What does a complex function look like?  

I never fully grasped why four dimensions were needed for complex functions, but the way it was explained here finally made it click

This has to be one of the clearest explanations from a SoME3 submission. Great work!

I love how this video is 20 minutes long and it felt like 5 minutes. Everything was well explained and it just kept getting more and more interesting. Congrats!

Love myself some #SoME3 in the evening.

I love your motivation at the beginning! You really hook the viewer with exactly what you're talking about, why it's interesting, and make promises for later in the video. It really kept me watching! In terms of clarity, I also loved how basic you started, letting anyone with even a small knowledge of imaginary numbers and what they are get by and understand the video. I think you could've done with maybe a short reminder of what imaginary numbers are and why were graphing their functions, but other than that it was fantastic. I also love your use of constant examples and animations, which really make sure the viewer is staying with you as your progress through the video. I think a couple of your steps could be more well thought through or explained, such as explaining polar coordinates, but even those were not bad and could be fixed with just a few seconds. In terms of originality and memorability, this was fantastic! I've never seen this covered before, but it's such an important problem with dealing with complex functions. I feel like most people take your first solution for granted when looking at the problem, so it's super interesting for someone to dive into possibilities we haven't considered for such a basic task.

only once did my teacher visually show the complex region on the white board, but this video helped it explain what it looks like. cool!

I love domain coloring, I used in a paper to visualize complex numerical solutions to a differential equation and people loved it too :) not only can you immediately see all poles/zeros/essential singularities/branch points immediately, but you also see the order of such just by seeing how many times the hue changes around a point Mathematica has a beautiful color function to visualize contour lines and at the same time lines of constant real/imaginary parts that I found the most complete

It actually is possible plot a 4D graph, for instance a surface with two parameters, in a 4D space. The solution is exactly what we do with plotting 3D surfaces on a screen. We project it onto a flat screen, and then rotate around in three dimensions to examine different views. You also can take 4D information (the surface), project it to a 2D screen, and similarly rotate around in four dimensions, examining the surface from different points of view in 4D space. For instance one view might show the real output on an axis perpendicular to the xy plane (the z-axis), another view would show the imaginary output on a different perpendicular axis (the w-axis). And you can rotate partially between the two views. The difficulty is interpreting what your seeing, which is a matter practice, but it certainly is possible.

Why doesn't anyone plot these like a vector field? Those plots often put little 2D vectors at points all over the domain. For many functions I think this provides a nice combination of accurate and intuitive.

I'm a mexican collegue student and i enjoyed the video, i always wanted to understand how to visulize a complex function and this is the first video that explain it well.

my colorblindness going wild on this one lmao

never could've imagined that by simply taking the sqrt(-1) we would ever get to 4 dimensions.

you could add a colour wheel in one of the corners in the coloured graphs to better convey what each colour means

Absolutely incredible video! Truly 3blue1brown level

Literally the best video I've seen about this.

Fascinating video, this explained everything in a way that i could understand, and I haven't even gone through calculus yet. Please make more videos!

Such a concise and well explained video!

You explained it perfectly, I understood Everything!

Outstanding discussion!

God bless you Awesome lecture

i hope not winning anything from some3 doesn't discourage you i loved the content and you should definitely make more videos

You just earned a subscriber

It was so original! though I have to watch this few more times to understand better

amazing!!

This IS parametric architecture. Complex calculus and complex analysis should be requisite for architects.

Gotta watch 24h version of spinning chip now

I was hoping at the end you’d plot ln(a) so we’d see the multiple layers!

This is so great. Subscribed 😁👍 Btw, how would you approach plotting quaternions? Are quaternion valued functions a thing even?

Just plotting the argument of the value when domain coloring gives enough information to work out the entire function up to a scale factor, as long as the function is differentiable.

thank you for the explanation, even though this is really a good video I still have a hard time to understand this, maybe because I'm still in high school

You got my sub just for simple "hope you liked bye"

10:35 looks like 3 quarks forming into a baryon

Morbius equation is a zeta function

Very insightful video. Can you tell the software you used so i can use it and get better understanding of the visualization?

A graph is just a set of input and output coordinates? So I guess that means that domain colouring _is_ a true graph of a ℂomplex function. It's a set of points (x, y, |f(x + iy)|, arg(f(x + iy))). Those last two coordinates then actually get mapped _up_ a dimension so technically it ends up as a _5_ dimensional graph used to plot a 4 dimensional function. Adding the height makes it a total of 6-dimensional, though with said dimensions not all be linearly independent, and then it gets projected down to 5-dimensions anyways in order to display on a flat screen. People talking about it being "impossible to plot a 4D graph" really do often forget that even their "2D graphs" are ultimately 5 dimensional, though it is true that the redundancy and less dense information can make it much easier to interpret.

Pretty much what I am saying in a round about way. Except I'm relating this to electro-magnetic light. You can represent your colors in two ways, potential energy or kinetic energy. The cone is a special case of kinetic energy when mass is zero. Potential energy is the flat disc but only when when there is mass (numbers between -1 and 1 on output can't be included). If there is mass, potential energy has a small lip on the edge. I have to make my own graphing calculator, I guess. If I want it done right. When kinetic energy has mass, it is a hyperboloid. If the mass is negative, it is potential energy. If the mass is positive, it is kinetic energy. At least that is my theory this morning. I'm going to fudge with the numbers and try to verify it.

9:40 So if I start at origin,(blackness) and walk right on red I'm in positive real number territory and walking left on cyan I'm in negative real number territoy; I don't understand why the yellow line represents 'positive' imaginary number (it is below real line) and purple represents 'negative'?(it is above real line) why is this the convention? Super nice video. Really makes the 4-d concept make sense. Thanks for your work.

Great ! What tool used for visualization / rendering ?

Yes

Once, I was playing with Geogebra on my phone with complex function and got one ovel-like function I liked it so much I took a screenshot (with out axis) Now it's my profile picture but I forgot what the function was. 😂

if you think imaginary numbers are insane, wait until you hear about quaternions

Can I know what programs you use for such video creations and thx

2:15 why is the domain a unit circle and not the whole plane? or you're just showing how that circle maps

Is this a collaborative project between a graph author who lives in the U.S. (spelling "color") and a narrator who lives in another country (calling Z "zed"?)

As someone with mild protean RG ‘color blindness’, the function colors without contours don’t perfectly match the contours 😂 @10:10

At 9:44 I would have expected the green line to have a negative imaginary part and the purple line to have a positive imaginary part. These are flipped in the narration. Not sure if my expectation is wrong or the narration is wrong

what software do you use to graph these?

Is this available in 3Blue1Brown's Python package?

very very interesting! we find fractal equations...

so z^2 is a potato chip nice to know

POV: You are Colour-Blind

Tw0 dimesion qua cubed

z^2 is a pringle chip change my mind

pringle

just one criticism: 16:31 you say that the graph of 𝑓(𝑥) is {(𝑥, 𝑓(𝑥)) ∀ 𝑥 ∈ ℝ} when in reality that only counts as the graph of a function 𝑓(𝑥) with domain ℝ, but not for any function whose domain doesn't span all real numbers. for the rest, the video is an amazing learning tool, hope you the best!

I thank God and then I thank RU-vid for recommending this amazing icon to me ((Free Palestine))

I am the 48th comment

Wow this looks really complex 🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣

This will kill me

you don't know what you're talking about