Oxford University Mathematician REACTS to "Animation vs. Geometry"

hello math fans Dr Tom Crawford here at the University of Oxford and today I'm reacting to Alan Becker's animation versus geometry this is the third reaction video I've done first of all we had animation versus math then it was animation versus physics and now we're back to my favorite subject we are back in the realm of maths we're talking geometry I have no idea what to expect the first two were genuinely some of the best stuff I've ever watched on YouTube so I'm really excited to see what this one has in store okay here we go uh play okay there's five stickmen oh it's an ad okay we started with an ad I thought it seemed weird cuz we were filming a screen okay here is the actual video oh okay so it was a DOT which is like 1D to 2D extending to a line all right I'm guessing this blob's going to become orange Stickman yeah there he is okay climbing out of the line as you do love it still 2D at this point we have an A and A B straight straight line between A and B all right so far so good ah another straight line okay appears to be infinite changing the angle ah and new angles I'm doing my first pause I said he was going to change the angle and then he moved it around and we had the angles up it's almost like I know where this stuff is going all right that was cool all right so we've got lines and angles so far so far relatively high school geometry but I'm imagining we are going to go into some weird places okay just right angles there yeah 180° on a straight line 360 on a circle very nice oh ratios okay right we're starting to talk ratios I'm going to make another guess this is bold because this is live and who knows where this is going to go we've started bringing in ratios so he's kind of got the whole line and then he's got a point in the middle and he's saying it's a to B and currently the dot is in the middle so that means the uh section labeled a is the same length as the section labeled B so he's written that A to B and he's written it as being a ratio of one I.E they're the same I'm going to hazard guess we're going to do something to do with the golden ratio and the number F I'm getting way ahead of myself I know it's going to come I'll talk about the golden ratio when we get to it but I'm expecting it to appear yeah he's moving it around okay different ratios yep ha there was a flash there as it went between one and three cuz like 1.6 yeah he's done it again okay there we go he's learning 1.618 golden ratio okay so I'll try and explain now the golden ratio is so he was changing the point um along the line so you're moving the line so you've got your section A which is on the left on the screen and your section B which is on the right so at the start the point was in the middle so a on the left was exactly the same length as B on the right so you would say their ratio is one: one they're the same that's why you had the number one now he's what he's done then is moved the slider across so he's made a bigger and B smaller and he was kind of playing around with it and he moved it you know at one point it was on three which means the section A was three times bigger than the section B and then you had this little yellow flash so this is the golden ratio um and what's happening here at this ratio of 1.618 which he has discovered this is when your uh section A is 1.61 18 times bigger than your section B but the reason this number is important and popular and appears a lot in nature uh it's talked about a lot you've seen the the spirals where you can like overlay the spiral on the Mona Lisa and all that kind of stuff that's the golden spiral which I'm pretty sure will come but the ratio itself the reason this number is important is section A is also compared to the whole line 1.618 * smaller so you kind of think of it as follows um you have the section B multiply it by 1.618 and what you'll get is the section A now if you then take the length of the section A as a ratio of the whole line a plus b that's also 1.618 * bigger to get the whole line so that's why it has this kind of really nice iterative property and through the way I've just described it of kind of continuing to multiply by the same number or adding together A and B added together gives you something in the ratio the whole line it's kind of hinting at Fibonacci isn't it and if you know anything about the golden ratio you'll know it's related to the Fibonacci numbers so that's what's going on that's my attempted explanation of the golden ratio I have no doubt we're going to to see spirals so let's carry on and see if they're going to pop up and there's F the symbol we use to represent the golden ratio okay so he's running away from F this this is bringing flashbacks to uh e in animation versus math all right oh go on Pythagoras go on Pythagoras it I feel like we may have just got our first Pythagoras there it is right angle triangle so we've got 180° total very nice angles in a triangle always 180° 90 30 60 ooh very nice okay see little bits of rotation and reflection uh-huh ooh ooh talking about parallel lines there so it had like a dash on two lines and then double dashes on the others um I think that was parallel lines or were they saying they were the same length they appear to be both parallel and the same length so this is kind of a notation thing when you're figuring out a geometry problem in maths if you have two lines that are either parallel or the same length or some kind of relationship between them you often will label them with numbers of dashes so the fact the two of the lines that have a single Dash are the same basically and then the two that have the Double Dash are the same um so again if you learn something about one you know something about it partner it's PE okay Square again we didn't actually do Pythagoras did we okay Square to pentagon hexagon increasing the number of sides very good okay so we're doing polygons more and more sides to our shape Circle theorems Circle theorems I'm going to pause because I'm going to admit that I'm terrible at Circle theorems uh so if you've seen any videos on my channel you have likely seen me attempting High School maths exams apparently that's what I do and you all seem to enjoy watching me struggle through them I always forget my circle theorems because it's something at least in the UK you learn at GCSE so you're 15 or 16 you learn all of these properties about which angles are the same within a circle things about chords which is a line across the circle right angles all this kind of stuff really nice geometry theups but then you never use them ever again uh so when I actually did the gcsc last exam a couple of years ago um filming it for YouTube I forgot my circle theorems and so I dropped like five marks because I just couldn't remember which angles matched but anyway so popped up with a few Circle theorems there which is very nice back to squares in the ratio with five very good Square again okay oh I think we just drew the golden Square uh so the golden square is a square where the side length of each side is 1.618 I.E is the golden ratio um so I think we've got the golden possibly the Golden Triangle down the bottom as well where one of the side lengths is clearly F you could work out what the other one is using Pythagoras I think they labeled it but I may have missed that but we're starting now not only just to draw General shapes like a triangle a square a pentagon at the moment it seems to be drawing those shapes but where the side lengths are in the ratio of the golden ratio so rather than being length one or length two they are length 1.618 because apparently it's really nice to look at for us as humans we like the golden ratio why everyone uses it to make things look nice oh here we go Pythagoras surely yes celebrate for Pythagoras right there I thought we were going to do it earlier when we drew the first right angle triangle uh but we didn't we just drew a triangle and talked about 180° being the total the angle we just seen Pythagoras's Theorem so Pythagoras's Theorem probably or uh hopefully you will all remember this it kind of sticks in your brain a squ + b sare is c s so you have a right angled triangle if you take the length across the bottom and square it it take the length at the side and square it then the total is equal to the diagonal the hypotenuse the length of that squared so A and B are usually the two bits next to the right angle the diagonal one you label c a s+ b square is c^ s so it's really helpful because in a geometry problem or even a real life situation if you know two of those lengths you can work out the third and this is the power of maths right having that equation tells you you only have to measure two things and you get the third one for free which is really nice um all right so POS I'm going to has a guess we might do trigonometry next let's see p is pretty strong lifting up that square okay oh it's proving Pythagoras very nice showing you that the areas of the sa very good well done well done fire and Stickman you have just proven oh God now we're being attacked by some crazy ass shap which looks like a graph Theory type thing okay felt like it all been quite tame until this point now we're getting attacked oh oh oh really nice spot so you've got this like grass um thing so you've got all these nodes which are the circles and they're all joined by the lines so we call that a graph it's whole area of mass called graph Theory very closely related to Geometry uh and also really good for plotting uh and understanding networks so you could think of it as like you know it's like a social media Network where the dots the nodes are people and the lines are whether they are friends really nice example and application of graph Theory so for some reason graph theory is attacking orange sck man and our beautiful proof of Pythagoras but as it's attacking the squares it's like eating into the shape and as it's doing that it's actually leaving behind a pattern so you have all of these like Square uh holes behind it and that's actually something called the menas sponge which is an awesome uh area of mats called fractals uh and the idea is that what you're doing is originally you're dividing up your square into nine pieces so 3x3 grid and then you remove the Central Square then all the bits that are left divide them up like the remaining eight squares divide them up into 3x3 grids and remove the center and you keep repeating this pattern so it's an iterative procedure you sort of get to one step to get the next step you just do what you did before you keep repeating that procedure and it generates these self-similar patterns which we call fractals which means that if you zoom in on the shape it continues to look the same forever so there are quite a few fractals and I've made a video about one of my favorites called the snowflake but the mener sponge if you keep doing the same pattern and then you think about the area that remains um so as you're removing all these squares you can figure out the area the area that remains is actually going to be finite and the perimeter so the distance around the edge if you were to like draw in chalk a square around every time you remove a hole you would actually need an infinite amount of chalk so you get these crazy shapes in fractals you have a shape that has an infinite distance around the edge but yet inside it is very much a finite value so the shape doesn't go on forever uh which is crazy so fractals are like really cool really interesting stuff and if you never come across them go down that rabbit hole trust me it is a fun one the function it's not quite the function gun but there's like a slingshot with five it's back okay we're sprinting way down the the infinite line fire is faster than orange stick man oh lovely bit of continued fractions there uh so you can create fi through something called a continued fraction which is like an infinite fraction that just keeps going on you're like divide this by this and then part of it by this and part of it by this and it creates that kind of Step shape so F transformed into that to allow orange thickman to pick up all of these various dots that's pretty nice you blinking you'll miss it moment right there I nearly missed it right we're shooting it again awesome very oh there we go golden spiral I knew it was coming okay so the golden spiral thing that appears on all of the pictures like the Mona Lisa and the Taj Mahal and all that Donald Trump's hair if you really want to go there so it's a spiral that's in the ratio of five now as you're drawing that first kind of Arc then what's happening is the next one is reduced by a factor of five so you'll sort of Imagine drawing a circle of some size and then the next Circle which is going to start a quarter of the way around is 1.618 the ratio times smaller and then the next one is again after another sort of um quarter Circle you go a little bit smaller and then a little bit smaller and you're going to generate this beautiful spiral so often you start from a starting point and go bigger and bigger and bigger so as you spiral out the ratio of your spirals they're getting five times bigger each quarter turn it's kind of how it works uh but we've done it in reverse uh so I'm guessing maybe something will happen when we get to the middle let's see okay no we're just going to start partway through all right fine cool still the Mena thonge in the background that's very nice okay yeah nice little triangle using fire like a hoverboard and apparently a way of attacking a giant graph Theory thing oh they did the the Star Trek um logo theorem it has a proper name I call it the Star Trek logo theorem about circles if you have that shape in your circle the angle at the top is half of the angle in the kind of bottom bit so the angle at the bottom is bigger than the pointy bit at the top and that ratio is two the angle at the bottom is twice as big as the poter bit at the top and it looks like the Star Trek logo so I call it the star Tre logo theorem that's true when all of your Edge points are inside a circle so again blinking you'll miss it but lovely little reference okay now we're paragliding naturally oh it's shooting at us so we've got like mini graphs lot happening nice kite shape good yeah cool like a diamond oh okay we are now drawing a pentagon where the outside so the five sides all appear to be uh golden ratio length so five in length and then we've got like a pentagram the little fiveo star inside the Pentagon um which are also bits of that are also in the ratio five so this whole thing is very focused on the golden ratio things are popping up yellow and gold when we get these beautiful looking shapes this is also uh for those of you interested in graph Theory the shape we've drawn here is what we call a fully connected graph because every corner every point is connected to every other corner on that original Pentagon so that's why it looks really nice as well and we get a smaller Pentagon in the middle that's in a ratio to the larger one eyes everywhere oh it's an army of FS brilliant I'm glad we're back to the battles I feel like the physics one o o o I'm getting carried away but I think I can see some kind of fractally pattern so again the graph the blob is attack I need a better name for it the graph Theory blob is attacking the golden Pentagon and as it's doing it it's kind of hitting it and creating these fractal patterns it's really nice okay they're all shooting it we're shooting squares we're shooting triangles so much going on orange stickman's nice and safe huh okay we're doing the fractal again I think it's like a pentagon version of the square the sponge that they're creating oh we trapped him in a pyramid very nice so we have a triangle based pyramid a tetrahedron trapping The graphy Blob now it's become in octahedron is the eight-sided shape there going to add some more yeah of course he is where are we going with this the cube Cube inside the octahedron very nice that now means we've seen three of the five platonic solids so I have these tattooed down my leg they are beautiful mathematical shapes they are possibly the most beautiful mathematical shapes because they're really symmetric um they're three-dimensional uh and you have three rules so you say um I want a three-dimensional shape rule one I want all of the faces to be a regular polygon so what that means is if I have a triangle then all the faces are triangles but not just that if I have a triangle I want the most regular triangle so an equilateral triangle where all the Angles and the side lengths are the same so it would be a square rather than a rectangle you would have a nice regular pentagon so everything's really symmetric um so you have 3D shape all the faces are the same shape and every corner you have the same number of faces joining together now sound like three relatively simplish rules and they're not that restrictive doesn't sound like they're that restrictive however and this is a really cool proof from a Greek mathematician called Theus thousands of years ago he showed there are only five shapes that satisfy those three really simple rules and then Plato kind of jumped in and did some of the work with them about the universe and stuff so they're called the ponic solids but it was Theus they should be the thean solids after the original person who showed their only five such shapes so we've seen the tetrahedron which is the triangle based pyramid the octahedron which was that kind of diamond shape with eight triangle faces and we saw the cube which is also known as a hexahedron which is a name no one uses wonder why the cube in the middle there with the six square faces so the other two that we might see are the do decahedron which is Pentagon shaped faces and there are 12 of them and then there's also the icosahedron which has 20 Faces and they're all equal lateral triangles so we haven't seen those last two but seeing as we've seen the first three there and they kind of lit up and were glowing I'm going to hasard a guess we might see the full set of atonic solids how exciting he's very much trapped in the Cube ooh the spinning things getting into that third dimension properly now okay there it is there it is I cost aedon fold it I'm weirdly predicted a lot of this video this is awesome and that's trapped him come on do decahedron we want the Pentagon shapes come on come on our ins stick man yes he knows he's telling him there it is the do deedon I can't believe I I somehow predicted the story This is Wild boom there we go is that it he's trapped in all five onic solids it's got to be it it's got to be it more fractals uh this looks kind of like a dragon curve I wonder if this is the highway Dragon curve it could be I actually had one of my student interns make a really nice video about the highway Dragon curve so it's something called an L system which is a really simple set of instructions that you can again you just iterate all of these fractals you create a simple rule and you're like keep doing it get something more complicated do the same thing you just keep iterating the same procedure and you get these really cool things it does kind of look like a dragon curve uh I'm I'm going to let's let's say it's the highway Dragon curve it's very similar and again it's creating that kind of infinite complexity in a mathematical shape so they're kind of spiraling out from our trapped um blob graph Theory blob is trapped inside the fonic solid so cool this could well be my favorite one yet but I've still got a to go let's see okay yeah oh oh God I feel like I was wondering when we were going to get into higher Dimensions I feel like this is suggesting some kind of higher dimensional space which of course is very difficult to visualize um but one way I think about it is our world is 3D so so if you want to create a four-dimensional shape what we do is we just kind of generalize what happens going from 1D to 2D to 3D so in a way you just think about well if I've got a DOT and then I go to a circle and then I go to a sphere then what's kind of happening is you're creating a shape with a very large number of sides that's incredibly smooth so a four-dimensional sphere would just be a shape with a very large number of sides that's really smooth okay you can't visual ize it because we don't live in a four-dimensional world but you can generalize those properties to higher dimensions and that's like how we make sense of higher dimensional spaces in maths you just think what happens going from 1D to 2D to 3D if there's a consistent pattern there the same thing should happen when you go from 3D to 4D from 4D to 5D so you spot those patterns this is what mat is pattern spotting and then you generalize them to higher Dimensions so even though we cannot see higher dimension we would expect those patterns to continue and that's how we can try and get a grasp on higher dimensional spaces so I think that's what's happening here we're beginning to enter potential science fiction let's see ooh okay all the fonic solid blobs trapped blobs flying around we're definitely in some kind of oh it's physics man it's Space Cowboy no way no way I've been waiting for them to actually link them together okay so I think what's happened H hypothesis he's kind of Trapped The Blob the graph Theory blob inside the platonic solids gone into this higher dimensional space which is kind of I I again I've kind of weirdly predicted it thinking about like science fiction Multiverse multi-dimensional realities and now we've brought back in physics Space Cowboy man riding his rocket which you all told me very helpfully was a Doctor Strange Love reference but I'm still calling him Space Cowboy so he's trapped in there cuz he fell into a black hole and all kinds of weird stuff happened amazing is there going to be a link to maths as well let's see or is that it he's just waving Space Cowboys waving at him okay I've also just had another thought I remember reading a paper which is published in nature it's a real bit of science which talked about a theory that the Universe could be in the shape of a do decahedron of course it's really difficult to prove this but we know it can't be linear we know it isn't spherical etc etc so you hypothesize what shape could it be and this paper talks about it being a do decahedron which is the uh 12 faces all Pentagon shaped so I wonder whether this is a little reference to that paper here because he's kind of Trapped The Blob the graph Theory blob evil graph Theory blob bad name for a villain he's trapped it inside the batonic solids but it was the do decahedron that 12-sided Pentagon one that actually trapped it and I'm wondering whether that's a link to it kind of being trapped within the universe and that's why physics Space Cowboy man has popped up there's my theory is that going to be it are we we're going back to the circle and it's going full on the loop again to where we started the end amazing honestly I I didn't think these could get any better and yet here I am with my face hurts from smiling that was exceptional um I I I need to rewatch it like 10 times I I probably missed so many details but there was so much awesome math so many cool references I absolutely love the whole universe thing the whole throwback to Space Cowboy and animation versus physics that was genuinely incredible so Allan I've said this every time you and your team so much respect that is so entertaining and as someone who is a math educator is a mathematician teaches maths just the work you're doing is amazing so you might not see this but if you do so much respect absolutely loved it um I'm going to go and watch that again and probably several times to try and see what else I can discover but this was my genuine first time watch through so hopefully it caught my excitement and uh everything else my actual real reaction to that video thank you to all of you for letting me know that animation versus geometry had just come out so I think I've done this one a little bit quicker perhaps than I did the last ones so thank you for the heads up really appreciate that and as always thank you for watching the video please do remember to subscribe to my channel if you've enjoyed this obviously go and watch this on Allen's Channel subscribe like Etc to that video because if we show him that we really like this stuff then hopefully he will continue to make more but that's it from me I'll see you all very soon take care

Channel: Tom Rocks Maths

Views: 123,197

Keywords: animation vs math, animation versus math, math animation, alan becker, tom crawford, tom rocks, tom rocks maths, tomrocksmaths, tom rocks oxford, dr tom crawford, tom crawford maths, oxford mathematician, oxford professor, oxford university, oxford university maths, oxford university professor, oxford maths, mathematician reacts, professor reacts, reaction video, TRM, animation vs geometry, animation versus geometry, platonic solids, golden ratio, phi, golden spiral

Id: hIS_0zAiNiY

Length: 31min 6sec (1866 seconds)

Published: Fri Jul 05 2024