Twisting the Plane with Complex Numbers
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- Опубліковано 20 вер 2024
- A computer animation by Jason Schattman that shows how complex-valued functions warp & twist the plane in stunning-and sometimes violent-ways. For example, the function f(z) = z^2 shown at the start of the video bends each vertical line of the grid into a left-facing parabola, and each horizontal line into a right-facing parabola. The square-root function (shown at 1:58) literally rips the grid in two! But my favourites are the reciprocal function f(z) = 1/z (at 1:23), which turns the grid inside out, and f(z) = sin(z) (at 2:20), which...well I won't give it all away. Just watch and enjoy. :)
More detail for the mathematically inclined
********************************************
These animations are illustrations of conformal mapping. A conformal map of a complex-valued function f(z) is a graph on the complex plane that shows how each vertical and horizontal line gets transformed by the function. (The complex plane is the set of all complex numbers of the form a+bi, where i is the square root of -1.)
For each point z in the grid that is being mapped, the program computes f(z) (think of this as the final destination of point z). The program then animates the journey of that point between z and f(z) over time. This is done using linear interpolation. At time t, where t ranges from 0 at the start to 1 at the end, let g(z, t) be the interpolated point between the starting point z and the ending point f(z). The program computes point g using the formula g(z, t) = t(f(z)) + (1-t)z, so that g(z, 0) = z at the start, and g(z, 1) = f(z) at the end, and g(z, 0.5) would be halfway between z and f(z).
To see more of my mathematical animations, check out my videos on...
Animated string art: • String Art with Comput...
All 6 Trig Functions on the unit circle: • All 6 Trig Functions o...
Optical illusions with rotating cubes: • Optical Illusions
Amazing epicycles using tumbling batons: • Amazing Epicycles
Sound waves in an oval room: • The Physics of Sound W...
Fly through the 3D Sierpinski pyramid: • Fly Through the Sierpi...
Drawing on a spinning white board: • Amazing Spirograph
Cool mathy fact: Watch how the quartic function f(z) = z^4 maps the vertical lines of the grid directly on top of the horizontal lines! Can you see how this relates to the fact that i^4 = 1?
I coded these animations using the Processing programming language. You can download the code at drive.google.c...
With the code, you can easily create your own complex-valued functions and make new animations with them.
This means that i am a complex person, because every time i wake up my blanket is twisted in the same way as shown in the video
Imagine if we could show these animations to the Ancient Greeks. Forgetting all other modern notions Plato would be bewildered by and in awe of, these would certainly peak his geometrical interest such that I think he’d proclaim the gods are real.
Beautifully said!
*pique
Honestly quite incredible
Thank you! I'm delighted that you enjoyed it.
Great, another math channel I have to watch 😄
Ha ha! Same!
They are so hard to find.
Meanwhile the dementia things taking over the entire media, I hate that so much.
your account is thirteen years old and you are still active cheers dude
Same
@@dadogwitdabignose Thank you!
I suck at explaining things, but it's quite cool how our brains will map the grid out, as if it is a 3d image and not what it started as; a 2d grid.
Perception and human psychology are what mathematics beautiful to us, and not merely a dry utility.
This makes me feel like it's possible to visualize conplex functions in 3 dimensions. Having some input space in the complex plane and the output be another plane projected upward, with a "dot plane" of lines connecting each input to an output.
I wrote a little program, for every point in the plane you plot the modulo |z| on the z-axis and you assign a color for every phase from 0 to 2*pw
My blanket every time I wake up:
Ha ha!
I like how √z tried to tear apart the complex plane
That's my favorite one, too.
Dude, this is so exciting to look how outer borders (where x and y going to infinities) stretches and overlap each others🙃
Thank you! I'm glad that aspect of it stood out for you.
Math always produces the most confusing but stunning works for me
The wonderful lthing about mathematics is that it can be enjoyed just for its visual beauty without having to understand everything about it the visuals were made. Kind of like being able to drive a sports car without needing to know how the engine works.
@@beautifulmath5361 idk man. You would still need to learn to drive for that sports car analogy. Opposed to just observing the mathematical art, since we already know how to see. 😅 lol
Best viewed with 3D glasses.
Ha ha, agreed!
Or after smoking weed.
@@masacatior it'll just feel like smoking weed if I saw this on VR
absolutely beautiful thank you for sharing :)
You're very welcome! I am enraptured by the beauty of complex numbers.
imagine living in a universe with planes like these
I feel like planes exist like this all throughout our universe but we humans can’t experience them without dying
@@SIX6VI.Underworld or rather we can't perceive them
@@skyfire299 yes we cannot
it feels like im being hypnotized while watching this
Complex numbers are hypnotically beautiful!
I understand complex numbers and complex maps well, but how are you getting the 'intermediate' parts of the mapping?
For each point z in the grid that is being mapped, the program computes f(z) (think of this as the final destination of point z), then linearly interpolates between z and f(z) as an animation. Specifically, let g(z, t) be the interpolated point at time t. g is computed as g(z, t) = t(f(z)) + (1-t)z, so that g(z, 0) = z and g(z, 1) = f(z), and g(z, 0.5) is halfway between z and f(z).
@@jasonschattman2336 ahh so it's literally just interpolating linearly between z and f(z)
@@glitchy9613 It does appear to be linear interpolation, but geometric interpolation might result in prettier animations due to the polar nature of complex numbers. That is, where linear interpolation from a to b by the factor t is L(a,b,t)=a(1-t)+bt, one may define that the geometric interpolation is G(a,b,t)=a^(1-t)b^t.
I don’t know if this is actually called geometric interpolation, nor have I ever seen it before, but I call it geometric based on the name of geometric means/averages as opposed to “normal” means/averages. In fact, the geometric mean of a and b is exactly G(a,b,0.5).
@@JordanMetroidManiac oh, I actually recently made a couple desmos graph thingies based on complex maps and one of them used linear interpolation after I found out about that being the way it was done here, maybe I should try this 'geometric interpolation' for a map and see how it looks.
It depends which function it is, but for certain functions such as f(z)=z^2 it is intuitive to generate the functions as f(z,t)=z^(1+t). There are different interpolations that can be made for different functions, in general you need to find a continuous way to go between z->f(z).
I was guessing what each transformation was, the first one was x^2 but I didn’t get there rest until I read the comments and looked so hard my eye started to hurt but I eventually was able to find it
This is simply beautiful. Conformal transforms are a true spectacle. The geometry of complex numbers and calculus.
Unlike the other guy, I don't understand complex numbers nor complex plane. However, is like we're watching it bending into a higher dimension.
It's eerily similar to watching hypercubes in 4d!
Strange, ominous, perplexing... this made my brain smile🙂
These visuals and the mathematics that underlies them evoke these same emotions in me, too. Complex analysis is truly beautiful.
This should be on display at modern art galleries.
Thank you! I am delighted that you see the aesthetic beauty in these structures just as I do. 🙂
How wonderful !
I've been waiting for a video like this.
So glad you enjoyed it
f(z) = √z was my favorite because of how uniquely chaotic it seems compared to the rest
For me as well! It is definitely more jarring than the other functions, which all appear smooth.
maybe because of the square root of i compounds
part of it may be because sqrt(z) doesnt map to the whole plane, unlike the other functions. A similar function is the natural log, which squishes the plane into a horizontal slice.
some of these resemble a 2 dimensional bent into a shape, sorta like something you'd see in topography class or something, and I think thats cool!
Yes, this reminded me of topology too.
It makes me wonder if complex math can explain extra dimensional travesment through normal spacetime by folding it.
there's definitely some relationship there. I don't know much, but I can tell that these are conformal mappings, meaning that they preserve angles.
@@zokalyx Yes, these are conformal maps. :-)
Just a note, and I know this video is rather old so you’ve probably had this thought yourself by now - it would be wise to have a bit of empty space at the end for the video’s end cards to take up, like 3 seconds or so of nothing. The end cards (at least on mobile) block the last animation and there’s no way for me to hide them.
Good suggestion, thank you. 🙂
Try to turn off auto play?
@@roygalaasen it’s not an autoplay thing. its the videos that pop up before a video ends, autoplay doesn’t change anything about it
@@darksentinel082 yes I thought perhaps that is what you meant. I just happened to accidentally turn autoplay on the other day and it confused the out of me.
On the other side, I do not have those things you are popping up, so I guess there must be a way to remove it. I do see them on my Samsung TV, but not on my iPhone.
Maybe you are on Android, and the version you have is different? I would try to look through the settings, perhaps there is something there? I will have a look myself to see if I find any settings on my phone.
Edit: I did actually find a setting to turn it off. The weird thing is that is is turned ON on my phone. Is it a switch to “enable in-video info card”, found under settings and general.
Yet another fantastic example of how to make really complicated and somewhat... Esoteric math much simpler by just graphing it.
Agreed! All good mathematics has a compelling visual interpretation.
1:22 I can hear the plane screaming as its intestines are being turned inside out
Reciprocal functions are vicious that way. :-)🙀
Really incredible!
Thank you! So glad you enjoyed it. 🙂
So, now I can understand what could be the equation of the whole universe beyond The Earth, ultimately is how this universe is all formed, is how all the trajectories, planets, comets, asteroids, stars, galaxies, etc are placed on in this multi-plane universe!
I feel like a medieval peasant who's looking straight to hell.
Oh but this is pure heaven! How could hell be so captivatingly beautiful??
Finally found some decent wallpaper for my cell phone
😅
Now i know why it's called complex number
There is perfection in the imperfection
Interesting way to think about it!
I added 4th dimensional tesseract cubes to each square represented in this plane in my mind while watching this. Wowsers!! I wish I could share what it looked like. 😮
That sounds amazing! The best part about mathematical animations is the imagination it sparks in the viewers that often transcends and surpasses the actual visual.
Imagine being a nurse and seeing a heart monitor turning into this
Oof! Yep, time to call the doctor!
Mesmerizing
Thank you! I'm glad you found it interesting.
I now know why we call them imaginary numbers now because I see these shapes when I'm tripping on acid
That's the best kind of math!
Pls show the plotted graph of any of these functions using complex numbers
That's exactly what these animations are showing! The final frame of each animation is in some sense the "graph" of the function. See the description above the comments for the details of how this works.
@@beautifulmath5361Yes, one can assume that these aninations are showing in _some sense_ "graphs" involving complex nos...and they are visually very appealing and intriguing too. But how do we plot complex nos. on a graph (sheet) as we do for whole nos., integers and fractions? I have never seen anyone doing it. That's precisely why I requested for a sample graph of any one function involving complex numbers. Otherwise, one cannot know, ordinarily, just from these aninations, about the real nature, complexity and beauty of the complex numbers!!
@@amaratvak6998 A complex number a+bi can be plotted on a grid like an ordered pair (a, b). So the number z = 3+4i would be plotted as the point (3, 4). The number i is (0, 1), because i = 0 + 1i. A complex function f(z) takes a given complex number z, and produces another complex number somewhere else on the grid. Thus, the graph of f(z) somehow has to show where every point z on the grid gets mapped to. This animation shows where every line of points in the original grid gets mapped to.
“f(z)= sin z / z “ is my favorite tbh
I'm glad! I like that one too. 1/z is mine.
This is living up to your channel name
Thank you! The visual and intellectual beauty of math is what inspires me to learn it and immerse myself in it.
I feel bad for people trying to recreate these in Desmos😂
This would be very hard to do in Desmos. It's much easier in a proper programming language, though even that it's still quite challenging.
This is what my topology in blender looks like
Ha ha!
Wow! I REALY want to trip to this
Ha ha! The number of psychodelic references on this video is hilarious.
@@beautifulmath5361 yeah jajaja and thank you very much for your videos by the way!!
I can hear the plane screaming for suffering on every tranformation!!
Ha ha! Poor plane. Some of them are harsher than others.
Very difi-cool!!!
Thank you! 🙂
I loved this, thanks man ❤
thank you! So glad you enjoyed it!
Thanks
Is this sort of how time travel would work
That's an interesting question. I think not, though, because time travel is something no one has yet figured out how to do, whereas the mathematics that these animations are illustrating has been known for a long time.
Something truly beautiful is in euler constat i loved it
I agree! Euler's formula is my favourite formula in all of mathematics.
1:39 it's like magnetic field representation
That's an interesting connection I hadn't thought of!
Wait a minute. Guys I think I found something...
Other graphs: Slow, simple, fun to watch
Graphs involving e and/or the reciprocals of z: AHHHHHH *pukes the plane*
😂 Totally agree! Great way to explain it!
Thank you for sharing this
Thanks bro, now from these equations i can tie my shoe laces
Ha ha!
Very interesting and knowledgeable.
Thank you! I'm glad you enjoyed it.
interestingly trippy❤
Ha ha, thank you! Complex analysis is the trippiest of the mathematical branches.
math is beauty
Instantly subscribed 🤩
Wow, thank you!
1:58 when you get your order.
Which kind of order do you mean? 🙂
Try z²+z which is similar to the Mandelbrot reccurrence
Oh that is a cool idea! I want to try that next. I do love variations on the Mandelbrot set.
I heard the word mandelbrot for the first time today right here and it just so happened to be mentioned in brithemathguy's latest video, wth?!
everyone talks about mandelbrot
I play these videos while I'm tripping
Ha ha! Thanks!
So this is what it feels like to be one of the corner folk.
Can you say more about what you mean? 🙂
@@beautifulmath5361 Corner folk is a video from the UA-cam channel Mister Manticore (formerly Alex Kansas). It is an analog horror video about cornerfolk, a species that lives in another dimensions and whatnot, and their world looks somewhat like the lines in this video.
This is me when im a flat plane and someone inputs complex numbers to my formula
Sorry, I have to reboot my system...
Great video!
So glad you enjoyed it!
Estoy seguro que estoy viendo algo que se supone que no debería ver; como un vistazo a las infinitas posibilidades y aplicaciones
Gracias!
The 1/z one is like if you were inside a Riemann Sphere and looking up
Interesting comparison! I'll have to read again what a Riemann sphere is.
Thought I was gonna listen to some John carpenteresque synth wave.... Make the change.
does anybody know if these animations could be expressed in terms of an homotopy or something similar? 🙄
I'm not an expert in topology, but based on how I made the animations, I believe it can. A homotopy between functions F(x) and G(x) from a space S to a space T is a continuous function H: S x [0, 1] --> T such that H(x, 0) = F(x) and H(x, 1) = G(x). That's kind of what these animations are doing. In this case, for a given complex-valued function c(z), for any complex number z in the initial grid, F(z) = z and G(z) = c(z). The animation transitions between F(z) and G(z) by linear interpolation using the formula (1-t)F(z) + tG(z), so that when t = 0, the animated point is at F(z) (z's original location on the grid), and when t = 1, the animated point is at G(z) (the final mapped location c(z)).
Very interesting the fact about the interpolation . Thanks for the answer!
“This edible ain’t shi-“
I imagine the plane screaming in pain
Ha ha!
I am in the right part of UA-cam again
Ha ha, thank you!
No way bruh x^4 made two balls and a thing pointing sideways
An explaination would be nice, of its usefullness and how to defone those special planes
Oh, just seen the functions on the top left, thats nice
There's also a lengthy explanation in the description section, above the comments.
@@beautifulmath5361 thanks
Oh no my head... But I also want to understand this. Math is really beautiful
There's a detailed explanation in the Description above the comments of what's happening.
I love the subtitles
I don't understand what the shifting plane actually represents. Can somebody explain?
I just now added an explanation for this in the Description section. Give that a read, then ping again if you have more questions after that. :-)
Wouldn't it be nice to transform images like that
Wow, that is a fabulous idea! I will have to try that.
I’m thinking about getting the z^4 at 1:18 tattooed 😅
Send a photo when you have it done! I'll make it the profile pic of my channel! 😅
Square root of z is some trippy shit
I know, that one is my favourite!
@@beautifulmath5361 reality ripping 😨
What does f(z) = log(z) + log(cos(z)) - z^0.994 look like? As an iterative map, it draws a fish. Yes, a fish. But I wonder what the gridlines mapped under f(z) once look like.
Interesting thing to try! Where does this function come from, and how was it motivated?
@@beautifulmath5361 Discrete attractors. I was playing around with different maps. This map has no globally attracting fixed points and has an infinite set of points which “draw up” a fish-like shape, where some points are more frequently visited than others. The density of the points mapped by the function can be seen as brighter colors whereas lower density of points may be dimmer. Overall, it creates a wonderful image. I parametrized the function where the 0.994 is. At 0.995, the function acquires globally attractive fixed points/cycles and no longer produces that fish-like shape. At somewhere around 0.75, there are fixed points again, but I forget the actual number there. The lower this value, the less defined this fish-like shape is. So at 0.994, it’s sharply defined. My main point of interest is how your grid warping animations might geometrically explain something about the bifurcation value of 0.994 :)
@@beautifulmath5361 Oh, come to think of it, some of the maps I’ve found would be perfect for your channel to display, if you’d like to use them. I’ve given names to a few of them based on their associated shapes. And I edited my original comment to fix the map, as I remembered it incorrectly.
@@beautifulmath5361 Some other interesting maps are the following:
“Owl Donkey”: f(z)=log(cos(z))+log(z)-z^0.806
“Galaga Ship”: f(z)=log(cos(z))+log(z)-1
“Rings”: f(z)=log(cos(z))+log(z)-0.618^z
(The exponentiation order is swapped there)
“Chevron Balance”: f(z)=g(z)*0.437+0.307*(g(z)+log(cos(z))) where g(z)=log(z)-z^0.994
There are more, but they compose these functions together and get very complicated. My favorite is the “Primordial Fish”!
@@JordanMetroidManiac All fascinating! The high sensitivity of the outcome to the parameter 0.994 reminds me of chaotic systems.
this just looks like non ecludian space
Interesting. I'd never thought about that connection. But with all those hyperbolas forming (like with sin z), it wouldn't surprise me if there was a mathematical connection between them.
Quite funny how people pay so much money for acid while they can just study complex analysis for free
I know, right! 🙂
I swear mathmaticians are not studying reality, but rather how far can they can go professionally without anyone knowing they take lsd and write equations about their experience
Ha ha, interesting interpretation. I've never taken lsd or anything else, but I do enjoy the psychodelic imagery that comes from elegant mathematics.
Wonderful! Thanks. Is the processing programme language easy for the average person or do you need to be mathematical or know something about programming?
Yes, a background in programming and complex analysis is needed to create animations like this.🙂
amazing
I'm delighted you enjoyed it
You do all realize the simplicity that you are seeing as called complex is actually anything but. Its a simple grid 1x1in given defined parameters. Then a cpu prgm with a separate set of defined parameters distortsthe 1st parameters and they commingle. From there you can press play and pause at any point and let the software calculate the new parameters from the two sets. So you see it is actually very simple integration. I cannot write code but I can cypher my own view and thoughts.
I think you misunderstood the word “complex”. It refers to a type of number, not how complicated some idea is
@@harambesson1098 they dont get any more complex than octonion numbers and they are fundamentally complex and yet simple and elegant compounds. I did not misunderstand the word complex. This word itself is contradictory all on its own that is why you were able to make your statement against my statement to begin with. Good joust do you yield or do you say more?
@@jarrodvrbetic6503 it’s complex in the sense of “apartment complex”, it’s composed of two parts. Has nothing to do with complexity in a colloquial sense once again
@@harambesson1098 so why not accept the word compound and leave your mental complex out of it? That is simply elegant all by itself. Accepting one word and its definition rather than needing another word to expand on a definition that introduces complex controversy? Again I did not misunderstand. If you keep insisting I do you must prove this to be true without objectionable evidence. If you understood this you would know you cannot win and I cannot lose. So if you say ok I can understand that and then attempt to understand that you might see you are only taking a side in a debate for the sake of arguing. I do not argue to win. I simply state facts and what is a fact?
Nice & thanks! :)
You are very welcome!
Прекрасно
Спасибо!
Look like antenna field topography lines.
That's an interesting connection!
My function isn't braining
You could try with the Newton fractal formula or the Burning Ship formula
Interesting idea. I'll have to look up what those are.
I know this video has nothing to do with the Backrooms series but the fact that the kind of music used here is extremely similar and the video is also about warping space itself, means it keeps coming to mind haha
I'll have to check out that series. (The music is just a free track provided by UA-cam that I picked from their list.)
2:59 pov: flandre scarlet 2x
So ... square root of -1 is a rotation in Geometry? And about square root of -2 (the "half" of a square, in other words, a triangle ...)
Multiplying a complex number z = a + bi by the number i (square root of -1) gives iz = ai + bi^2 = -b + ai. If you plot both the original a + bi and -b + ai on the complex plane and connect both points to the origin (0, 0), you'll see the lines form a right angle! Thus multiplying a complex number by i results in a 90 degree rotation.
Now do the Riemann zeta function.
I totally should! 3Blue1Brown has a beautiful and visual video on that.
@@beautifulmath5361 yeah but his is too fast to see. I want to see a super detailed exploration of it, or maybe an online toy that lets me play with the grid parameters.
You’re telling me, a complex number twisted this plane?
At some level, yes. For each animation, a complex-valued function changes the initial grid lines into twisted ones. The animation is showing the transition from one to the other. I give a detailed description of how this works in the Description section above the comments. 🙂
i’m not well versed in complex analysis so this might be a stupid question, but how are the transitions chosen, are they just aesthetic or is there some variable thats changing?
Each animation is generated by a different complex-valued function, which are shown in green text at the upper left. The first animation for example is generated by f(z) = z^2, the next by f(z) = z^3, then later by f(z) = 1/z, etc.
@@beautifulmath5361 sorry i meant how are the transitions from f(z)=z to f(z)= chosen
@@noahgilbertson7530 For each point z in the grid that is being mapped, the program computes f(z) (think of this as the final destination of point z), then linearly interpolates between z and f(z) as an animation. Specifically, let g(z, t) be the interpolated point at time t. g is computed as g(z, t) = t(f(z)) + (1-t)z, so that g(z, 0) = z and g(z, 1) = f(z), and g(z, 0.5) is halfway between z and f(z). Does this help? 🙂
@@beautifulmath5361 yes thank you!!! This looks so cool I love it :)
wow so points are actually travelling in a straight line
1:38 As a physics enthusiast, it looks to me as if they're magnetic field lines.
Yes they do remind me of that too!
What programming language did you use to create the animations?
I used a language called Processing. It's ideal for creating graphics and animations.
solving psychedelics, on step at a time.
can someone briefly explain why complex numbers twist planes?
There's a detailed explanation in the description section above the comments. Give that a read, and if you have any questions about it, I'm happy to answer.
Can you please tell in which software you tried this?
I coded this using the Processing programming language.
@@beautifulmath5361 Hi, have you tried changing the Z to (omega)? a=-1/2, b=+-root 3/2