Animation vs. Math

To be clear, my lead animator is the math nerd behind all this. And as always, watch DJ and I talk about it: https://youtu.be/dRj3X7IFCjY

If you could turn this format into a video game, you'd have an incredibly powerful tool to teach kids math.

Utterly delightful!

10:45 orange you've been here for 11 minutes and you've already made a all destroying death laser out of math how

I love the surprise Euler identity early on when just playing with simple addition and subtraction, because it’s just like when your playing with a simple concept in math and stumble across something bizzare/that you have no clue how to understand yet.

0:07 introduction to numbers 0:11 equations 0:20 addition 1:24 subtraction 1:34 negative numbers 1:40 e^i*pi = -1, euler's identity 2:16 two negatives cancellation 2:24 multiplication 2:29 the commutative property 2:29 equivalent multiplications 2:35 division 2:37 second division symbol 2:49 division by zero is indeterminate 3:05 Indices/Powers 3:39 One of the laws of indices. Radicals introcuced. 3:43 Irrational Number 3:50 Imaginary numbers 3:59 i^2 = -1 4:01 1^3 = -i = i * -1 = ie^-i*pi 4:02 one of euler's formulas, it equals -1 5:18 Introduction to the complex plane 5:36 Every point with a distance of one from the origin on the complex plane 5:40 radians, a unit of measurement for angles in the complex plane 6:39 circumference / diameter = pi 6:49 sine wave 6:56 cosine wave 7:02 sin^2(θ) + cos^2(θ) = 1 7:19 again, euler's formula 7:35 another one of euler's identities 8:25 it just simplifies to 1 + 1/i 8:32 sin (θ) / cos (θ) = tan (θ) 9:29 infinity. 9:59 limit as x goes to infinity 10:00 reduced to an integral 11:27 the imaginary world 13:04 Gamma(x) = (x-1)! 13:36 zeta, delta and phi 13:46 aleph

THE MATH LORE 0:07 The simplest way to start -- 1 is given axiomatically as the first natural number (though in some Analysis texts, they state first that 0 is a natural number) 0:13 Equality -- First relationship between two objects you learn in a math class. 0:19 Addition -- First of the four fundamental arithmetic operations. 0:27 Repeated addition of 1s, which is how we define the rest of the naturals in set theory; also a foreshadowing for multiplication. 0:49 Addition with numbers other than 1, which can be defined using what we know with adding 1s. (proof omitted) 1:23 Subtraction -- Second of the four arithmetic operations. 1:34 Our first negative number! Which can also be expressed as e^(i*pi), a result of extending the domain of the Taylor series for e^x (\sum x^n/n!) to the complex numbers. 1:49 e^(i*pi) multiplying itself by i, which opens a door to the... imaginary realm? Also alludes to the fact that Orange is actually in the real realm. How can TSC get to the quantity again now? 2:12 Repeated subtraction of 1s, similar to what was done with the naturals. 2:16 Negative times a negative gives positive. 2:24 Multiplication, and an interpretation of it by repeated addition or any operation. 2:27 Commutative property of multiplication, and the factors of 12. 2:35 Division, the final arithmetic operation; also very nice to show that - and / are as related to each other as + and x! 2:37 Division as counting the number of repeated subtractions to zero. 2:49 Division by zero and why it doesn't make sense. Surprised that TSC didn't create a black hole out of that. 3:04 Exponentiation as repeated multiplication. 3:15 How higher exponents corresponds to geometric dimension. 3:29 Anything non-zero to the zeroth power is 1. 3:31 Negative exponents! And how it relates to fractions and division. 3:37 Fractional exponents and square roots! We're getting closer now... 3:43 Decimal expansion of irrational numbers (like sqrt(2)) is irregular. (I avoid saying "infinite" since technically every real number has an infinite decimal expansion...) 3:49 sqrt(-1) gives the imaginary number i, which is first defined by the property i^2 = -1. 3:57 Adding and multiplying complex numbers works according to what we know. 4:00 i^3 is -i, which of course gives us i*e^(i*pi)! 4:14 Refer to 3:49 4:16 Euler's formula with x = pi! The formula can be shown by rearranging the Taylor series for e^x. 4:20 Small detail: Getting hit by the negative sign changes TSC's direction, another allusion to the complex plane! 4:22 e^(i*pi) to e^0 corresponds to the motion along the unit circle on the complex plane. 4:44 The +1/-1 "saber" hit each other to give out "0" sparks. 4:49 -4 saber hits +1 saber to change to -3, etc. 4:53 2+2 crossbow fires out 4 arrows. 4:55 4 arrow hits the division sign, aligning with pi to give e^(i*pi/4), propelling it pi/4 radians round the unit circle. 5:06 TSC propelling himself by multiplying i, rotating pi radians around the unit circle. 5:18 TSC's discovery of the complex plane (finally!) 5:21 The imaginary axis; 5:28 the real axis. 5:33 The unit circle in its barest form. 5:38 2*pi radians in a circle. 5:46 How the radian is defined -- the angle in a unit circle spanning an arc of length 1. 5:58 r*theta -- the formula for the length of an arc with angle theta in a circle with radius r. 6:34 For a unit circle, theta / r is simply the angle. 6:38 Halfway around the circle is exactly pi radians. 6:49 How the sine and cosine functions relate to the anticlockwise rotation around the unit circle -- sin(x) equals the y-coordinate, cos(x) equals to the x-coordinate. 7:09 Rotation of sin(x) allows for visualization of the displacement between sin(x) and cos(x). 7:18 Refer to 4:16 7:28 Changing the exponent by multiples of pi to propel itself in various directions. 7:34 A new form!? The Taylor series of e^x with x=i*pi. Now it's got infinite ammo!? Also like that the ammo leaves the decimal expansion of each of the terms as its ballistic markings. 7:49 The volume of a cylinder with area pi r^2 and height 8. 7:53 An exercise for the reader (haha) 8:03 Refer to 4:20 8:25 cos(x) and sin(x) in terms of e^(ix) 8:33 This part I do not understand, unfortunately... TSC creating a "function" gun f(x) = 9tan(pi*x), so that shooting at e^(i*pi) results in f(e^(i*pi))= f(-1) = 0. (Thanks to @anerdwithaswitch9686 for the explanation -- it was the only interpretation that made sense to me; still cannot explain the arrow though, but this is probably sufficient enough for this haha) 9:03 Refer to 5:06 9:38 The "function" gun, now "evaluating" at infinity, expands the real space (which is a vector space) by increasing one dimension each time, i.e. the span of the real space expands to R^2, R^3, etc. 9:48 log((1-i)/(1+i)) = -i*pi/2, and multiplying by 2i^2 = -2 gives i*pi again. 9:58 Blocking the "infinity" beam by shortening the intervals and taking the limit, not quite the exact definition of the Riemann integral but close enough for this lol 10:17 Translating the circle by 9i, moving it up the imaginary axis 10:36 The "displacement" beam strikes again! Refer to 7:09 11:26 Now you're in the imaginary realm. 12:16 "How do I get out of here?" 12:28 Don't quite get this one... Says "exit" with 't' being just a half-hidden pi (thanks @user-or5yo4gz9r for that) 13:03 n! in the denominator expands to the gamma function, a common extension of the factorial function to non-integers. 13:05 Substitution of the iterator from n to 2n, changing the expression of the summands. The summand is the formula for the volume of the n-dimensional hypersphere with radius 1. (Thanks @brycethurston3569 for the heads-up; you were close in your description!) 13:32 Zeta (most known as part of the Zeta function in Analysis) joins in, along with Phi (the golden ratio) and Delta (commonly used to represent a small quantity in Analysis) 13:46 Love it -- Aleph (most known as part of Aleph-null, representing the smallest infinity) looming in the background. Welp that's it! In my eyes anyway. Anything I missed? The nth Edit: Thanks to the comment section for your support! It definitely helps being a math major to be able to write this out of passion. Do keep the suggestions coming as I refine the descriptions!

Masterpiece :_AlanHeartblue::_AlanHeartgreen:

09:58 I love seeing my logo so powerful!

If math lessons were like this, math would for sure be everyone’s favorite subject Edit: well, this blew up fast. Thanks!

It speaks to Alan and his team’s talent on a number of levels that they can even make me feel sympathy for Euler’s number.

never in my life would I have ever thought I would see something tactically reload a math formula...

10:02 WAAAAAAAH!

I love how he goes from learning basic operations to university level maths

The reason why I love this series so much isn't just because of the animation and choreography, but because rules of how the world works are established and are never broken. Regardless of how absurd fight scenes play out there's a careful balance to ensure that not a single rule is broken.

I love how people dont even know what is the imaginary number and still watch this Imaginary Number (written as 'i') is equal to the square root of -1 which we considered the second dimension in our Real Number Line, the Real Number line with the Imaginary Number line is known as Cartesian Plane, which is basically a place used to graph equations. (More nerdy stuff at 25 likes)

A meganalysis of this: 1: a number representing a single entity Equal: a sign representing the sum of a equation Plus: a positive function symbol representing addition 1+1=2 Double equation: this is when you put two plus or minus. 10: a even number Parentheses: 2=(1+1) 3=(1+2) Minus: a negative function symbol representing a take away in math Negative: a negative function symbol representing the negativity or the lowest of a number e i pi: a mathematical equation equaling 0 Multiplication: 3x3=9 (1+1+1+1+1+1+1+1) 3×(4) = (1+1+1..) Division: 6/2=3 12/6=2 81/9=9 Exponent: 6+2²=64 4+4²=16 Fraction: ½=20% Square root: sqrt(4)=2 sqrt(16)=4 i.: -1=n i=1. Radian: also known as rad rad=4.8 maximal rad = 6.2 Pi: symbol representing 3.41 tau: symbol representing t below pi Summation: a symbol representing Σ a calculus sum. Function: symbol representing f f(x) f(sin) Integral: the symbol representing the integral of a number so far good Ratio: r=100 Idk i quit😊

The sound design is a masterclass on its own

So far, this is the best action movie in 2023!

Wow. This animation is as cool as reality if I had to rate it it will be 10/10👍