How to Fold a Julia Fractal
acko.net
acko.net
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)
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.
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
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).
Author uses MathBox which he appears to have developed: http://acko.net/blog/making-mathbox/
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.
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.
"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 ?
p irrational:
z^p = a1 z +a2 z^2+... =
= a1 k1+a2 k2 +...
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.
z = x + i*y = e^(a + i*b) = e^a*(e^(i*b)) = e^a(cos(b)+i*sin(b))
b
b + 2*pi*t
If p is a rational number represented by m/n, then
360 / (m/n) * m = 360n
Now p is irrational, what happens? According to what you said, there are so many different graphs that can not be merged into just one.
Now p is irrational, what happens? According to what you said, there are infinite different graphs that can not be merged into just SOME.
> 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.
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.
Check http://get.webgl.org/ to see if WebGL works for your system.
Also, very nice graphics.