Secret Kinks of Elementary Functions

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What happens to graphs between degrees of polynomials? How can we draw complex inputs and outputs in 2 dimensions? And what will we see if we try?
00:00 Intro
01:14 The Messy Powers
04:41 About Complex Numbers
07:15 Importing a Function into the Complex Plane
12:19 Overshooting with Euler
17:27 Roots
23:30 Flower Pressing
25:47 Down to and Around Zero
27:06 The Big Bang
29:34 Enjoy!
30:53 Bonus Functions
Correction: 03:26 The first x in the expansion is raised to 1.4. It should be raised to 1.6.
DESMOS GRAPHS:
===============
5th degree polynomial
www.desmos.com/calculator/e9t...
x^p
www.desmos.com/calculator/eob...
x^x
www.desmos.com/calculator/um2...
base Gaussian
www.desmos.com/calculator/sl4...
More on Complex Numbers:
• Counting in Imaginary ...
Music by:
@timkuligfreemusic (intro)
@Lisayamusic (the rest of the video)
Notes:
At 2:46, when we start converting fractional powers, the fractions must be reduced to lowest terms before we use them to assign the degree of root and power under it.

Опубликовано:

9 июн 2024

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Комментарии : 496

@Procyon50 6 месяцев назад

When you revealed the "3d" structure of the cube root function by rotating (24:26), wow. I actually gasped.

@tiagobeaulieu1745 6 месяцев назад

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!

@imaginaryangle 6 месяцев назад

Thank you! I was also surprised!

@sigmascrub 6 месяцев назад

I'll be honest, I read "Secret Kinks of Elementary" and I was _very_ concerned about how that was gonna end 😅

@dsdy1205 3 месяца назад

💀

@Imperial_Squid 6 месяцев назад

It's a shame this video wasn't released as part of 3blue1brown's SoME series, you'd have blown everyone out of the water!

@imaginaryangle 6 месяцев назад

Thank you for saying that! I guess I needed more practice to iron out the **ahem** kinks 😁

@sleepycritical6950 6 месяцев назад

@@imaginaryangleclap…clap…clap

@williestroker3404 6 месяцев назад

@@hyperduality2838 yeah the number two shows up a lot...almost like it's the commonest nontrivial number.

@williestroker3404 5 месяцев назад

@@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?

@ewthmatth 4 месяца назад

@@hyperduality2838 sir, this is a Wendy's

@AvanaVana 6 месяцев назад

This channel is one of my secret kinks. Thank you!

@lexacutable 6 месяцев назад

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.

@imaginaryangle 6 месяцев назад

Thank you so much!!

@ER-uq9jw 6 месяцев назад

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.

@imaginaryangle 6 месяцев назад

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.

@Verrisin 6 месяцев назад

whow, redefining plotting as `z = x + f(x)i` is such a brilliant idea!

@ironicanimewatcher 6 месяцев назад

didn’t know elementary functions were freaky like that 😳

@BingusDingusLingus 6 месяцев назад

This is what I was here for

@lumipakkanen3510 6 месяцев назад

I just love how the visualized elements jiggle when you mention them. "He's talking about me, yay!" ❤

@imaginaryangle 6 месяцев назад

Gotta show them affection 😉

@dolthhaven8564 6 месяцев назад

this is literally the best day of my life

@WhattheHectogon 6 месяцев назад

That was just wonderful....very enlightening, you should be quite proud of this work!

@imaginaryangle 6 месяцев назад

Thank you so much! A lot of love went into it!

@WhattheHectogon 6 месяцев назад

@@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

@imaginaryangle 6 месяцев назад

@@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?

@dadutchboy2 6 месяцев назад

yeah, this is awesome

@nicholassullivan6105 6 месяцев назад

@@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!

@111wxnderland 6 месяцев назад

wow I didn’t know elementary functions could be kinky

@ILSCDF 6 месяцев назад

Wow, this channel is so underrated

@imaginaryangle 6 месяцев назад

Thank you! 🤗

@elilevio6948 6 месяцев назад

I completely agree Amazing

@wiltondewilligen3595 6 месяцев назад

Criminally so

@ILSCDF 6 месяцев назад

@@enantiodromia maybe, but my comment speaks 100% truth

@ILSCDF 6 месяцев назад

@@imaginaryanglebtw, congrats, this is now your most viewed video

@Sanchuniathon384 6 месяцев назад

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 6 месяцев назад

Thank you! That might come up at some point if I find a good story to tie it together.

@j.b.4090 6 месяцев назад

@@hyperduality2838but then why do we exist in three dimensions and not a multiple of two?

@samuelyigzaw 5 месяцев назад

@@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.

@SOBIESKI_freedom 6 месяцев назад

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. 👌👍👏

@imaginaryangle 6 месяцев назад

Thank you! As for 3D, my animation skills are not keeping up with everything I'd want to show. But I'm learning!

@alexanderparshin5979 6 месяцев назад

Surprisingly low amount of subs for this channel, the vid is really insighful and clearly a lot of effort is put into it! Thanks!

@mcmaho17 6 месяцев назад

This is excellent! Thank you for making it.

@imaginaryangle 6 месяцев назад

My pleasure! 😊

@EdTLive 6 месяцев назад

I have no idea what any of this means, but I like the smooth, soothing and synergistic voice of the narrator.

@AThagoras 6 месяцев назад

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!

@AThagoras 6 месяцев назад

@@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

@srandom3867 6 месяцев назад

Comment for the algorithm, this is pretty great and I love how you made the visuals work even under constraints

@wargreymon2024 6 месяцев назад

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.

@lexacutable 6 месяцев назад

agreed, it would be a clear prize winner I think

@imaginaryangle 6 месяцев назад

Thank you so much! It makes me happy you think that!

@salsuginusrex5196 5 месяцев назад

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.

@Alexus00712 6 месяцев назад

Yup, I'm now a sub after having bared witness to all these new secret kinks of elementary functions!

@stevenlaczko8688 6 месяцев назад

YOU NEED COMPLEX INFINITY MORE VIEWS WOW That was such a beautiful video, thank you!

@imaginaryangle 6 месяцев назад

Thank you too!

@douglasstrother6584 17 дней назад

"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.

@user-yb9ol8sz7o 5 месяцев назад

AMAZING, BRILLIANT, INCREDIBLE, that's just FANTASTIC. Best mathematics video I've seen for complex numbers. Amazing beautiful video THANK YOU.

@shantilkhadatkar1195 6 месяцев назад

This is good stuff. Should put this in submission for Jjjackfilms.

6 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.

@fightocondria 6 месяцев назад

Im seriously looking forward to more deep dives from you

@tirthankarmishra1420 6 месяцев назад

You deserve millions of subscribers. Keep up the good work!

@imaginaryangle 6 месяцев назад

Wow, thank you so much!

@sander_bouwhuis 4 месяца назад

Wow... just wow. This video is incredible! This exemplifies what I love about maths. There are so many deeper layers of understanding.

@thekutay25 6 месяцев назад

This is the best video I have watched in a long while, teaching me something elementary yet revealing about math. Looking forward to other videos!

@purple_sky 3 месяца назад

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!!

@flyingcroissant8555 6 месяцев назад

Reaching the end and watching everything come together was really cool to see

@Alexus00712 6 месяцев назад

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..

@jhatzi99 6 месяцев назад

Truly amazing video, both in terms of explanation and aesthetics! Providing the desmos links is a nice treat as well.

@gebrehiwotewnetu358 5 месяцев назад

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.

@pineapplequeen13 6 месяцев назад

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!!!

@thp4983 6 месяцев назад

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.

@imaginaryangle 6 месяцев назад

It was exactly the feeling I had in school! An itch that just wouldn't go away.

@magicorigamipiano21 6 месяцев назад

This was absolutely brilliant the build up to the “flower pressing” was jaw dropping. Immediately liked and subscribed! Keep up the great work!!!

@imaginaryangle 6 месяцев назад

Thank you! Will do!

@broiled_lemming4533 6 месяцев назад

Wonderful and intuitive visualization! Thank you!

@enigmaos 6 месяцев назад

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

@SandipChitale 6 месяцев назад

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.

@imaginaryangle 6 месяцев назад

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" 😉

@SandipChitale 6 месяцев назад

@@imaginaryangle I guess "imaginary" frees one to be more creative :) So in that sense I agree.

@b.rogers8452 6 месяцев назад

As a non-mathy person, this was fascinating and your way of explaining worked very well.

@GameplayAndRelaxation 5 месяцев назад

Absolutely sublime work.

@ghostchris519 6 месяцев назад

Actually one of my favorite videos on RU-vid

@tristinbell 6 месяцев назад

It's so amazing to see new high quality math channels popping up! Keep up the great work!

@imaginaryangle 6 месяцев назад

Thanks! Will do!

@lexscarlet 6 месяцев назад

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

@JojiThomas7431 6 месяцев назад

Beautiful video. Nice presentation and ideas.

@BoutiqueLaTrice 6 месяцев назад

This was AMAZING! Thank you for sharing! I’ll be watching this on the big screen next time!!

@0FAS1 6 месяцев назад

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!

@craftycurate 6 месяцев назад

Beautifully presented and highly engaging, and was I able to follow it.

@ramziabbyad8816 6 месяцев назад

Polynomials and rational functions is all I ever needed

@LukePalmer 6 месяцев назад

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!

@SomebodyLikeXeo 6 месяцев назад

Complex plane is just so wonderful. Thank you for making this!

@mattiasaxner9841 5 месяцев назад

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.

@Qreator06 6 месяцев назад

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

@tylerboulware6510 6 месяцев назад

Very cool, and amazing visuals! I learned some things. I think I'll have to watch at least one more time to understand it better.

@KineHjeldnes 6 месяцев назад

Woooow this was amazing! I learned a lot, and suspect I will rewatch many times

@mikemac5070 6 месяцев назад

This is the best video on complex numbers since welsh labs blew my mind w his series on them. This is just what ive been waiting for. Godspeed!

@imaginaryangle 6 месяцев назад

Thank you so much! Those videos were amazing!

@OfficialFoliLucker 6 месяцев назад

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)

@imaginaryangle 6 месяцев назад

Thank you!

@MrSilverSerf 6 месяцев назад

Wow! That was amazing journey! Thank you!

@hongkonger885 6 месяцев назад

nah, this is *the **_true_* definition of _underrated_

@alex_creeper2752 6 месяцев назад

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!

@HypocriticalElitist 5 месяцев назад

If one of these days I ever study complex numbers, I'll rewatch this video and let it blow my mind for more than just the pretty visualizations.

@imaginaryangle 5 месяцев назад

Maybe check out my video about them: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-nlqOQ0vJF0Q.html

@MarkTimeMiles 6 месяцев назад

Beautiful, thank you. 🙏

@Wagon_Lord 6 месяцев назад

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!

@disengronkulifactice 6 месяцев назад

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.

@Wagon_Lord 6 месяцев назад

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})

@twelvefootboy 6 месяцев назад

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.

@DavidAspden 4 месяца назад

Beautiful. Well Done.

@sertacatac0 6 месяцев назад

This is the channel that I was looking for, great content!

@imaginaryangle 6 месяцев назад

Welcome aboard!

@ClaytonMurzyn 5 месяцев назад

Thank you. This was amazing insight 👏

@brandonchelstrom2260 6 месяцев назад

Absolutely Beautiful

@zxdasfsa3221 6 месяцев назад

honestly one of the coolest videos I’ve ever sene

@DeJay7 Месяц назад

Absolutely fascinating!

@imaginaryangle Месяц назад

Thank you!

@carloscampello8406 6 месяцев назад

Very good video. Much interesting the graphics.

@Seiffouri 5 месяцев назад

Such an amazing journey. Thank you so much for your hard work.

@imaginaryangle 5 месяцев назад

Much appreciated, I'm happy you enjoyed it!

@EPMTUNES 6 месяцев назад

Very awesome video, I love the color scheme and great explanations😊

@juanpablo2097 6 месяцев назад

Extremely value content, this remains me that everything in nature are dicted by mathematical laws, simply amazing

@3g0st 6 месяцев назад

17:03 so pretty!I left a longer comment but I got shy 😆 Basically, you are a great teacher, and you wield the visual tools effectively. The alt girl on top of the map example got me lol.

@imaginaryangle 6 месяцев назад

Thank you!

@swapnilshrivastava116 6 месяцев назад

I wish I could add more than one like and view. Amazing video.

@imaginaryangle 6 месяцев назад

Thank you!

@KStarGamer_ 6 месяцев назад

This is great! For what it’s worth, monodromy is a great way to formalise some of the things you mentioned.

@xemnes6494 5 месяцев назад

great video man. thank you for your work

@imaginaryangle 5 месяцев назад

Thank you!

@sleepygrumpy 6 месяцев назад

Wow this is f*cking amazing -- easily the top math video of the last several months - -instant sub

@imaginaryangle 6 месяцев назад

Thank you and welcome!

@wyvy7013 5 месяцев назад

coolest math video ive seen in months!!

@imaginaryangle 5 месяцев назад

Thank you!

@QP9237 2 месяца назад

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.

@AlanKey86 6 месяцев назад

Great visualisations!

@AlanKey86 6 месяцев назад

Hi @imaginaryangle - if you're ever looking for music for your videos, let me know!

@imaginaryangle 6 месяцев назад

Thank you! I subscribed to your channel, I will keep it in mind 💙🎼

@Coolbluepinguen 6 месяцев назад

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

@HyperFocusMarshmallow 6 месяцев назад

Great video!

@Akash-._ 6 месяцев назад

Amazing video! It’s really amazing how you visualise the mathematics and how you explain it! :o

@imaginaryangle 6 месяцев назад

Thank you!

@hamzahafez7346 6 месяцев назад

That was an amazingly well done video

@BMXaster 6 месяцев назад

Oh my god, this is absolutely beautiful

@josephyoung6749 6 месяцев назад

beautiful animations

@assassinosoldato92 6 месяцев назад

Majestic job

@charlybrown9024 6 месяцев назад

So nice. So neat. Subscribed.

@imaginaryangle 6 месяцев назад

Thanks and welcome!

@jelenahegser445 6 месяцев назад

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) ).

@imaginaryangle 6 месяцев назад

That's no accident 😉

@enviroptic3342 6 месяцев назад

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.