
How to Fold a Julia Fractal - bpierre
http://acko.net/blog/how-to-fold-a-julia-fractal/
======
pdkl95
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...

[1]
[http://www.youtube.com/watch?v=PbwaFQ2r2c4](http://www.youtube.com/watch?v=PbwaFQ2r2c4)
(the entire video is great, but the true insanity starts about halfway
through)

~~~
alphonse23
I felt the same way. A lot of appreciation goes to the writer of that piece!

------
Gracana
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.

~~~
jonahx
my new macbook pro got too hot for my lap within about 30 seconds....

~~~
Cowicide
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?

------
lwhi
The interactive illustrations are brilliant. They really aid comprehension.

~~~
bsilvereagle
Phasors finally clicked after watching the first one.

Author uses MathBox which he appears to have developed:
[http://acko.net/blog/making-mathbox/](http://acko.net/blog/making-mathbox/)

------
greggman
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.

~~~
vitd
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.

~~~
sigterm
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.

------
darsham
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 ?

~~~
zygy
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.

~~~
darsham
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.

------
j2kun
Excellent post!

> 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.

------
ii
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!

------
roadnottaken
This guy is so talented it's actually depressing!

------
NAFV_P
Dare I mention hypercomplex numbers?

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.

------
fjcaetano
Stunning visualisation. Really eased the comprehension of what's being
described.

------
woah
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".

~~~
puzzlingcaptcha
previous discussion:
[https://news.ycombinator.com/item?id=6268610](https://news.ycombinator.com/item?id=6268610)

------
8iterations
Can you place links at each of your figures so that we can directly link to
those of interest? This is great stuff!

------
hcarvalhoalves
That's blogging on a whole new level.

------
coreymgilmore
Great read. Numbers are great, and there is so much hidden between/under them.

Also, very nice graphics.

------
regularfry
The visualisations are broken for me, Chromium 34.0.1847.137 Debian 7.5
(268882) and Iceweasel 24.5.0.

~~~
bsilvereagle
The visualizations are WebGL dependent.

Check [http://get.webgl.org/](http://get.webgl.org/) to see if WebGL works for
your system.

~~~
regularfry
Yep, WebGL works on my system. Both browsers show the spinning cube fine.

------
nicholassmith
I always love reading these, even when I don't understand some of the maths
involved.

------
Jemaclus
Some of the math was over my head, but the visuals on this are amazing.

------
KhalilK
Anyone has an idea how to create such smooth animations?

~~~
pornel
I presume it's been made with the author's own library:
[https://github.com/unconed/MathBox.js/](https://github.com/unconed/MathBox.js/)

------
dvdplm
Amazing work. Love this.

------
mitosis
Heh: "Achievement unlocked: Refresh Prince of Bel Air".

