It makes infinitely more sense when you stop thinking of TSC as a stick figure, and start thinking of him as the numerical value that he is: *frames per second.* He is innately math meeting visuals. That’s why, for example, the multiplication sign speeds him up: it increased his play rate.

I think the reason symbols work on TSC is because he's using the number on his attributes, such as speed and position, the two attributes he edits in the animation. He isn't a number, but he's composed of them, like atoms.

You see that a youtuber makes a really masterpiece when even the university teachers are talking about it.

There's a lot of people who are going to finally understand concepts by seeing them in visual form. This is incredibly well done

10:04 something interesting about the integral is that it leaves behind a trail because the integral is the area under a curve

I'm always so shocked to see the attention to the tiniest of rules and details in their videos. Most of the tricks we saw in the Minecraft series could be done in-game, which is insanely cool as the videos serve an "educational" role too in that regard. Same with this video, just nothing but tremendous praise

It’s insane that this animation about math is not only flashy, but also makes sense! Props to Alan Becker’s team for making this animation, and to you for giving an in-depth analysis!

This reminded me of why I started to like math in school, before college ruined it. Feels nostalgic.

Imagine this is like a game, where you discover maths and the dialogue explain to you endlessly

While I don't like maths all that much, I used to and this brings a smile to my face. This is amazing.

as someone who hated doing math but loved learning the concepts and what math can do, this video is amazing; visuals are so important for learning and being able to see it in form helped me learn what I couldn’t in class. Your analysis really helps!!

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. (l 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 withx= 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 Fora 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 K-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 +de net tnderstand, nfertunately... 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 logl(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 fo 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 Den'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! Comment credit goes to @cykwan8534

5:04 in this case, TSC Is considered as X, since he is not a number, the "math dimension" has to do something with him if he include himself in an equation of sort, so TSC is X, making X rotate 90° on the axis, so watever his position was (if x was a point on the axis), it is rotated by i

0:27 I think you should’ve added the fact that the “motion blur/blending/in-between” frames actually have an equals sign! I find that really neat and fascinating, because they took the 1 = 1 concept and smudged it in with animation!

Im sure the original video gona become an Internet timeless classic.

As a physician who loves physics and maths, I absolutely love this gem of a masterpiece ❤️

I knew that there was a lot of math in this video that was going directly over my head, but I trusted the animator to have done their research. I'm glad to see that I was correct. I'll have to send this to some of my engineer friends and see what they think.

Of all the analysis videos I've seen on this animation so far, this one is definitely the best

Honestly I was scared looking at all this without an explanation, fearing I forgot “how to math” but once I saw this I understood I had an understanding of the math because I recognized it, I’m filled with calm now that I can understand this level of math, thank you?

Python with Prosper also covered this animation frame by frame and with some historical explanation. The effort being put into the animation and analysis is insane.

You have managed to condense trigonometry, algebra, introduction to calculus, and all the fundamentals required for those subjects within a single animated video with an entertaining plot of 14 minutes. Outstanding work. Definitely will be sharing this as a reference for anyone I end up teaching some math to.

That's a really good video! It explained everything in a good way, and was the first one that came in the recommendations that actually says something smart about the math. As a big math fan, I learned today some new stuff. The Tailor series, the small integral references etc. were all incredibly helpful. Thanks for the video!

Thank you!!!!! I’ve been waiting for someone to do this! ❤ I loved the Aleph at the end. I have dyscalculia but love the concepts of math. So frustrating! It was so nice to see all this laid out like it was and I was just hoping someone would label all the different functions and formulas!

Thank you for this! I loved the original video but knowing more of the context behind it is great

10:32 had me stumped for a while, but I think the interpretation is that he's feeding every point along the circumference of the circle (sinx + cosx) into the tan function simultaneously, so every point along the circumference of the circle is emitting the tan death ray at once

Everything Alan Becker touches is given full respect of the concept

I was thinking the video was going to explain more the maths, but its very cool like this !

8:51 got me dead 😂 " someone touched that radius again"

Now when it's explained like this, i would love to have a game that makes us use maths like they did in the animation. Learning maths like that would have been way more fun!

You sir, are a hero, spreading our word of math to the world. Goddamn, now everyone can appreciate the beauty of math :)

Best explanation on the video so far, a lot of others I found missed more intricate details and the ending concepts

This was really nice. I recognized most of the math in the animation, and at least understood what sorta math most of the rest was, but this really filled in the blanks in an understandable way, thanks!

4:34 I believe this is mostly a velocity thing where instead of TSC’s speed Accelerating by let’s say 1.2units(or Meters)/second, by adding a Multiplication Symbol to their legs, TSC’s Acceleration is now x1.2ups instead of +1.2ups. 7:26 This is fun because the Number just normally clashes with the Arclength/Radii, unlike the Number Sword Clashing Earlier. The Radius Length is a defined term, and therefore cannot be “deducted” or other similar variables would also have to adjust to this truth. However, the Arclength(of r=1), is as strong as a 1 +/- sword, and will be deflected by a 2 or higher.

so glad this is here! i'm really happy with how much i was able to recognize the first time (aleph, complex plane, ∞-dimensional ball) but the slightly more in depth explanations of the sum figure and the integral staff helped a lot!

There is still a lot of stuff with which I struggle in here (everything that involves the radian gives me a headache) but there are some stuff that I finally begin to grasp when given visual form. Alan truly outdid himself with this one and your explainations are very welcome.

Please make more of these, I know you didn't animate this but it adds so much to the video, really cool

3:16 power is repeated multiplication (which itself is repeated addition) 5:05 the bow that TSC uses is actually just '2x2=' oriented differently (idk how to explain it), which is why the projectile is '4' (answer)

My question is how TSC learnt math so fast, enough to use things people who have been studied for years cant remember

Dude that was boss. Loved it. I was happy to have already learned 99% of that stuff otherwise it would have be way harder to follow along. Still had to slow down the speed to .5 to really grasp what was happening though. made me so happy watching this.

Wild, might use this to refresh myself on basic stuff before college classes start this week

I still can't believe that is literally a lot of math explained just on one video!

8:25 I think you might have put sin(x) and cos(x) the wrong way round? Still the best explanation of this I’ve seen, with so many easy-to-miss details!

this is literally the best analysis of it I've seen so far

Let’s explore the differences between Parabolas and Hyperbolas in a simple way: 1. **Parabola**: • A Parabola is like a graceful curve ⤴️ that looks like a smile 😃 or a frown 🙁, depending on its orientation. • It’s the shape you get when you graph a quadratic equation, like . • Picture throwing a ball into the air 🏈- its path forms a parabola as it goes up and then comes back down. • Parabolas have a special point called the VERTEX, where the curve changes direction. It’s like the peak of a hill 🏔️ or the bottom of a valley. 2. **Hyperbola**: • A Hyperbola is like two mirrored curves🪞⤴️ that stretch out to infinity ♾️, forming a symmetrical shape. • Imagine two branches of a tree 🌲 that grow apart from each other, but NEVER touch ❌. • Hyperbolas have two SPECIAL LINES called ASYMPTOTES, which the branches get closer and closer to but never actually touch ❌. It’s like chasing after a dream that you can never quite reach. (Differences: • Parabolas have a SINGLE curve ⤴️, while Hyperbolas have TWO distinct branches ↩️↪️. • Parabolas can open upwards, downwards, left, or right, while Hyperbolas stretch out horizontally or vertically. • Parabolas have a Vertex, while Hyperbolas have Asymptotes.) (Similarities: • Both Parabolas and Hyperbolas are types of conic sections, which are shapes formed by SLICING a cone. • They’re both used in mathematics to model various phenomena and in engineering to design structures like satellite dishes and reflectors.) In summary, while both Parabolas ⤴️ and Hyperbolas ↩️↪️ are curved shapes, they have distinct characteristics that set them apart. Parabolas are like graceful smiles or frowns 😃🙁, while Hyperbolas are like mirrored branches stretching out to infinity🪞📏♾️.

Was completely expecting text to show up at 1:00 saying "Math sends you to the void" or something lol

This was exactly the mathematical breakdown I was looking for. Thank you so much for posting it!

You are the best explaining everything in this video and helping me understand this!

I learned new things out or Alan's animation and your explanation, thank you very much!🎉❤

5:04 I think he uses i with his arrow to make the translation upward (2x2xi) but since he was running so it makes an arc. That also explains why he can't sustain his elevation like e and falls down right after.

9:59 this my favorite part of you commentary, really nails what happened

I have watched many analyses on this subject, but none of them noticed the exponential relationship with dimensional shapes like you did. Impressive stuff.

Honestly, There is SERIOUS work done in alns video, the insane attention to detail, its almost like itts full of MATH REFERENCES? Wasn't expecting this from the channel with stick figures on a windows desktop

10:00 One point i think you missed is right after the integral appears, there's some expressions that appear on the left and right of it. The integral is 5 separate integrals, which are in the exponents of the e^...i on the left of the integral, each of the top 3 evaluate to π/2, the fourth goes to 3π/8, and the last to π/3, meaning that each of the five expressions evaluate to e^iπ. Also at 8:30 i didn't realize why the tan function was cancelling out the e^iπ since I didn't see the π that was multiplying in the tan function, good job spotting that. And at 13:04, while i knew the formula for the volume, It just didn't connect for me, so overall great job of explaining it.

I still remember calling math an easy subject when I was 1st, 2nd grade etc. But oh boy! Match is much harder than I thought it would be.

This really helped me understand the math founding this animation. Thank you for your work!!!!

Thank you for doing an analysis of this.

I personally thought the reason why the circle increases is because more E^i[pi] enters it, thus 'adding' to the radius. I only noticed with multiple watches, but as more enter the circle, it increases in size. I don't think any of them are actually touching the equation-just that their mere presence is adding into it!

Seriously somebody better make a game based on this animation as I'm pretty sure it can be a real entertaining way to teach kids of any age Mathematics how i know I'm pretty sure mathematicians and math teachers would agree to the idea

"The smallest cardinal infinity (hence why its so big)" you realize how insane that sounds right /pos AVM has me in a chokehold man. And watching this video explaining the things in it just made me love it even more. Thank you so much!

Amazing. this is being added to my favorites.

This is by far one of my favorite animations of all time, because it combines two things I love: math and epic battles.

respect for this guy for putting hard work for this so the kids can understand some stuff

THank you, i really needed that. I was actually wondering so this js good.

thank you for this analysis,very entertaining,is a very cool additive to the original video

I'm surprised how much of this I actually totally understood after just a little explanation lol. And THANK YOU, the mystery of Aleph has been itching my brain for days. Couldn't figure out how to even search for it by visuals.

Finally someone that can explain it easier and straight to the point, great job!

That's like impressively well put together.

Okay, thank you for breaking this down. Most of it's over my head, but I love how you explained everything.

As someone who took honors math classes throughout high school, the fact that recognize almost all of these mathematical concepts both amuses me and horrifies me.

I love it that you chose this form of anaylsis with editing in text instead of stopping it every time something comes up, much better

more things to note and perhaps clarify, 4:22 graphically, using a negative sign on the x coordinate of a point in space flips it about the y-axis. Here, it flips TSC around. This happens again at 8:03 but relative to the graph's (0,0). 6:01 θ and r are polar coordinates. Where θ is a phase and r is a magnitude. The equation θr equals the arclength the dot travels from a reference direction. 7:04 the highlighted area is equal to the area of the unit circle. This becomes more significant at 10:38, when it projects into an area of effect. 7:30 subtraction of radians when depicted in the complex plane results in a clockwise rotation, which is the direction that the slash arcs travel. 7:45 The formulation for this is a little confusing. 2πr is the equation for circumference, while πr² is the equation for a circle's area. The way TSC forms his shield suggests that his shield has a circumference of 4, and an area of 4π, which isn't possible. A circle with an area of 4π has a radius of 2, and the circumference of a circle with a radius of 2 is 4π, not 4. (But interestingly enough, a circle with a radius of 2 has the same circumference and area.) 8:26 Personally, I would depict e^(-iπ) as cos(π) - i*sin(π) because that negative symbol bears a lot of significance when working with signals. Sine and cosine have this weird relationship with negative symbols. cos(x) = cos(-x), but sin(x) ≠ sin(-x). Instead, sin(x) = -sin(-x).

A best content always get a explanation that some people dont get it. Thanks man

Noone's talking about this, but in 5:53 TSC multiplies itself with the radian(or seems like it) making another copy of it impling that TSC's value is 2 in the "math dimension". Just theory crafting over here lol

This video made me realise how much detail they put in ! Even the having the "bullet" make a tangent wave !

12:59 - I'm not sure if it's intentionally, but when e is stood next to the circle and beckoning TSC to enter, the "iπ" part overlaps with the circle and looks like it says "in", which is where it wants TSC to go.

Note that at the third appearance of Euler's Identity, when they're fighting, TSC uses the arc of the radian, which is the radius, of 1. This is why the swords cancel each other out.

4:41 "By the power of addition, i compels you -e^i(pi) !!"

Seeing all of this makes me realise just how visually striking this animation is in terms of what it conveys, such as 8:15 and 9:57

i was looking forward to such a video

1. **Coefficient**: • Coefficient is like the number that hangs out 😎 in FRONT of a Variable ❎ in a math expression. • It’s like the price tag 💲🏷️ on an item in a store - it tells you HOW MUCH of the Variable you have. (For example, in the Expression 3x, 3 is the Coefficient of x.) 2. **Base**: • Base is like the foundation of a math operation, especially in Exponentials and Logarithms. • It’s like the bottom of a building 🧱 - everything else rests on top of it. (In the Expression 2^3, 2 is the BASE.) 3. **Exponent**: • Exponent is like the little number floating above ☁️ the base, telling you how many times to MULTIPLY✖️the base by itself. • It’s like the power that makes things Grow ⬆️ or Shrink ⬇️ in Math. (In the Expression 2^3, 3 is the Exponent.) 4. **Variable**: • Variable is like the mystery number in a math problem that can CHANGE or VARY 📈📉. • It’s like a box that can hold different things depending on the situation. (In the Expression 3x + 5, x is the Variable.) 5. **Constant**: • Constant is like the unchanging part of a math expression, always staying the SAME ✅. • It’s like the fixed number that NEVER MOVES in a game. (In the Expression 3x + 5, 5 is the Constant.) 6. **Monomial**: • Monomial is like a simple math expression with just ONE term, like a single ingredient in a recipe. • It’s like a SOLO player🧍‍♂️in a game, doing its OWN THING without any partners ❌👫. (For example, 3x or 5y are Monomials.) 7. **Polynomial**: • Polynomial is like a more complex math expression with MULTIPLE TERMS added or subtracted together. • It’s like a team of players working together to solve a problem 👫🧑‍🤝‍🧑. (For example, 3x + 5 or 2x^2 - 3x + 1 are Polynomials.) 8. **Relationships and Differences**: • Coefficients, Constants, Variables, and Exponents are ALL PARTS of Expressions, while Base is specifically related to Exponentiation. • Monomials are a specific type of Polynomial with just ONE TERM, while Polynomials can have MULTIPLE TERMS. • Coefficients and Constants are similar in that they’re BOTH FIXED numbers, but Coefficients are associated with Variables while Constants stand aline. (Tips and Tricks: • Remember the “C” connection: Coefficient, Constant, and Constant Base (in Exponentials). • Think of Variables as the “variable villains” that can change their value anytime! • Monomials are like “mono” (single) and Polynomials are like “poly” (multiple) - simple and complex, respectively.) In summary, these Math Terms are like building blocks that help us understand and manipulate expressions and equations. They each have their own role to play, but together, they create the rich tapestry of mathematical concepts and problems we encounter.

That was amazing. Thanks for the analysis, I learned something from this.

WHERE'S THE VIEWS THIS IS SO GREAT i subbed

Thanks for making this commentary! Once they started dealing in calculus, the original animation lost me, 😅 but your explanations really help!

extremely amazing the way you expresses all these

I loved this analysis!

6:22 When a circle is stretched like that, it turns into an ellipse. So, any ellipse with major and minor axis greater than or equal to the radius of a circle and be created by stretching said circle.

I knew id find this video! Thank you so much

Even tho I knew Alan did research for this video (by understanding most of the math method he used) I'm still amazed at how he made a great animation by keeping such logic in it, truly a masterpiece

Instructions unclear. I am falling off a bridge now. Help!!!

9:57 "Integrals can handle infinites". Bro is saying it like he's a marvel supervillain or something

Alan Becker and his team are Genius 🤯🤯

Good job on going viral! you deserve it

2nd comment: This analysis video is incredibly amazing. It made the animation more impactful knowing what it is happening and what it all means. Most of all the last part explaining what was the big thing is and that gave more impact to the animation. 10/10 analysis video

Im glad I found this its the only review/breakdown video I could find that properly covers everything that happens thanks a ton for the explanation of all this it was amazing to learn

i understand it better now. please do more review like this.

*brought me up into a whole new level*

i love how you could throw this into the first math video ever made and people just learn it right there makes math so much easier

was hoping someone would make something like this, love it! it's clear you know your stuff

I appreciate this more with all the context

I like this explanation the best. He knows the character the best, and the info of the entire animation. and silly whatever around it.