a shadow of a 3d cube a 2d cube a 1d cube and a 0d cube now can you spot anything else on this page that looks familiar yep there's our formula for the number of md bits in an nd cube now one of my main missions today is to turn you into hypercube masters and prove this bits and pieces formula to prove that the x plus 2 to the power of an expansion trick really always works but before i get to that let me show you some very pretty consequences of that formula [Music] here's an idea why don't we substitute some numbers for x what would be a good number to try here how about one let's see what happens well in the brackets on the left we get 1 plus 2 equals 3. and on the right all the powers of 1 are equal to 1 and so we can just zap them now the right hand sides are just the sums of the edges inverters and so on so what this tells us is that in total an nd cube consists of 3 to the power of n bits and pieces for example for our 3d cube we get those familiar numbers 6 and 12 and so on and it all adds up to 3 to the power of 3 is equal to 27 in a square it's 3 squared is equal to 9 etc cute isn't it and probably not something most people would have noticed or considered to be of any significance that worked pretty well so why not try again what other values of x come to mind well x equal to zero is always worth the shot but you can easily check for yourself that nothing interesting comes from this choice if you do this everything wipes out right it turns out the most interesting choice is x is equal to minus one then in the brackets on the left we get minus one plus two that's just one and raising one to any power still just gives one okay what about on the right well even powers of minus one give one and odd powers give minus one so we get alternating plus and minus signs in the sum there we go hmm interesting how these numbers on the right cancel out isn't it still here asked what's the big deal well let's see first those leading ones on the right which correspond to the highest dimensional blob are common to all the equations since they're not special in any way let's move them out of the way and move them over to the left okay that's zero there's a two that becomes a zero two and a zero zero two zero two forever i'm sure some of you will have had an aha mod by now but let's keep on going until everybody is a high remember red stands for the number of vertices aqua for the number of edges brown for the number of faces and gray well we haven't called it anything but 3d blob so far let's give it a more respectable name let's call it a cell 0 is equal to 0 at the top not much of an equation so let's zap that one also we can make these equations look nicer by reshuffling them back to front now we're getting somewhere for example the third 3d equation tells us that the number of vertices minus the number of edges plus the number of faces of a cube is equal to 2. a lot of you will recognize this formula and many of you will also know that what's amazing is that the formula works not only for our cube but for all convex 3d polyhedra that is vertices minus edges plus faces equals two for any convex so no indentation solid object whose surface is made up of a finite number of polygon phases for example for the stoichiometry here the number of vertices edges and faces are 12 30 and 20 and 12 minus 30 plus 20 really is two and for this pyramid we have five minus eight plus five which is again two v minus e plus f equal to two is called euler's polyhedron formula there the swiss mathematical superstar euler the man with the towel head strikes again by discovering another one of the top 10 most beautiful and influential formulas in mathematics next time you buy an elaborately cut diamond for your beloved and she won't be happy unless you also tell her how many vertices edges and faces it has remember that you only have to count two of these numbers the third follows from euler's formula what about our other zero two zero two formulas well believe it or not these work for all convex polyhedra in all dimensions this was discovered by another swiss mathematician ludwig schlafly no towel hat but man that be it if i was a hair man i'd be jealous okay can we actually prove that the euler-schlafly formulas really always work well the 1d equation is super easy the only convex 1d poly thing is the line segment it has two vertices proved neat i can tell you're really impressed right no ok well then let's up a dimension the 2d formula is v minus e is equal to 0 which says that for convex 2d polyhedra the number of edges is always equal to the number of vertices well that's also easy to see a convex 2d polyhedron is a 2d blob surrounded by a necklace of vertices and edges and since the vertices and edges take turns along the necklace there are equal numbers of them proved again we're on a roll up one more dimension now okay so now we're back to euler's formula for a 3d blob and proving euler's formula is a lot trickier but a mythologized sketch of one particular proof does not take more than a minute or so and so let's do it take an arbitrary convex 3d polyhedron i'll illustrate using a cube again as our example of course we know exactly what we'll get with the cube but it should be pretty clear that the same process will work with any convex 3d polyhedron we start by placing our 3d blob in space so that one of its faces on top is parallel to a plane then we place a 3d light source close to this face and by doing so we create an intersection-free shadow on the plane the resulting network has the same number of vertices edges and faces as the original shape well with one proviso i think it's clear that we get the same number of vertices and edges in terms of the faces of the cube it's also clear all have counterparts inside the network except for the one we shine the light through on top right that phase of the cube corresponds to that phase of the network and that one corresponds to that one on the other hand the shadow of the top face is the whole network now to compensate for this one missing phase inside the network we simply count the outside of the network which currently does not do anything as the missing phase slick mover so in our example those are the six phases of the network their faces one to five are inside the network and phase six is the outside main thing we have six faces just like in the cube we started with now we modify the network by suitably adding and subtracting vertices edges and faces first we chop up every phase into triangles by adding some edges like this okay then we prune away from the outside like this okay there there there prune prune prune prune proof until eventually we end up with a triangle now at each step of this process we're changing the network and so we're changing the numbers of vertices edges and faces but and this is the nifty part it is easy to see that with each step the quantity v minus e plus f does not change so for all the networks that we create along the way the quantity v minus e plus f is the same but that means that the v minus e plus f of our original blob is the same as the v minus e plus f of the triangle and what is the b minus e plus f of the triangle well it has three vertices three edges and one two phases and so the v minus e plus f of a triangle is equal to three minus three plus two is equal to two and the exact same argument works no matter which 3d blob which convex 3d polyhedron we start with the v minus e plus f for all convex 3d polyhedra is equal to the v minus e plus f of a triangle and so is equal to 2. now that's a very pretty proof isn't it a proof that you won't forget for the rest of your life i hope again what makes this proof work is that we shadow squish our 3d blob into a network and then by a number of mini steps we transform the network into a triangle and both the squishing and each of these mini steps keeps the v minus e plus f the same i said the sameness is easy to see but let's have a look at one such mini step to see how it works so for example at this stage of the process we're about to prune away the vertex and the edge indicated by the arrows but what's the effect on v minus e plus f if we remove one vertex and one edge well minus one minus minus one that's zero and so the net effect is zero and we can be sure that the new network has the same v minus e plus f as the original network very nice isn't it and very similar cancelling arguments work for all the many steps in the argument bit of a challenge for you go through the individual steps of our transformation convince yourself that the v minus e plus f really never changes of course for a completely rigorous proof there are still eyes to be dotted and t's to be crossed but there is no hidden trickery promise what about proofs for the higher dimensional formulas for example the 4d formula is supposed to apply to things like the mysterious and monstrous 40 120 cell with its 600 vertices 200 edges 720 faces and 120 cells that the shadow over there 600 minus 1200 plus 720 minus 120 equals zero works at least for that one well the 40 and higher dimensional formulas can also be proved by elaborate squishing and pruning just as we did prove euler's formula a lot harder to keep track of all the multi-dimensional bits and pieces but basically the same thing now before i move on let me mention that these formulas form the starting point of a lot of very deep and powerful mathematics in particular the left sides of these formulas can be calculated for many mathematical spaces apart from convex polyhedra and the resulting numbers tell us a lot about these spaces the possible shapes of the universe we live in and so on great stuff and something that my colleagues and i really get very excited about now i'd like to finish off the video by proving that our x 2 to the power of n expansion trick and the beard man formula really work i'd also like to give you a hands-on intro to high-dimensional cubes and their spectacular shadows again this formula is supposed to give the number of m-dimensional cubes in n-dimensional cubes as you'll see it's actually surprisingly easy to prove and the proof is very pretty that be it this is great [Music] sick of cubes yet not me i love cubes and i could go on for hours but i can see that some of you are flagging a little so okay here's another kitten this time in a cube very cute feeling revived and ready for an ultra satisfying a-ha experience it's a bit of work but 100 worth it and by the end of it you'll all be hyperspace masters okay when studying cubes it makes sense to focus upon some particularly nice example like the one over there its vertices are all the points whose coordinates are either plus or minus one nice and symmetrical the corresponding one and two dimensional cubes are these there plus minus one everywhere for all the vertices and what about the higher dimensional counterparts well these hypercubes may not exist in any physical sense but mathematically there's absolutely no problem to define them in a meaningful way that meshes in and extends perfectly what we observe in dimensions 0 1 2 and 3. for example we declare the vertices of our prototype 40 cube to be all the points in mathematical 4d space with these plus minus 1 coordinates obvious right there is no other reasonable way to do this then how many vertices are there well there are four coordinates each of which is either a plus one or a minus one and so in total our 40 cubed has 2 times 2 times 2 times 2 is equal to 16 vertices just as the biermann formula predicted in exactly the same way an nd cube has to the power of n vertices now some pairs of vertices are connected by edges and others are not what's the rule in the lower dimensions which we can then use to also define which pairs of vertices are supposed to be connected by edges in dimensions four and up it's really quite natural and simple let me show you okay here two vertices connected by an edge when you move from one vertex to the other along the edge exactly one of the coordinates changes in the case of the edge over there since we are moving parallel to the x-axis it's the x-coordinate that's changing and that's true in general for all the lower dimensional plus minus one cubes two vertices are connected by an edge if and only if they only differ in one coordinate here's another example there all coordinates the same except for the last now we use the same rule to define we're really playing guard here we're extending what's there in lower dimensions when two vertices are connected in higher dimensional cubes for example here three vertices in our 4d cube the vertices on the right only differ in the last coordinate and therefore are connected by an edge the top two points only differ in the second coordinate and are also connected by an edge but there is no edge between the bottom two points because they differ in two coordinates easy right so now we know what the zero d and 1d bits of higher dimensional cubes are that's the vertices edges what about the 2d bits that's the faces 3d bits the cells and so on well that's also easy to get the general super rule let's rephrase our edge rule like this two vertices form an edge in the nd cube if n minus 1 coordinates are the same right different in exactly one coordinate is the same as saying that the remaining n minus 1 coordinates are the same let's type this out there two vertices of an n-dimensional cube for my 1d bit if they share n minus 1 coordinates okay actually the 2 in front really stands for 2 to the power of 1. now to get the rule for faces just replace all ones by twos for example over there are 2 to the power of 2 is equal to 4 vertices that form a phase of a 3d cube so here the n is 3 and three minus two equals one and indeed all these four vertices have one coordinate in common right the z coordinates are all the same works in the most general form of the rule the two gets replaced by an m and now the whole thing reads like this cool and based on this rule it's actually not very hard to show that the beard man formula is really true which then implies by the binomial formula that the x plus 2 to the power of n trig really always works maybe give this a try yourself as i said proving the formula like this is pretty easy but sadly proving like this is also a bit boring my friend tristan had a better idea for a much more direct and visual proof an absolute killer to mythologize but totally worth it have a look [Music] beautiful the way our cubes grow from a single point right and of course you can go on like this algebraically to grow all our high dimensional cubes now let's start again from the beginning but this time we also grow the numbers of bits and pieces in parallel sounds intriguing doesn't it well it definitely is have a look there that's the point again the zero d cube it consists of just one vertex and on the right we record this number effect as we did at the beginning of the video [Music] [Music] so [Music] very pretty right and the algebra carries on all the way to infinity and it gets even prettier again one more time from the start okay let's call the polynomial expression that counts the bits and pieces in the nd cube pn then this is p0 now let's calculate p1 the expression for the 1d cube again important here is that the second expression is the first expression times x let's set up first on the right we've already done this on the left we have one p0 plus another p0 that's two p zeroes plus x times p zero so that's x plus two times p zero so that means that p one is equal to x plus two times p zero can you see where we're heading with this almost there anyway with exactly the same argument we can now show that p2 is x plus 2 times p1 that p3 is x plus 2 times p2 and so on and here's the finale [Music] aha and that's why the x plus two triple of entry works how satisfying was that well thanks again to kristen for this very beautiful argument very pretty and sort of a natural place to stop but i just could not resist i just had to add another mini chapter to round things off [Music] it's all very well to define these abstract high dimensional cubes and to make sense of them in the way i described but how can something that does not exist in the physical sense have a solid 3d shadow to finish off let me explain this little miracle to you the explanation is based on our special plus minus one cubes and the shadow picture that i and iron man showed you earlier in this shadow picture we see a 3d cube being projected onto a 2d plane using a light source above this plane the shadow is that square inside a square network which is clearly the 2d counter part of the cube inside a cube for the shadow that we're trying to explain now i created a projection picture in mathematica of course on the computer all this projecting and visualizing is all done in terms of coordinates now in this picture the z-axis is vertical and i've chosen the light source to be the point zero zero three on the z-axis the shadow is formed on a plane parallel to the x y plane at depth z is equal to -3 now to mathematically create the shadow of the 4d cube we simply have to add one dimension a w coordinate then the process is essentially identical so replace the zero zero three coordinates of the light source by zero zero zero three replace the plus minus one coordinates of the 3d cube by the coordinates of the 40 cube and replace the 2d plane by the 3d hyperplane w is equal to -3 then push the execute button and voila as if by magic the shadow of the 4d cube appears out of nowhere on your computer screen let me show you the programs for the 2d and 3d shadows side by side to drive home the point how easy it is to create these nice 3d shadows of 40 cubes [Music] [Applause] [Music] as you can see apart from putting in some more vertices and replacing the occasional three by four there is not much of a difference between the two programs if you want to play around with these programs you can download them by the link in the description real mathematical magic isn't it well that's it for today i hope you enjoyed today's selection of mathematical miracles as a special reward for everybody still watching here is some extra magical shadow play featuring our 3d and 40 cubes spinning in space enjoy until next time [Music] [Music] so [Music] [Music] [Music] [Applause] [Music] [Applause] [Music] [Music] you

Channel: Mathologer

Views: 210,591

Rating: 4.9575257 out of 5

Keywords:

Id: E2l95ttmJOg

Length: 30min 57sec (1857 seconds)

Published: Sat Aug 28 2021