A Mathematical Explanation of Animation vs Math

I'll be giving an explanation and review of  the math as presented in Animation versus Math released by Alan Becker about six hours  ago. So I know a little bit about math but I don't have quite the same understanding as some  other people like math Majors or math students, but I'll try my best to explain at  least some of the concepts that I might come across or other cool things  that might show up. So here we go. And yeah, we can skip ahead or kind  of fast forward through some of the parts t

hat are pretty obvious. So yeah, this one  just appears out of nowhere he adds them together. Learns addition. He's like oh okay. Here's a two,  a different symbol. Oh here's a three, and so on. And then here's a second digit that appears in  the one's place. A zero huh. Okay that is 12, 2 he learns to expand. 4 20 (blaze it). And he adds  it all the way up to 100. Just kind of shows the vast expanse of this universe here. He's like I'm  bored. Sits down. He accidentally compresses it. Seems lik

e tapping the equal sign here kind  of compresses the numbers. Learns subtraction. Oh yeah, go to 98. Down to 0. -1, huh? Shakes it.  Becomes e to the i pi. Wow e huh? What's e? Wow, it's animate! Why is that? Escapes off to  the imaginary world or the complex world. It becomes -1. He's like oh I'm gonna get that e  guy again. Okay well I'll just explore. -1. -3. +3 Yeah, because I guess he flipped  that sign. Right? Kind of cool. Tries to turn it into multiplication. Oh here  we go, yeah. 3. He

expands it. Here we got a grid-like representation of multiplication plus  one. * 4. He flips, yeah he flips it, doesn't he. This first number is the kind of column number.  Yeah. Cool. He once again taps that equal sign. Yeah just to compress it. So tapping the equal  sign kind of compresses the number and simplifies it. Turns it into division. Here we have...it  takes 3 2's to add up...or sorry it takes 3 -2's to subtract from (should be add to) 6 to become  0. And this gray number is like th

e counter. So it becomes 3. Expands it. Yeah once again. Ooh  here we get some 1's. Takes 6 -1's to add to 6 to become 0. So yep. He gets 0. 6/0. And you  cannot add enough 0's to become 6, so it's just infinite 0's basically, and all these 0's are  falling but they don't reduce the number of 6's. So yeah, I thought initially like when I was  first watching this video this division by 0 would become like a climax of sorts...like a climax  yeah in the video, but it didn't actually end up happenin

g. So we have exponentiation. 6 times 6 is  6 squared, which is 36. He adds 2. It's 8 squared, 64. He doesn't simplify it though.  Then 6 minus 2 is 4. 4 squared is 16. 4 to the power of (2 plus 1). 4 to the 3,  this kind of extends to the third dimension. So here we have kind of like a cube if  you can kind of visualize that. Here we have a four-dimensional kind of thing. And  it's fifth dimensional with columns so it's here: 1, 2, 3, 4 that's one dimension. 1, 2,  3, 4 that's another dimension

1, 2, 3, 4 within each column: that's another dimension. So here  we're at three (dimensions). And then it's 1, 2, 3, 4 that's another dimension. And he extends it  down so it's four of those kinds of I guess 2 by 4's so yeah. You see that he taps the equal sign.  Presses it, simplifies it to 1024. Learns about uh different types of exponents like negative  exponents. That's a fraction. He's like ooh, okay. Yeah fractional exponents, that's the square root.  He simplifies it. Doesn't simplify i

t all the way. He puts square root of 4. Square root of the 2.  This is an irrational number. Square root of one. Square root of NEGATIVE one. Oh, i. i came from  e to the i pi, so okay that's cool. So now he can add them together, this is a complex number. Oh,  i times i. It's like, "what is that?" he scratches his head. So yeah i squared. That's -1. That's  just a property of i. That's a kind of a defining characteristic. Here we find e to the i pi. So i  times i is -1, and we saw earlier that

e to the i pi is -1. So you have i attached to the e to the  i pi. He's like oh [ __ ] let me get out of here. It's like a kind of cornered animal. He runs away.  It's like "I wanna escape!" The Second Coming kind of chucks i at e to the i pi times i. So kind of  a cool thing. i times i is -1. -1 times e to i pi is going to be 1 because e to the i pi is negative  one, so it's all going to be...It's kind of like i to the 4th power. So he attempts to escape,  but then i multiplies it so then he b

ecomes a real number, so he's back into the same real  plane. And I guess. Yeah it simplifies into a negative number. He expands himself. This  is called Euler's formula. Let's see, yes! e to the i x is equal to cos x plus i sin x,  so you can read into that if you're interested. It's a really weird formula, but there's a  3b1b video that explains it pretty nicely. Let's see if I can find it. Yeah, this one. So  this is a good video if you want to understand this formula and specifically in the

context  of rotations. Throws a negative sign and hits The Second Coming. Flips him because negative  signs kind of flip things. And then he, or this e kind of expands itself. Or what  does it do...it adds a negative pi...And I think there's an error  here actually because yeah...there should be a parentheses around the pi  minus pi. It should be more like... Let's see if I if I can graph this. Okay well  desmos doesn't have imaginary numbers in it, but it's still nice to kind of show what's goi

ng  on. So, I'm saying it should be like this rather than e to the i pi minus pi. Because like order of  operations. Like multiplication happens first. So this is what they're saying, but really it should  be like that. Because otherwise, this is the same thing as like...let's say it'd be the same as e to  the i pi over (e to the pi). Yeah this is not 0, this is going to be something else. Or sorry  this is not e to the 0 which would be one. So I'm saying all these...these three things  are equa

l, but this is what they mean to say. So yeah e to the i pi Let's see. e to the i pi starts off as a -1, and  he evaluates to e to the 0, which is 1 so he kind of goes from, in the unit circle, from negative  one to one, which is a kind of a cool arc. He's like oh! Let's see...multiplies his speed.  Flips him so that like stops his momentum. Then he like catches him...he's like okay let's  fight and this is like a cool sequence, like a little sword fighting sequence.  Will have to slow down for

this. So he pulls out a 1 out of his ass. He fights,  they fight with 1's. He has a -1 so -1 + 1 equals 0 which is kind of this clash you see. They fight,  they fight some more. Some more 0's. So then he expands it to like a 4 so it's like more power. So  yeah it just kind of knocks him over, but if you notice, a -4 plus 1 is equal to -3 so that's kind  of cool but I don't know why this is still a 0. Yeah it keeps shrinking  the stuff ever so slightly. Then he just like duplicate his +1's so he

has  two swords. Oh more power. Converts it to +2. The +2 converts into a bow. 2x2...2x2 is 4, so  it shoots these arrows. These 4rrows I guess. So he takes that 4. Converts it to a fraction. e  to the i pi over 4. That's kind of like a rotation in the complex plane by 45 degrees because pi  over 4 is the same as 45 degrees. And he runs off Shoots more arrows. And he's like running hella fast while shooting arrows which  is pretty impressive I'll say. And then he multiplies himself by i. And mul

tiplying a number by i kind of rotates it  90 degrees in the complex plane, so he kind of launches himself up there. Shoots  more of these kind of arrows. Let's see. Then he discovers kind of this  coordinate system. Huh, what can I do with this? Oh, imaginary numbers? The complex  plane? The real plane. The real number line. Then he makes this kind of unit circle. And then  he discovers like radians. So radians...let's look at the definition. Yeah this is the definition of  a radian. You could

look at that on your own time. But then it turns out you need 2 pi  radians to kind of cover the entire circle. He's like okay, let's take this kind of  1 radian arc or 1 radian sector I guess. And it has a length of 1 because that's by  the definition of a radian. He multiplies it. He has 2 of these arcs now. Expands it. The  radius is greater. Theta r equals something. r equals 5 right now, so it's 1, 2, 3, 4, 5 units.  Minus 2, so now it's 7 units. Now it's three. And then it's theta. He shak

es it. He kind of  shifts it. This is not really mathematical. This seems more to me like some creative interpretation  by the artist, or the animator. So yeah, this is also another creative thing like just  splitting the, or kind of tilting this theta line Kind of cool though. r plus 1. Division sign. So  r is 1 as defined before. So that's just theta equals 0. He flips it. Kind of a nitpick, but  this doesn't quite align with this. Like this bar within the theta doesn't really align with the 

kind of line in this circle. At least until here. It's like some uh...yeah it's a nitpick. So then  here we have theta equals pi. It's like "oh yeah, pi. I remember that guy." He splits it into  cosine and sine because I don't know...trig functions. And then he's like fighting with it. He  makes a kind of function. He kind of draws it out. If y'all don't know what sin looks like,  it's like this basically. Starts off at 0. So at time equals 0, the evaluation of  sine of 0 is just 0, and we see t

hat kind of here. Although kind of weirdly enough,  when we have sine of pi/2, it should be one, but it kind of looks to be 0, or sorry, -1 over  here so that's just another nitpick I would say. Then he starts with cosine. It goes vertical  this time for some reason. Cosine of x. Yeah it starts off at one when time equals 0.  Time t equals 0. So yeah, that's kind of what we see here. Pretty accurate. Then he hits both  of them at the same time. And it's like okay, "let's make it i." So as I said

before,  multiplying by i kind of rotates it 90 degrees. Although I don't know why cosine  is kind of vertical to begin with. And we have cosine, sine. Cosine x. We plot them  both together. It kind of creates that little pattern here. And if you kind of envision this  in 3D, it looks sort of like a helix, but...Let's see. He combines them. What is it. Cosine  of pi plus i sine of pi. Well that's just the Euler's formula right? We saw it from before.  And then Euler's identity actually is just

an application of Euler's formula specifically when  you plug in...when you plug in x to be pi. You can see that because cosine of x...let's see...cosine  of x...let's see...cosine of x is just -1, so that's going to be -1. And then  i times sine of x...sine of pi is just going to be 0. So it'll be -1 plus  0 is equal to e to the i pi. Okay so then he finds e to the i pi and kind of just spawns him or  summons him. He's like, "let me out of here man!" Fights him with the arc. Some more sword  fi

ghting. Oh this is cool. Again they're forgetting the parentheses here. It should be  e to the i times...what is it...okay I'll just say it. e to the i times (pi minus 2pi). So then  he can kind of draw an arc. Goes from -2pi to pi. Which overall would mean e to the i...e to the  -i pi to e to the 0. So once again that's going to be kind of going from -1 to 0 (should be  1) so it gets this kind of half arc here. Or half circle. He does more of that. They  fight. So then he gets like a big kind o

f sword number so the swing is massive. It  sends him flying a little bit. He uses the bow a bit more. This is the Taylor  expansion of e. So e Taylor expansion. Uhhh. No, I don't want that. Here, this is good  enough. So you can look into this, but here we see e to the x. In this case, you could plug in  i pi. So if you use. If you go to this kind of function. Instead of x, you would use  i pi, and that's kind of what we see here. There's probably another nice YouTube  video about this. You can

look it up. So he launches the first few terms of that  Taylor expansion. X to the 0 over 0! That's what we see here. And then he gets to 1, 2.  And then 1, it kind of splashes when it hits. The actual part of it kind of expands. So 0!  That's 1. i pi to the 0th power is just 1, so that's just 1. And here we get...yeah you can  kind of slow down the video to see for yourself. He multiplies by 4 of that kind of quarter  circle. Creates a full circle. Multiple by pi, create shield. Multiplies tha

t  circle by 8. Uhhh, cool sure. Let's see, we can slow this one. Let's see e  to the i pi is -1. -1 times -1 is a plus, so that's just r plus theta. So he's expanding theta,  although for some reason the radius is increasing. Although it's R plus a constant is equal to R plus  a constant, so what should be happening is...I don't see why r, it still isn't  obvious. He takes 8 divided by that Well 8 divided by (8 pi r squared)  should actually be one over pi r squared, so it shouldn't return back

to a unit circle. It  should be like something weird. It should...yeah whatever. And then he gets kind of pulled in.  He's kinda spinning like a madman. He's using the number 7. 7 is sharp. Got a little pointy  end. Oh this is the coolest gamer move by the way. He like just uses the negative sign  on himself just to flip him to the other side. Yeah, goes back to Taylor expansion mode. He's  shooting all these numbers. All these terms. Oh he's like okay, I gotta get up. Pole vaults  himself with

the 4. Takes the coordinate number. Uses baseball. Damn, sin wave. Kind of like a ray  gun. Okay so this guy. He's like he's really angry now. Gonna expand myself. Gonna exponentiate.  Gonna, you know undergo exponential growth So here...what does it do. Sine over  cosine, that's just tangent. This is like some weird kind of function. Creates  tangent. I don't understand what this thing means. I looked it up earlier it's like the  what, latex. That's the left harpoon arrow. Sure. You can you pr

obably explore that, but I also don't know what this dot  means exactly if it has any meaning in math. But my guess is that it just creates  a function. So he get is cocked and loaded. This whole army of e's. They keep cloning  themselves. So these are tangent curves. Tangent looks like this. So yeah, that that  looks pretty accurate. And we saw earlier, this function was just tangent. So tangent  collides with e to the i pi turns it to 0. I'm not sure why that's the case because if you like do

f  of -1 for example, that's just kind of arbitrary. Oh wait it's actually 9. Ok, so yeah I don't know  why that's happening. And then even if you plug in like e to the x, it's this bogus thing. You can't  really see it. Maybe, maybe uh yeah. Actually I don't know why this tangent of x is kind of like  the antithesis, like an antidote to e to the i pi. So he's launching it, shooting him. For some reason, this thing kind  of expands this this kind of domain He shoots, creates a line. Yeah it's pi

, so the angle of this line is going  from 0 to pi. Yeah gets that plus sign to the pi. Okay cool So this whole army of Taylor-expansion-mode kind of e's show up and shoot him. This  whole swarm, it's like an alien vision. Just spam that [ __ ]. Okay here we go. I think he  just adds 7i, which goes up vertically 7 units in this complex kind of plane. Shooting it, takes  the infinity. This guy ain't gonna die yet. He gets this kind of infinite cannon. I don't know  the mathematical reason behind

this like...well, okay. It's just cool I guess. Like  a function taking in Infinity? Yeah that's gonna kill everything. And  then yeah. That...they get absolutely annihilated. Then here, I didn't really  understand this. This is like linear algebra or something. Yeah so you can look up linear algebra  spans and whatnot. Span. That's what it is. And I'm guessing x1 and x2, those are like the basis  vectors for this, whatever this plane this is. Oh this is the equal sign okay. Yeah,  okay. So thes

e are basis vectors, although I think this is just like some artistic thing.  There's not really a connection here maybe. So he escapes he's like "omg I'm abouta die."  Luckily they have 4 of them. You only need one of them. You know what, I would have had  like maybe a couple of me just like you know, stashed away in the complex  plane (dimension) if I were e. So Taylor expansion, Taylor expansion,  Euler's formula, euler's formula. Expands sine of...yeah...sine of pi. That can be  expanded. So

this is like the exponential form of sine of x. Let's see...yep. And the derivations,  you can probably look them up on your own too. Uh this. What is that. That's a Taylor  expansion of cosine of x. Cosine of x... Yeah here we go. So instead of x, they use  uh what is it? Something else. They use pi. Yeah pi here. Pi goes in there, so pi to  the 2n. And they use parentheses here. Nice. Let's see, this is some kind of identity. It's  saying i pi is equal to whatever this power is. So we can say

pi is equal to 2i time log  of, or the natural log of that fraction. So let's see. pi equals 2 i  natural log of this fraction Okay what was it. Oh 1 - i. Yeah, and  then...yeah, someone already asked it before, but that's how you can prove it.  You can also look at Stack Exchange. Yeah they have some nice proofs  looks like it involves... Wait that's the question. The answer Okay yeah, you multiply by the conjugate,  you simplify to become - i, and then yeah. Looks nice. Okay, cool. So that's

that identity,  and in general, for the rest of these identities that show up, you can just kind of probably just  do what I did. Just kind of plug in the value online or you can even ask chat GPT, but yeah.  Maybe ask GPT-4. That's going to be more accurate. So yeah, it's kind of getting this this whole  mech suit-up. I imagine you can make a whole video about this. I'm not gonna go through all  of it. Shoots him. He blocks it. The limit goes from one to infinity because they're all kind  of co

mpounding. Creates this integral sign. I don't really understand what these kind of  things mean on the side. Let me think... I guess, okay these are constants, so like  e to the...this could be taken out of the integral since it's to the left of it.  I don't know the significance of this, but someone else can explore that and also, cool  thing is that this integral sign...Goes from like a tangent right? This is like a tangent. It gets  compressed, and those tangents if you saw earlier, they kin

d of look like integral signs  if you kind of squint hard enough. Yeah kind of. So yeah, he's got this  God staff. Tiny scream. They fight omg, getting destroyed. This guy has like an arm cannon  plus like a sword, he's so cool or a spear, omg. What was that, he held up like his arc, yeah. Dam. Get's absolutely launched. He's like ok well uh... Let me come up with something. He goes  up nine units vertically. So then he's going up. Summons all his minions. And  then he creates this massive canno

n. Oh look at it. There's fire in this mathematical  world it looks like. Yeah he just gets ooof... Okay set the radius to 100. Then  massive Cannon coming in. Kablamo. Okay this guy runs away, he's like  oh let me chase this guy. He gets a circle multiplies his velocity by like  82,794. Get's launched by that cannon. Chaes after him. Explodes. eEscapes into the  complex or imaginary world it looks like. That's why everything's flipped, you know why  everything is rotated by 90 degrees once agai

n. And then [ __ ] everything's getting kind  of obliterated. It gets cracked over. Here we see more negative numbers, or rather imaginary  numbers. So this is like, this could be like 2i or something. This is gonna be something. Nice. "Oh  [ __ ] let me save you." Plus 5. Why would plus 5...I guess this is still kind of like some  real-world stuff so adding plus 5 would move to the right plus 5. Multiplies by i so he can  escape into the other portal. The real world it's like "oh, um, hello." "

Hi, yeah,  uh I need to ask you something." "Oh um wow that was a lot of death and destruction."  "Yeah thanks." "Okay let me help you huh?" "Okay, um yeah, so I need something. I need to get out  of here you know. Uh do you understand? Okay uh Exit you know? English huh? You know this is  a mathematical creature, I hope you understand English." "Exit? oh yeah! Okay I know where that  is. "Okay let's go to the imaginary world" "Um not that one let's go to uh the real world. Let  me show you how

this works." Okay because what, because i to the 4th power is just going to be 1. So he ends up back to where he was. He's like "ah  okay, I didn't need that. I want to escape out of here, not just go back to where I came from"  "Uh okay let's get this function, this weapon of mass destruction. Actually let's extract  it. Let's get this kind of thing over here. e to the i pi. Okay, just stand in there.  Let me Taylor expand myself. Let me uh convert that factorial to the gamma function.  So this

is what the gamma function looks like. Yeah that's the definition. The Wikipedia  page kind of explains it decently well. So it turns out that if you evaluate  this integral, it kind of ends up being the same thing as this thing over  here. So, what was it. n! is equal to gamma of n plus one, which if you add 1 to the  arguments of both sides, then you get gamma of n plus 1 equals n!, which is yeah, that's  correct. He uses even numbers. This part, I didn't quite understand I think it has  some

thing to do with the gamma distribution maybe. Let's see Something with the 2n. Well, I'm not sure, but someone else can look into  it. Then he creates this kind of portal for him. He's like, "Okay well, see ya!" Multiply by i.  Someone with a math major could probably better explain this. Yep, evaluates, becomes a -1. Get  sent off to a different dimension. And then yeah, he's not alone actually, he has friends. So  you have xi. Not Xi Jining. More like the letter. That's what we want. From Wik

ipedia. So in  math, what does that mean. We want the lowercase so... Yeah these are all the different ways you can find  Xi or xi. That's phi, that's the golden ratio, that's another important constant. It  shows up everywhere. It's also like flux in physics. And then delta. Delta math  symbol. Let's see what does that mean? Uppercase. No we want lowercase. So  these are all the uses. probably like the Kronecker delta. That's what  I'm most familiar with. And then there's like little giant in t

he figure.  Or a huge giant. It's aleph. Aleph null. That's what you can...okay actually you know  what, it's not even aleph null, it's just aleph. And in math that kind of represents cardinal  numbers or like numbers beyond Infinity. So, or numbers beyond... yeah it's just Infinity.  Like Vsauce made a nice video about this. How to count past infinity Yeah, and then when does it come up? Oh here we  go. Aleph null. Yeah. So you can watch that video, and then it kind of went away. The plus end. 

And uh yeah that's my analysis of Alan Becker's Animation versus Math, although apparently...what  was it...his lead animator kind of made it all. Hope you enjoyed my analysis of this. I didn't  understand everything, especially that last part and also that integral earlier, but I hope  you enjoyed my analysis. All right. Thank you.