Amazing Math Graphs
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- Опубликовано: 11 сен 2024
- Why did I make this video? I don't know, visualized math is satisfying to me. Why not give it a try yourself, especially if you are a pre-calc student like me and know most of the trig and some basics of calculus? It is really mesmerizing.
This is the first part of the series.
Playlist: • 🖋️ Maths
Hi Shibacchi - / @hishibacchi5357
Graphing calculator I used: desmos.com/cal...
enderman.ch
![Chlöe - Shake (Feat. Jeremih) [Official Visualizer]](http://i.ytimg.com/vi/aDs_1ufpfv8/mqdefault.jpg)
For those who are unaware, most of the jagged lines/dots/shapes in the more complex curves are not actually 'real', they appear because the function changes so rapidly (or where the computation involves such large/small numbers) that Desmos' numerical solvers stop working properly.
They are complex?
@@oosmanbeekawoo It's complex in the sense of literal complexity, not the complex plane. (This isn't to say that there aren't complex solutions; that just isn't what I think OP meant)
@@primalaspie welllll ackshually they are all complex since R ⊂ C ...... 🤫🤫🤫🤫🤫🤫🤫
at some point it has a hard time dictating where lines should be drawn, and where they should be separated too.
Aliasing
The problem with desmos is that when equations get too hard to process it starts processing less points. This can be avoided by just zooming in, you get less of the equation but if it is not crazy hard to calculate it will be accurate. For example tan(x^2 + y^2) = 1 is an infinite series of circles with the center (0,0) with their radius approaching the previous ones radius. Zooming in this becomes evident but when zoomed out it just becomes a jumbled mess. If you start increasing the number 1, the rendering becomes so hard that desmos limits the points of the equation calculated so that it looks like there are just a few random dots (points). After a certain number nothing at all is rendered.
I just made my own graphing software in python that's only limited by hardware lol
@@feepants4495 lol, same here but used gm2.
5:28 seems like a really easy way to generate alien alphabets. It's crazy how each column looks like a fully fleshed-out alphabet that you could easily imagine seeing scrolled in some ruins on a distant, deserted, and desolate planet out floating in space.
Absolutely loved the way you think. Nice One.
00000000OOOOOOOOoooooooooooooo
Thought the same, maybe it's the one which amazed me most
damn that's what I thought when I first did that
@@Tzlil-jw1fg exactly, so don't use the proper graph, use the desmos one
for anyone who wants it, the music is "synchobonk" by steventhedreamer, who's the father of a youtuber named 3kliksphilip
Thank you!
Teacher: "The exam is going to be easy!"
The exam:
Yeah
Is this hell?
Guilty.
it's not very hard to throw around random functions in desmos to create some cool stuff like this
But some of them still can solve and drow with derivation
such as "x^3/y = x" ,and " y = e^sin x + 1"
(I don't say it's easy to solve)
4:43 *me waiting for thr sorting algorithm video to start*
real
When I was still in school, I imagined this must be what university calculus must look like.
Was not prepared for the epsilons
We signed up for Extreme Integration 5001 not this epsilon crap!
@@alganpokemon905 Hehehe exactly
I imagined this could be Calculus 4 if it existed 😭
@@mynameusedtobelong What do you mean?
@@mynameusedtobelong epsilon looks like this ε, and is used commonly in real analysis (and plenty of other analysis fields I presume)
POV: you closed your eyes in math class for 5 minutes
1:18 I mean you can, you just have to separate the case where x=0 beforehand. You get y=1/x², and along with x=0 that makes the whole graph.
Damn, didn't see yours... commented the same lmao
Can't you moltiply by 1/x and it becomes y=x^2?
@@fabrizioperini288 multiplying by 1/x is same as divinding by x, and you cant divide by variable if you dont consider the x=0 case (unless x cant be equal to zero for other reasons)
@@neijrr ok now I understand thank you
@@neijrr in equality you can divide by variables. You cannot divide by variables in inequality
I tried the graph at 5:40 out myself and it's really simple; It just looks trippy because the calculator's having a bit of a stroke. It's supposed to be just a bunch of circles of center (0,0) with increasing radii.
yeah it's not loading pixels properly
Some of these equations might have been drawn wrong by desmos. Still, awesome.
Are there any ways to find the true shape?
@@cara-seyun I'm not sure.
@@stutavagrippa8690 possibly if you run it multiple times on different devices, you’d see what remains the same and get an idea
The main problem is the function varying too fast and desmos interpolate wrong
For example tan(x²+y²)=1 (thumbnail) should just be infinite concentric circles centred at origin getting arbitrarily close to each other as the radius increases, but desmos can't interpolate it properly.
i used to do something like this, but with 3d graphs, in school. instead of paying attention in math class or whatever, i'd find cool patterns and shapes. i made snowflakes and very surreal aqueduct-like designs. at some point i had a somewhat intuitive understanding of what caused what. zooming in and out would garner "unique" results within the same function. it's very fun to mess around with!
Some of these look like a slice of a 3D object (similar to a human tomography). I wonder if there's something going on in the complex number axis that we're unaware about. Beautiful work anyway.
Exactly what I was thinking. This is related to elliptic curves. If Desmos used arbitrary precision and supported complex numbers, domain coloring, and maybe a 3rd axis, we would get the full picture
Good point
They actually are (at least most of them), but this has nothing to do with complex axis. If You look for example on the equation tan(x^2+y^2)=1 it is just a set of points where plane z=1 in the three-dimensional Euclidean space crosses the graph of the function f(x,y) = tan(x^2+y^2), which itself is a two-dimensional surface "living" in a three-dimensional space. And this can be applied to the other graphs too. If You have for example the equation y=cos(x^x), You can rewrite it as y-cos(x^x)=0, and then the graph is just the intersection of plane z=0 with the graph of the function f(x,y)=y-cos(x^x). There is actually an entire branch of calculus dealing with the behavior of such curves, called implicit function theory.
these can be continued to the imaginary numbers so yes
Oh man, if it's a mess in real axis, imaginary would be inimaginable
the way you put them in video, the message between, and the music choice. they all, together, make this video feels like good old online flash game. totally love it. what a nostalgia
Was worth the pre-hemorrhoid
5:43 Huh, this one reminds me of the rippling patterns formed when you approximate a sphere in discrete voxel space
That's very closely related, as the look of the graph is very much the result of Desmos discretely approximating circles.
In polar coordinates the function reduces to tan(r^2) = 1, so it should be infinite concentric circles with less space between them as you go away from the origin. (which is what you see when zooming in on the graph and reducing Desmos' workload). In fact, for each y value that the line x = 1 passes through the graph at 5:55, there's a circle centered on the origin with that length of radius. In other words, the circles get _very_ dense _very_ quick.
That's what i thought about when i saw the thumbnail
me too
glad someone else saw the connection. wonder if there's something deeper at play, and also if it's possible to recreate the effect of the circles on the voxel sphere "flowing" the more detailed the sphere becomes, but in this 2D space. somehow.
Did you see tze RUclips Video?
"Dad, how are babies made?“
Dad: 1:03
excuse me what-
Alternative title: brain hurt% glitchless math sub category wr
Is that kliksphillip's music?
The one tan(x2+y2)=1 is pretty curious though cause it can be solved using polar variable change. We got r=sqrt(pi/4+kpi) so the solutions should be circles of different radius
And they are. Any zooming changes the where the “circles” are. It’s just a combination of the estimations of Desmos’s solver in infinite detail and screen limitations on the same.
Amazing
@@bruceleenstra6181 the link works
true, i used my algorithm from scratch to plot this equation and it did just a bit better than desmos
strange that "Abs(left / right)
also here's a showcase to the equation...
pasteboard.co/7W885ydbhwMY.jpg
2:16 yes i am feeling cosy, thanks for asking
Do you mean cozy?
5:38
In this one, the graph is actually many concentric circles but the graph plotter is not able to render it properly, so it looks like this. 😁
what i find more interesting is instead of the circles becoming more defined they actually dissapear
so actually you are wrong
@@hexagon8899 except he is not wrong. The software used is just having a hard time rendering everything properly. solving tan(x^2+y^2) = 1 by hand isn't hard, and what you'll find is that the graph will be concentric circles of radius sqrt(pi*(k+1/4))
@@hexagon8899 The graph is obviously radially symmetric so it's easy to see that the graph's wrong
@@mathieugouttenoire9665 i forgot what concentric meant when i commented that
Some of the later equations are not nearly as complicated as they appear.
For example, tan(x²+y²)=1 can be thought of as y=tan(x) intersected with y=1 in 1D, then rotated around the z axis.
The one before can also be described pretty thoroughly: you can easily find a (sort of) grid of points and show each point is encircled by a shape that is itself bounded near the coordinates of the point, although it's only visible towards the lower left.
x = tan(y²) looks impressive, until you realize it's y=tan(x²) rotated 90°.
Others however are genuinely man-boggling, and I'd love to study them more in-depth!
My favorites are 2:23, 3:36 (which both have intriguing similarities) and the two afterwards, which I just can't wrap my head around!
My favourite is y=sin(1/x) (not in this video) because it has an infinite number of turning points in a finite region of space
if you type:
f(x) = sin(1/x)
f'(x) = 0
you can actually see them
@@lesarXD Yeas bró
f(x) = sin(1/x) and f(0) = 0 is actually a great example of non simple-conneced but topological connected space!
You here?
that graphs pretty troll it just oscillates to 0 xD
I'm really happy to see this music being used, and it's perfect use of it. Kudos, great video
Where can I find this music?
@@becausewhynot8004 The song is synchobonk by SteventheDreamer.
Reminds me of 3kliksphilipp videos
It gets even crazier if you mix in the hyperbolic trig functions, those are always fun to explore
Imagine mixing complex numbers or z axis…
I also know very complex maths, simaltanoes equations and that
@@xoxoheartz wait I thought complex numbers don't exist in the normal cartesian coordinates. Pls explain me
@@newaccount-cz6tb i do not remember but there was a method to graph complex numbers using Cartesian plane but it was really specific.
@@Xarr3 same, I'm 6th and understand all trigonometry
5:00 i tried saying this one out loud and the furniture started floating
3:50 gows crazy it looks like an impractical futuristic gun
I'd never thought I would hear 3kliksphilip music here
same hahah
I was gonna say the same thing 😂
Your video makes me, a student very love Maths, now is more interested in my subject and even your Windows videos. Thank you so much, Andrew (or Enderman)!
Minecraft "Cave sound" will be even more fitting than music 😅
5:52 this one is my favorite. I remember just randomly finding it and being amazed
I had worked on many math art equations , but never seen such big collection of terrifying equations. its awesome
*Paste in desmos:*
\left|\frac{xy+a}{x}\left(0.01+x^{b}y^{c}\left(\sin\left(x^{d}
ight)+\cos\left(y^{d}
ight)
ight)
ight)
ight|
5:38 the graph is actually very simple - it's a bunch of concentric circles - all circles are just around the origin!! and are perfect circles. desmos just draws this very wrong. proof:
tan(x^2 + y^2) = 1
substitute theta = x^2 + y^2
tan(theta) = 1
theta = pi/4 + pi*k for some integer k
x^2 + y^2 = pi/4 + pi*k
right hand side is some constant which is at least sometimes positive, so
x^2 + y^2 = c^2
the equation of a circle. different k values correspond to different radii.
I feel like you could by playing with these equations enough you could stumble upon an entirely new form of math
cos(x^tan(y))/sin(y^tan(x))=0.5
It looks so grainy??
@Progreshbar ok
@Progreshbar which one?
To all who want to know what those graphs mean:
1:34 Gauß' approximation of prime numbers
3:51 some manufactured small parts
4:33 A OCD-Test (the lines are not parrallel)
4:40 Your Routers Bandwith Diagram
5:10 T̵h̷e̴y̷ ̸A̵r̴e̷ ̷C̵o̷m̴m̸u̵n̸i̶c̵a̷t̸i̴n̵g̴ ̴W̵i̵t̵h̸ ̷U̵s̶
5:19 A Map to T̸͔̙̖̩͇̓͂̓̑̕ͅh̷̥̭̲̻̀͗̎̽̚͜͝e̶̞͖̟̫̋̉͛̆m̶̜̆
5:32 First 2 Coloumns are Wingprofiles
5:43 A 2D-Wave on a 3D-Object
How do you write those symbols on top the letters?
5:41, tan(x² + y²) = 1
or, x² + y² = tan^-1(1) = π/4
or, x² + y² = π/4
But in 1st equation tan function repeats itself along x axis.. thus generating such effect when having a complicated multiple variable inside it. Another way to get the logic is that you have to pick any random possible value for x² and y² pair such that the tangent of their sum will always be 1.
@TheMainataur study math
So we know that tan(π/4)=1
its also: tan(π/4)= tan(π/4+πk) with k being a Rational number
so when removing the tan we get:
x^2+y^2=πk+π/4 try it in desmos it will look the same as the first formula
Credit 3kliksphilip for the music in the comments
The factorial function is also fun to use!
Nice use of the exclamation point
I like how the strictness and precision of maths still creates such natural and random looking shapes
This is very cool, thanks for uploading the video! When I was 14 in 1980 my dad brought home a TRS-80 Color Computer running Microsoft BASIC with 16KB of RAM, no internal hard drive, it required a cassette deck for recording programs or playing games. There was a game port and a place for two joysticks but many games were available on cassette. Anyway, I mainly used it for animated graphics and because of the limited memory you had to choose between having more colors and more pages but lower resolution, or higher resolution and fewer pages and colors. To draw a circle you would input the center of the circle on the column and row graph based on the resolution you chose, then you sin/cos to draw the circle and you could make an arc by putting a certain number between 1 - 360, if you wanted to do a full circle you could then add a command to fill it with a color, but if you wanted to do an ellipse you could add a command for the ratio of height to width, then you could add the command to fill that with a color, that was always the last command. These commands were always line numbers like line 10 line 20 line 30 and you spaced them apart by 10 so you could go back and add additional lines if you wanted to tweak it by adding sounds or GoSub commands, although GoSub's were usually already planned. GoSub was a subroutine that you built in like a macro to do something after your main program did most of what you wanted, like if you had it pick lottery numbers the subroutine would put them in sequential order and then display them.
I'm now studying analysis on manifolds , and there it says that it's a fact that {(x,y):f(x,y)=C} is a manifold if the Jacobian at all points in this set is of full rank. This means there is a way to do calculus on some of these graphs. Imagine doing calculus on these graphs
Algebraic geometry?
Welcome to general relativity
yep all C^1-graphs look locally like when you draw it with pen with measure=1, some theorem from calculus about measure of jacobian=0 points=)
is this music from 3kliksphillip's dad?
No one:
The netflix intro: 4:14
holy shit i love hi shibacchi
Nice to know that you're making one as well!
Thanks! As someone who does generative art as a hobby, I really like these.
In x/y = x^3 : of course you can divide by x on both sides, it's y=1/x^2. This is never negative.
It's pretty cool to actually graph these equations in 3D. Meaning if you have f(x, y) = g(x, y), graph z = f(x, y) and z = g(x, y). You see not only the intersection (whose projection on the xy plane is these graphs) but also so much more.
You can take advantage of the imperfections of your visualizer instead of being a victim of it!
On my calculators, I used to plot cos(x), sin(x), -cos(x) and -sin(x) on extremely wide ranges in different colors, and it drew very varied figures; sometimes they would be like woven carpets, sometimes they'd be like mellow waves, it was really fascinating.
(I even managed to figure out what kind of approximation my calculator was using from the fact that lower frequencies waves seemed to appear in the larger picture. It's incredible what you can learn from imperfect information!)
I remember experimenting like this during my first year of university. This is really cool and fun!
The weirdest part is seeing some of these patterns and recognizing the real world physics they resemble
that's super interesting, can you give an example?
^
4:35 I crashed my GeoGebra tab 😂😂 Literally, Chrome showed the "tab is not responding" pop-up for a few seconds before finally graphing the equation
It's just so many almost straight up and down lines close to each other, it can't handle it
It looks like enchanting table language at the end of it at 5:06
6:00
me: *spams "zoom out" button*
my laptop: *explodes*
One of the simplest equations I've found that Desmos doesn't like:
y = cos(y^x)
Does some fun stuff at xcos(y^x) and |xcos(y^x)|, as well.
Just found tan(x^2+y^2) = xycos(x)sin(y). It gives you a cool stair-step effect around distorted circles.
5:44 so that's one way to map the sphere to the Euclidean plane... (referring to Henry Segerman's video of circles on cubic approximations of a sphere.)
Credit to henry
Cant wait to ask these questions to my tutor
12 yrs old:
WTF IS THIS?????
14 yrs old: y axis is up and down and x axis is left and right
Square root of 2 is 1.41blabla
That is all i know
This is honestly amazing I wonder how some of these would act in any 3d environment
Most of the later ones are just desmos being glitchy, they don’t actually look like that. Like tan(x2+y2) should just be a bunch of concentric circles around the origin
Yeah I was thinking the same thing. Also two variables functions are in a 3 dimension space and we only see a plane that cut the graph on the origin so those 2d graphs can be misleading sometimes
@@yuyy8565 I made a typo; the equation is tan(x2+y2)=1, so in theory a 2D graph is fine (We're not plotting a function's output, but rather the set of values in R2 where that function is equal to 1)
And around x^2 = π etc. by periodicity, right?
As a math student, I’m pretty sure I got some of the easier ones or similar functions to check continuity and if they’re differentiable 😂😂
They’re always impossible to draw but since we “only” had to check limes, they were always so so complicated
Its good if we take screen shots afyer going to the settings, remove the axes grids and reverse the colours itll be awesome wall papers
As a student in the 90s I used to have a graphing calculator, and I use these kind of equations for sequence to use with Takens attractor recontruction, because I liked pretty patterns with fine structure. I felt pretty clever. My calculator was stolen from my coat pocket one day when we were playing Laser tag before a physics exam. I felt pretty stupid, never really recovered.
Doesn’t matter that you were careful with the plastic battery panel
0:33 Fun fact, the point where the two of them intersect is ( _e_ , _e_ )
which is also (2.7182,2.7182)
But why?
Just makes you realize that a single equation can store a whole map worth of info. Also that basically anything you draw can be represented by some sort of equation.
If you haven't seen it yet, have a look at this: ruclips.net/video/BFld4EBO2RE/видео.html
5:35 looks like an Alien language ,out of the earth
I graphed tan(x^2+y^2) = 1 on a graphing program I wrote in Python, and it's just a load of concentric circles. I think the many different circles all over are artifacts of Desmos.
Congrats, you made me even more curious on the trigonometric graphs I will abuse them as much as I can now
The one at 5:29 seems like a time travel representation of a mountain that evolves itself in a city full of skyscrapers with floating objects that connect the earth to the sky, millennium after millennium.
When I first saw the graph of y = x! It actually made me want to learn why it looks the way it looks.
@nnnoooo Not y = x, it's y = x!
("!" Means factorial)
It's amazing how simple some of these equations are for what they create in a graph
I know this video is old, but there's definitely something wrong with some graphs. For example, 4:16 is intended to be a graph of a function that doesn't define Y at all, so all the lines should be infinite and vertical. Great video nevertheless!
True, it wouldn't surprise me if most of them were wrong actually. Desmos just isn't good enough for these sorts of graphs. It would be interesting to compare this with how other software draw these graphs
The equation x^3 + y^3 = x^2 + y is equivalent to x^3 - x^2 = y^3 - y. This equation can be solved by finding the factorization of the left side and setting it equal to the factorization of the right side. The left side is (x^3 - x^2) = (x-1)(x^2 + x + 1). The right side is (y^3 - y) = (y-1)(y^2 + y + 1). Thus, the equation is equivalent to (x-1)(x^2 + x + 1) = (y-1)(y^2 + y + 1).
4:50 and 5:39 and 5:51 seem like such cute, peaceful, harmless equations. and yet...
5:39 and 5:51 can be quite easily sketched by hand and so can 4:50 with a little more thought, assuming you are familiar with the behaviour of cos(x^2)
@@user-ef8kc4rv7n i am not
Logs : i make most difficult graphs!
E : really?
Mod : don't listen to them!
Sin cos : lol have you even seen mine?
Tan : So cute
You not only corrupt Windows, you also like to corrupt the cartesian plane as well.
tan(x²+y²)=1 has just circles with center (0,0) & with radius pi/4+n(pi) , n is an integer
I don't know any single thing about calculus, just looking at the nice squiggles on the graphs. :)
Me too.
I think the graph in 5:36 is extremely misleading. This is because we have tan(x^2+y^2)=1
=>x^2+y^2=arctan(1)
=>x^2+y^2=π/4,5π/4,9π/4,..
So it should really be a bunch of concentric circles (centered at the origin) with radii π/4,5π/4,9π/4,...
Not sure why desmos gives circles centered in other places as well.
Are these well-known equations, or did you come up with them randomly, or...? Very cool and alien, thanks!
tan(x^2+y^2)=1
sin(x^2+y^2)=cos(x^2+y^2)
Because of trigonometric Pythagoras:
sin(x^2+y^2)=cos(x^2+y^2)=±1/√2
so:
x^2+y^2=π/4+kπ, k element Z (actually element N because squares are always non-negative in the real numbers)
So the concentric circles totally make sense.
I just don't understand why the centers appear to be at so many points.
2:31 when you program with Lisp
((((((((()))))))))
((((((((()))))))))
5:37 So if tan(x^2 +y^2) = 1 => x^2 + y^2 = atan(1), so we have the equation of a circle, whose centre is at the origin, (x^2 + y^2 = r^2) with r^2 = (4n + 1)π/4 with n = ± 1, ± 2, ...∞. Enjoy maths, numerical solutions are where the fun begins (they were for me). There's nothing like solving an ODE using gold old RK4th order with predictor correctors or a nice thermal PDE using finite differences in a spreadsheet.
2:50 if you do sin(x^y)>0 the result is weirder
(1.5(t-1.4sin(at)),1.5cos(2et)) works really well, set a to be between 0 and 20, press play and have fun.
I’m just waiting for someone to come out with an equation that maps out the whole earth
That is absolutely beautiful!
I also love to play with function and their graphs.
To make graphs even more insane you could add imaginary number to visualisation. I mean, if you f(x), try also real part Re(f(x)) and imaginary part Im(f(x)).
Good and interesting video! It was hard for me to comprehend and guess the behavior of the graph for most of them but I didn’t know graphs can be this cool!
5:22
me wondering HOW THE FUK did it got graphed, does the computer slaps all numbers in x and get an y and graphs the results? how did he got that point mess in the top of the graph lol
im not sure how they do it, but one solution could be to just base them off of the pixels on your screen.
It basically takes many samples over a region that you want graphed. Say you want the region to be [-1,1]x[-1,1]. It could then sample points (maybe one point sample per pixel, although more samples would be preferred) from this region to see whether they satisfy the given equation (with some small error) and if that error is small enough, then it knows that that pixel should be colored in. This works well for "standard" functions and equations, not so well in this case, so that's why a lot of these are not correct graphs.
5:17 this one is amazing, looks nearly like pure randomness at the top
Math:
Me: Task failed Successfully
I did not expect that music on this video
2:22 Desmos can make wood? What the heck
Overlay the following functions:
sin x² = y²
sin y² = x²
cos x² = y²
cos y² = x²
tan x² = y²
tan y² = x²
I know it's not only one function, but it makes these cool repeating radial patterns like a mandala.
4:40 Looks like it could use a sorting algorithm
Can't wait for the sequel, Eldritch math graphs
3:07 is absolutely an art
@5:30 I believe it says "Amenhoptut owes me 180 bushels of wheat and 60 cattle, to be repaid on the spring equinox next year"
Эти функции специально придумали, чтобы передавать сообщения на древне-каком-то-там
At 4:56 it reminds me of GPR (Ground penetrating radar) scan. Very useful in surveying.
that and 5:53 actually Did remind me of Anime for some reason... maybe large mobile machines and the like..
thought that wasn't weird! 😄😄
cos(x) > sin(y) is a diagonal checkerboard
cos(x) cos(a) > sin(y) sin(a)
cos(x) - cos(a) > sin(y) - sin(a)
tan(x) cos(a) > tan(y) sin(a)
tan(x) - cos(a) > tan(y) sin(a)