Never would I expect to find out the exponential function has a sort of focus, but its location does seem to line up with intuition. This was great work!
@@hyperduality2838 ok I guess it's a joke by now. But why can't you replace 'duality' with 'plurality'? Your enlightened BS is based on the most simply ignorant case it can be...why not obsess over trinity?
Holy cow, I'm so glad this popped up on my feed. It's not often I learn about an entire new way of visualising and conceptualising the behaviour of simple functions like this that I had no idea about. 3blue1brown hinted at this with the animations of remapping the complex plane, but this takes it so much further and with so many cool insights along the way. I'll have to watch it another one or two times at least to really absorb the details I think. Really nice work, and I'll share this around. I hope your channel grows because you deserve the audience.
So, uh, thank you, for this. I haven’t enjoyed a math video this much since I discovered 3blue1brown a few years ago. This was amazing. I really loved that you opened this topic with a seemingly simple question and explored its complexity and beauty, while building a sense of intuition. Also, the pressed flowers metaphor was everything. So good, I’m obsessed. Chef’s kiss. Fingers crossed will get to see an explainer on the irrational powers someday.
Thank you so much for sharing those feelings in such detail! It really reaches me and gives me so much joy! I hope I get around to this topic, it's one of my favorites.
@@imaginaryangle I'm curious to know if you have found any kind of formula for the location of those "foci" for functions other than degree 0, 1, or 2 or the simple higher degree ones like x^n with no other terms
@@WhattheHectogon No, just the simple cases you've listed here (you can see it in the Desmos graphs given in the description). I also didn't come up with an elegant approach to look for it. Do you have an idea?
@@imaginaryangle It seems like they happen where the derivative of x + f(x)i is 0, since the lines 'bunch up' around the kinks. So they would be the x where f'(x) = i, plugged back into x + f(x)i. Don't know about a general explicit formula though, and it doesn't explain why they would be like foci. Great video by the way!
This was awesome, I love seeing the Riemann sphere existing still hidden in this. Maybe you can do a video showing what's going on when you go into the extended complex plane with the Riemann sphere?
@@imaginaryangle yes the Riemann Sphere was the only thing missing from this video. Adding it would've been perfect! An entirely geometric interpretation of all complex numbers, while also making your "complex infinity" (really just an unsigned infinity like 0) make intuitive sense. Better yet, graphing on the actual Riemann Sphere would show what physically happens when your graphs like 1/x shoot off to infinity. Hint: 1/x is just a Mobius transformation.
Marvelous and gorgeous! Please produce more like this. Truly enlightening and edifying. It would be fantastic to see more of the 3D renderings, though. All becomes clear when you add more dimensions. Keep up the good work. 👌👍👏
Wow! I studied complex analysis many years ago, and while I understood it well enough to get good marks in assignments and exams, I always felt that I didn't really understand it. It's like a jigsaw puzzle where I have all of the pieces and I know which pieces connect to which other pieces, but I can't see the whole picture. This gives me a new way to visualize and think about analytic functions and see the whole picture. I find, that I can't understand anything in mathematics unless I can find a way to visualize it. Thank you!
@@hyperduality2838 I do agree. I came to the same conclusion myself. "Being and non-being create each other. Difficult and easy support each other. Long and short define each other. High and low depend on each other. Before and after follow each other." -- Lao Tzu
Visual representation of curves on complex plane, done beautifully and explained clearly in details. If this was submitted in #some, it will be easily at the very top.
This is brilliant! Very brilliant! This channel belongs in the same league as Mathologer and 3Blue1Brown. This has echoes in three other videos: Mathologer - Times Tables, Mandelbrot and the Heart of Mathematics - where multiple foci of cardioids appear Welch Labs - Imaginary Numbers Are Real [Part 13: Riemann Surfaces] 3Blue1Brown - Taylor series | Chapter 11, Essence of calculus I wish imaginary numbers had more real name (like orthogonal numbers or some such - this idea was suggested by Riemann I think) so that they will not get a short shrift and thus allow development of more intuition about them. I know "imaginary" is just a word but sometime sociologically it has an effect of apathy. I think you are helping us develop that intuition which of course should be followed by mathematical rigor. But creativity starts with intuition.
Thank you so much, this means a lot! And you're right, my focus is more on assisting the building of intuition. It's awesome that there's a whole ecosystem of math educators and each of us can dive into our own approach without fear that something important won't be covered. And I might be biased (I definitely am), but I like the word "imaginary" 😉
This is the most intuitive way anyone has tried to explain to me the connections that each power graph has to any other power graph. Using complex number space and compressing it to fit on two and three axes really does show a lot of what's hidden on the real number line. And it was done in a way that retained the shape and certain key features of each power graph. Bravo!!!
I've been playing around with polar and parametric graphing and functional analysis in both 2d and 3d desmos for the past few weeks on end, but never once conncted the dots to how the imaginary plane plays a roll in all of it. Seeing this was absolutely mind blowing, as it really just connected so much stuff I've already learned together in ways that are simply incredible to think about. Seeing all the relationships layed out in this manner was just absolutely mind blowing, and it really taught me a lot about how many different areas of math that I'm currently interested in were dreampt up and developed further. And truly, the foci are OP
Thank you for scratching a brain itch that has been itching since primary school! Very satisfying! My only note would be on the animations: in a lot of the animations the mid-point is where the magic happens, but it is also when the animation is fastest. So I'd suggest either inverting the speed, such that it is slowest around the half-way point, or just leaving it linear. Though linear animations always look a bit stiff. But the whole point, in my opinion, of this video is showing what happens at the weird transitions, so its a bit of a tease, that exactly that part is sped up.
"Complex Variables" by John W. Dettman (published by Dover) is a great read: the first part covers the geometry/topology of the complex plane from a Mathematician's perspective, and the second part covers application of complex analysis to differential equations and integral transformations, etc. from a Physicist's perspective. For practical reasons, a typical Math Methods for Physics course covers the Cauchy-Riemann Conditions, Conformal Mapping, and applications of the Residue Theorem. I've used Smith Charts for years, but learned from Dettman that the "Smith Chart" is an instance of a Möbius Transformation. The Schaum's Outline on "Complex Variables" is a great companion book for more problems/solutions and content.
Thank you, this is great! I get these glimmers of how beautiful complex analysis is, but it's too much to comprehend at once, I only get pieces. So thanks for helping me assemble a little bit more. I was dubious about the x.+ i f(x) trick at first but it's actually a pretty interesting tool for visualization. Thanks again!
Wowwww amazing visualisations! I love seeing functions' level curves and I don't think I've seen this used as a way of visualising complex functions before - I feel like I understand these functions better now, thank you so much for making this!!
This is good stuff. Should put this in submission for Jjjackfilms.
9 месяцев назад
So... I was washing my dishes looking at RU-vid videos because, you know, hardly anyone just enjoys washing dishes. And then I started looking at your video and actually stopped washing because my brain needs more processing power for your explanation. I stood there, hands wet, for the whole video. Great video sir. I liked it very much.
This is absolutely alien intelligence So so good man, excellent work. This is one of the best videos I've seen that translates math almost completely into art. You need to take these ideas and perspective shifts and put em up as AR assets because I would totally PAY to get to interact with knowledge like this. There's some that teach you foil, and there's some that incrementally bump up the understanding of everything you know anything about. This is totally core human curriculum
The form of the function you actually “graphed” f(x)=x+i•g(x) seemed kinda arbitrary at first but the you blew my minds at the foci it had when you include the the complex values with the same magnitude. I’m amazed but am still trying to interpret what I’m seeing here
Fantastic. One of the best ways to visualise, theorise and conceptualise so many different parts of geometry, rays, graphs, and ellipses I have ever encountered.
I love how this title attracted two very different yet equally interesting people, some might fall in the venn diagram intersecting both groups.. Like me who finds maths interesting, and would also like to know about how secretly kinky elementary functions are..
nice work, bro rather than pumping out ai shitty content, you chose the true path of making quality content. this way, you will gain even more subscribers. wish you much very luck. thank you for making math more loveable. sincerely, freshman year cs major student. also i smell russian accent. are you, by any chance, russian? (i am though.)
This is excellent. The visuals are really beautiful and everything is presented in a way that flows smoothly. Seeing cardioids (or cardioid-like curves) pop up in the quadratic case was interesting; I'm going to spend some time this weekend understanding why they're there. Thanks for the great video!
somewhere there's a really cool video digging into the Mandelbrot set and related functions that also points to some fascinating connections to where the cardioid shape in that comes from, as well as some other shapes you get when you generate Mandelbrot-like sets with exponents other than 2. These are somehow clearly related to what's going on in this video but I can't quite articulate it.
Thanks, I'll have to find it. I've found that a circle of radius R is mapped to the cardioid: r+2R^{2} \sin\left(\theta ight) = R But at this point my understanding is algebraic only. (Edit: Forgot to mention that it's technically mapped to that cardioid translated up by R^{2})
Mind. Blown. Division by zero equals complex infinity, a circle of imaginary radius zero with a direction undetermined. Fantastic video, especially if I ever decide to do mushrooms.
Amazing video. I have been wondering about just this (but on a very rudimentary level). Fascinated to learn about a deeper structure here and very good visualizations. Feels there is much more to know here... Thanks for god job with this video.
I tried playing with parametric representation on desmos to visualize space transformation when using complex variables, this really reminded me of that. In a similar vein I wanted to understand and play with raising non-unitary complex numbers (a+bi) to non-unitary complex powers which lead me to finding my favorite number: Gelfond's Constant (-1)^(-i)=e^pi.
Amazing video! Mind blowing stuff - love your initial explanations of imaginary numbers too, somehow it feels intuitive - we gain freedom (of rotation in this case) when negation comes into the picture. Thanks!
as someone who hasn't really learnt complex valued functions yet, this video was pretty colors with funky patterns. don't get me wrong, i enjoyed the whole thing fold and unfold before me, i just feel like there are more technicalities that would be nice to know. for example, i still don't really get why you used z = x + f(x)i. i see it's supposed to be in the form of a complex number, but what are the disadvantages of using only this? what are some nice things that we've missed out on just by choosing to represent values like this?
I get it, this video leaves a lot of the technical details out to focus on the visuals. The reason the z expression is used is that this way you get x values assigned to the Real axis, and f(x) to the Imaginary. For functions who's domain is the whole set of Real numbers, you just get the regular graph, but you also get the option to extend it beyond that. The disadvantage is hinted at when the cube root is discussed, there you get a glimpse into the parts hidden by this method. The entire multivalued truth of these functions in the complex world isn't something that can be drawn all at once, you always have to make some compromise. I picked one where we stay in 2D, while preserving symmetries and being able to track change in behavior. Check out my video linked in the description for a deep dive into these kinds of numbers!
I looked at z = x + f(x)i as a way to take a real-valued graph y=f(x) and "re-explain" it as a plot in the complex plane by forming a complex number z from x+yi. This is a neat trick, because it produces the familiar graphs for real-valued functions, but it generalizes to complex functions.
I always kinda viewed it in my head as a sort of 3 dimensional cylindrical system, with the x axis feeding in real numbers and then receiving magnitudes (y axis) and angles rotation (ø axis I guess, rotating around the x). there's some fun visualisations to be had in looking at the roots of unity or w/e for non integers, y'know of the form x^a=1. the way solutions pop in at and right after even integers and gradually repel each other as they spawn and split away is very pretty
This video is simply amazing! Since i introduced myself to complex numbers through some youtube videos and wikipedia articles, i always wondered what were these rainbow-looking images, that on some resources were shown as "graphs". For a high schooler, that did not really learn anything complicated about calculus, (not even mentioning complex "world") this was rather distressing to read the information in overcomplicated and scientific way that is shown in almost all articles and pages. This video just united anything that i knew about essense of graphs and complex numbers and i am absolutely love it! Dear creator, you really deserve more views and i wish you it! IM IN FOR YOUR NEXT VIDEOS!
This is an excellent representation of how graphs transform into another function in the complex plane and how would they behave. I might have to look deeper into this to understand what really is going on!! (Congrats in advance when reaching 5k subs btw)
Captivating. I gorged on number theory and numberphile videos etc.. until I got burned out on the talking head, crude construction paper or blackboard approach. As much as I liked the presenters, I needed a snappier pace. This was the fastest 30 minute math vid I've seen in a while.
I think I would call this a pattern approach. Normally before we had the advent of computer graphics, we weren't able to see comparative variables without extensive, time-consuming drawing. This approach gives us a better understanding of effects and patterns that develop by creating multi-variable or parameterized. The parameterized graphs shows the interesting points in the equations by displaying the patterns. Keith Devlin, a mathematics professor of Stanford U. recently mentioned that we should start thinking in patterns rather than simply in equations since the patterns give us more information on the effect of equations on many processes.
@imaginaryangle, i really like the intro to complex numbers from the channel Richard Behiel, title "complex numbers in quantum mechanics", it gives an intuition for complex numbers in terms of how one invents them to deal with sinusoidal patterns in reality and I feel like that justified their existence more cleanly to me than most other intros I've seen, so it seemed worth bringing up
what a beautiful video! the "spirals" from the negative exponents reminded me a lot of the graphs of trig functions in polar coordinates ( r(theta)=cos(a*theta) ).
also notable with the negative powers is folding the rings inside out. it's something i explored a bit as i tried to come up with some way to get the complex conjugate (with the purpose of flipping the phase shift of a filter) a while back (never got anything that works, and moved on to other things before trying to make an approximation), i think because it also changed the phase of the complex number, or it in combination with something else got me close. it was like most of a year ago so my memory is a bit hazy, but it was nice to see something about complex plotting again, very cool topic.
Just commenting to help promote this, fucking wild. I feel like many people have had the same question about the forms of graphs between powers. I got to the conclusion that its dirty and doesn't make to much sense because decimals (say x^1.7) turn into fractions (x^ 17/10) which turn into gross powers and roots (17th-root(x^10)), and that's it. Thank you for the new intuition. Most people here know 3blue1brown as the king of teaching like that, but man, this video is right up there with him. Edit: I recently rewatched Morphocular video: What Lies Between a Function and Its Derivative? | Fractional Calculus. I am VERY curious how someone smarter than me might relate these two concepts into the ultimate "graphing functions with powers and derivatives between whole numbers" video
I’ve always thought that it’s counterintuitive to identify points on the complex plane by the sum of their respective coordinates, although this might be simpler and it can work alright, it just makes more sense to identify it by the length of the segment from the origin to the point times i to the power of twice the adjacent angle (in radians) over pi.
😮 I thought I had seen beauty in maths but it seems I haven't seen anything. Thank you so much for this video. My mind is blown and my curiosity is overflowing. ❤
What do you mean you can't draw in four dimensions? It's possible (and _easy)_ to draw in _6_ dimensions on a screen, with 3 of those dimensions being "compact," and one of those dimensions being time. This very video is already 6 dimensional. _Every_ video since color displays has been 6 dimensional. Domain coloring is probably the best way to graph a complex function as it uses those 3 compact dimensions to represent the two output dimensions, with focus placed on the most important output dimensions, which isn't the Real or Imaginary components, but rather the phase (which is already compact anyways). Edit, with the style of graphing used in this video, color phase was basically used as an _input_ dimension to convey the magnitude of the complex number. It was still very cool to see, both how it manages to replicate the functionality of reflecting inverses across x = y, and showing off foci (and the paths they follow when parameters are changed) for functions that you might not initially think of as having them.
True. But the downside is that compact dimensions usually don't describe shape well, and if you've used up time for one of the dimensions, you no longer have it available for varying a parameter. In the particular case of this video, you're right, the visualization does add a few compact dimensions, namely magnitude expressed as hue, and by cherry-picking magnitude increments and line opacity, density also emerges.
@@imaginaryangleVarying a parameter with time is just using time as an input dimension rather than an output dimension. It's still being used as a dimension either way. With the final visualization of the power changing at the end, the time dimension mapped to that power input, though seemingly through a non-linear R->R function, as it paused briefly on integer values. When doing the 3D visualisation for the cube root, _that_ was a case of actually using 2D to try to convey 3D information, and just relying on the various tricks we've evolved to extract 3D information out of 2D images, helped by changing an angle parameter to observe 3D effects like parallax.
14:45 - That's why you take the principal branch of theta being in [0,2п). Otherwise x send to x^1.1 is not a function. The same goes for any power. 27:15 - We're approaching progective geometry now. lol
Math truly is the heart of beauty. Wonderful presentation all around! A couple ideas on the visualization, for the 3d portion, perhaps some blurring and dual-window cross-eye stereoscoping would be edifying... And pretty please can you make it so only the intersections of the rings are visible with little dots? My intuition says it would look especially cool at higher resolutions and perhaps highlight more complex "channels" through the structure, much like how the line traveled by the focal point jumped out to the eye.
Yes, 3D needs more effects to make spatial comprehension easier, but I'm not that good with tools that let me do that yet. And seeing just the self-intersections of the cycles is an interesting idea. Thanks!
I'd love to see more 3-d-ization of the real component with the complex components. I never knew that the two complex components are identical, but "compressed" into the 2d complex plane. Would love to see more than just the cube-root case.
They're identical in the sense that they are just collections of step lengths that can be taken in some unit, and then they apply those identical steps to different units: 1, -1, i, -i. I steered a bit away from too much 3D because lines hanging loose in 3D space are difficult to correctly visually interpret. Is it going towards you, away from you, how far... Maybe when I get better at 3D shading and environment building.
@@imaginaryangle I feel that motion in 3D graphs can go a long way towards conveying information and cleaning up what's being displayed. I was just about to write a separate post when I saw this one; I was going to ask if the rainbow lines (z-values?) could be plotted in the environment introduced at 24:02. I'm guessing they would form a complicated structure but I'd imagine it would be interesting...
@@imaginaryangle I think you made a good choice, restricting most of the concepts to a 2D representation I believe forced them to be more understandable (and kept the time under control). Plus, now I have yet another reason to keep checking back :)
It is one of the best video which relates the complex numbers with some interesting applications!!! From 29:55 the different graphs somehow reminded me of the d, p orbitals of electrons. I wonder if there is any connection?!
Amazing video. it was very fun & intriguing to watch. (might have to rewatch a couple times to digest it tho lol) I got like 40% of what was bein said (I barely know enough mafs to get thru highschool). The visual representations were top-tier. I don't think i would've understood anything without them.
Something like this but for visualizing elliptic curves as we vary the j-invariant would be amazing and afaik hasn't been done yet by anyone on RU-vid (This can also be done for elliptic curves over finite fields).
