I'm not sure if it was the visual aids (which were outstanding), or simply how the subject was approached and written about, but that article finally gave me an intuitive understanding of "complex numbers" that i've never quite gotten before. Sure, they were nice for some types of coordinate/vector math, but why they were nice always seemed to be something I missed. The warping of the coordinate system that you get when squaring is probably the key. It totally explained Julia sets for me.
Now if only I could get the same understanding for the incredible complexity of the Mandelbrot. After seeing[1] what happens down around 1/ 2^1116 (~3E353) - and remember that the set is supposed to be simply-connected - I'm not sure how to begin any kind of real understanding. There is simply too much detail there...
Ouch, this page absolutely destroys my browser (up-to-date Firefox on Windows 7.) 20 seconds of unresponsiveness on load, followed by very choppy scrolling and animation.
The worst part is that it's not the article, but the template that's killing it.
I love the visualizations, but I think the author needs to reconsider including the parallax trickery on inner pages. There is enough to look at, and, when it's not busy crashing your browser, this kind of eye candy is a distraction.
This crashed my browser for about 10 seconds too,
up-to-date Chrome on Windows 7.
I agree about the parallax stuff, its great, but when you're sitting there waiting for your browser to come back alive it sort of puts you off visiting the site again
For me this doesn't render correctly -- the visual transitions just flicker by. Tested with Iceweasel (Firefox) 29.0 and 30.0 as well as Chromium 35.0.1916.15 on Debian Wheezy.
Renders almost correctly on Android Chrome on my Note3 (but the screen is a little cramped, so while the animation/transitions work, parts of the text is garbled/layered on top of other text).
Weird, the old MacBook Pro with Snow Leopard I'm using with old Safari 5.1.x didn't even rev up the fans or anything. Which OS version and browser are you using?
FWIW, no problems at all on FF 30, Macbook Pro, OS X 10.9.3. Responsiveness and framerates were good, and no unusually high CPU logged for the entire time I was reading the article.
I wish my math classes in high school and college had stuff like this. It's amazing how much easier it is to understand something when you can see and interact with it.
I love the visualization on acko.net but for whatever reason, unlike apparently many of you, I still don't understand the topics he's trying to explain.
As just one example on step 18 of the first major visualization it says "And that's actually a remarkable thing, because it means our invented rule has created a square root of −1. It's the number 1∠90∘" WAT?!
I'm sure someone skilled at math understands that but as a lamer at math I have no idea how that explains that the rule invented the square root of -1. Nothing before that has mentioned square roots or how this vector + rotation has any relation to them.
I don't want to disparage the author's hard work, as it was quite beautiful to look at, but as someone who already understands what he's talking about, I felt this was a terrible way to present it. (Like I say, it looked great, but I don't think it would aid a new student learning about complex numbers at all.)
The notation he invents is confusing, too. That "1∠90°" just means a vector with length 1 and pointing 90° from the X-axis (which the author later changes to the Y-axis for no apparent reason).
In my opinion, polar coordinates (which is what the notation represents) are a fascinating tool that can be very helpful. I use them quite a bit. Conflating that with complex numbers is confusing if you don't already understand both polar coordinates and complex numbers, in my opinion.
Hmm I don't think he invented the polar notion. It is a standard representation of complex number to me and I have certainly used it back in school (I studied EE). This notion actually simplifies multiplication significantly.
I think that particular slide was poorly worded. He's actually introduction a consequence of the rules, but not saying it's an obvious consequence. Slide 19 shows how (1∠90∘)2 = -1.
I had the same reaction. There should be a few words showing that we now have a point which multiplied by itself (rotated by 90° around 0) goes to -1. So this number is the square root of -1.
Maybe someone can help me understand this. On slide 39 of the first slideshow, it says :
"For any irrational power p, there are an infinite number of solutions to z^p=c, all lying on a circle."
This means that most of the solutions have an angle larger than a full circle, right ? But if complex numbers can be represented as the sum of the real and complex parts, how can their angle be superior to 360 degrees ?
but there is 1 extra z'!=z which has the same value k1 for z'^2 as a solution z of z^2, 2 extra z' which have the same value k2 for z'^3 and so on. As long as mdc(a,b) is 1, for distinct a and b, z^b and z^b won't share all z' alternatives. You can then choose prime numbers s.t. you get infinitely amount of distinct solutions.
Exponential expanation: (more intuitive)
So we're taking roots of complex numbers: all z s.t. z^p=exp(a+jb). Without loss of generality, take
z^p=exp(jb) instead, since exp(a) is just a positive constant. A trivial solution for that is simply exp(jb/p). But remember that for any x, exp(jx) = exp(jx+2 k pi) for integer k. For example, exp(jpi/6) can be written either as that or exp(j 13 pi/6) -- that's because exp(jx) is a sum of periodic components cos(x) and j*sin(x). So we can add new solutions of the form exp(j(b+2 k pi)/p), for any k, which may or may not overlap with previous solutions. But if 1/p=m/n (rational), you can set k=n to get back to exp(jb/p+2 pi)=exp(jb/p); otherwise, you get infinitely many solutions, since there is no k/p=integer. However, you can arbitrarily close by an approximation p~m'/n'; in fact, you get arbitrarily close to any other rational exponent exp(jq), so that the solutions are scattered everywhere.
This is the 'collapsing' that he talks about. If one of the complex solutions had a magnitude of 1 and an angle of 395 degrees, e.g., it would be equal to a number with a magnitude of 1 and an angle of 35 degrees.
Thanks (and thanks to the other replies !). So a complex number has a single representation when using real and imaginary parts, but an infinity of representations if you use angle and magnitude (just add or subtract 2π radians). I guess that should have been clear from the article, I just needed to sleep on it.
> To understand the effect of c we need to make a Mandelbrot set.
In fact, the Mandelbrot set can be defined to be the set of c that make the Julia set connected. This is something new I learned today because I was suspicious about the claims of connectivity (proving fractals are connected is a highly nontrivial task).
Now I kinda want to start looking at MathBox for presentations.
To fold Julia fractal you only need to treat it like something simple and two-dimensional, project it onto a square grid, transform that grid, and then interpolate every cell using simple affine transformations.
EDIT: Removed unneeded emotional junk
EDIT 2: Affine transformations are just simple arithmetic operations between complex and dual numbers but nevertherless matrices are simpler!
The first ones you come across are quaternions, represented by set H. Something I learned a few years ago, in set C there are two solutions to x^2+1=0, but in set H the set of solutions forms a Lie group, or basically an infinite number of solutions.
Wow, I didn't even read the article at all, but the background and title of the site are amazing. Some kind of 3d parallax rendering on canvas. As I was examining it, a badge popped up, I had unlocked "Dat Parallax".
Now if only I could get the same understanding for the incredible complexity of the Mandelbrot. After seeing[1] what happens down around 1/ 2^1116 (~3E353) - and remember that the set is supposed to be simply-connected - I'm not sure how to begin any kind of real understanding. There is simply too much detail there...
[1] http://www.youtube.com/watch?v=PbwaFQ2r2c4 (the entire video is great, but the true insanity starts about halfway through)