
How to Fold a Julia Fractal - rosstex
http://acko.net/blog/how-to-fold-a-julia-fractal/
======
jfarmer
More than being an intro to Julia fractals, I think this post is a great
introduction to complex numbers and functions of the complex plane[1].

The way this is presented is very similar to how most math-folk I know picture
these concepts in their head. This is probably one of the toughest things for
beginners, who don't understand that (most) math-folk think in pictures like
this and not in symbols.

For example, starting at around Slide 29 in the first visualization, the
author actually paints a picture of a branch cut[2] without using that term.

Likewise, starting at around Slide 12 of the last visualization, the author
hints at the special relationship between complex numbers and differentiation
in the complex plane. The jargon-y stuff involved here are holomorphic
functions[3], the Cauchy-Riemann equations[4], and the very surprising-but-
central theorem of complex analysis: Cauchy's integral theorem[5].

    
    
      [1]: Functions from ℂ to ℂ are "hard" to reason about because there
           are 4 dimensions involved, at least if you're picturing ℂ as a
           2-dimensional plane.
      [2]: https://en.wikipedia.org/wiki/Branch_point#Branch_cuts
      [3]: https://en.wikipedia.org/wiki/Holomorphic_function
      [4]: https://en.wikipedia.org/wiki/Cauchy-Riemann_equations
      [5]: https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem

------
jimmyspencerjr
What a wonderful post this is! I first encountered Julia fractals while taking
a masters class in chaos theory, technically a physics class. It was a masters
of liberal arts program so you got a little bit of everything... I hadn't
taken math since high school, really, and the class utterly annihilated my
conception of the world.

I remember the Julia fractal in particular because it was so beautiful, and it
was around this part of the course -- maybe 75% of the way through -- that the
fractals and like topics started to blow my mind. Our professor showed us this
video that zoomed in on a Julia fractal, something like this,
[https://www.youtube.com/watch?v=gruJ0S3TTtI](https://www.youtube.com/watch?v=gruJ0S3TTtI),
and I remember watching it all day at work the next day. I also searched for
images of the most beautiful Julias to make as my desktop background, of
course.

Not only were they beautiful but so symbolic, as this article captures: Julia
fractals are part of chaos theory, which holds that even determinate, logical
systems can nevertheless manifest completely unpredictable and nonrecurrent
behavior. It's a straightforward equation that gets you these beautiful -- and
utterly terrifying, ceaseless, dreamlike -- images, when mapped in a certain
way. For me, that's a really beautiful concept because with "Enlightenment"
mathematics, Newton and Leibniz and co, you got this concept of a determinate
universe, which could therefore also be known and predicated in advance. Yet
chaos theory shows that even determinate systems can be impossible to know,
refusing to allow the complexity and variety of that which exists to boil down
into a boring pattern of predictable and even controllable outcomes.

------
skybrian
Nice to read this again.

Not nearly as pretty, but if you'd like to read more about complex numbers,
here's a slide show I put together a while back:

How to explain Euler's identity using triangles and spirals
[https://docs.google.com/presentation/d/1oMNjkDp-
LieSGnZEwNpc...](https://docs.google.com/presentation/d/1oMNjkDp-
LieSGnZEwNpceG8KTIvbnus9olu3KqnM5bg/edit#slide=id.i0)

~~~
rosstex
That's awesome, thank you! It really did "connect the dots" in my head :)

------
kzrdude
Previous discussion:
[https://news.ycombinator.com/item?id=7898883](https://news.ycombinator.com/item?id=7898883)

~~~
tricolon
Older discussions:

[https://news.ycombinator.com/item?id=5017078](https://news.ycombinator.com/item?id=5017078)
(4 comments)

[https://news.ycombinator.com/item?id=5036235](https://news.ycombinator.com/item?id=5036235)
(50 comments)

------
SCAQTony
WOW! The masthead alone is worth a scroll, you even get an achievement badge.
LOL

~~~
corysama
You would like the article on how they made it

[http://acko.net/blog/zero-to-sixty-in-one-second/](http://acko.net/blog/zero-
to-sixty-in-one-second/)

All of their MathBox-powered articles are wonderfull really.

~~~
skeuomorf
nitpick: _He_ not _they_. [0] Steven is also the person who developed MathBox.
[1]

[0] [http://acko.net/about/](http://acko.net/about/)

[1] [http://acko.net/blog/making-mathbox/](http://acko.net/blog/making-
mathbox/)

~~~
tomn
Thanks for adding more information, but that's a perfectly cromulent use of
the word they.

~~~
skeuomorf
Actually I think the correctness of using a singular they is disputed but I am
not a grammaticist, however the word _cromulent_ which you used is not even a
real word but I didn't mention the correction to do all that nitpicking, I
just thought since the person's information is accessible, it'd be nice to
refer to him correctly that's all.

~~~
pdkl95
> the word cromulent ... is not even a real word

Only to a prescriptivist[1] trying to keep the language static. "Cromulent" is
slowly making its way into descriptivist[1] dictionaries and is recognized
about as often as any other new word, so it just as much a "real" word as
other new words ("email", "google" (transitive verb), "truthiness").

> the correctness of using a singular they

The alternative is gendered pronouns which have several problems[2].

/* I'm not trying to nitpick your post; I just thought these two Tom Scott
clips were fun and relevant to these grammar issues. */

[1]
[https://www.youtube.com/watch?v=2qT8ZYewYEY](https://www.youtube.com/watch?v=2qT8ZYewYEY)

[2] [https://www.youtube.com/watch?v=46ehrFk-
gLk](https://www.youtube.com/watch?v=46ehrFk-gLk)

~~~
skeuomorf
That was kinda my point, I was not nitpicking -relative OP's- grammatical
usage, I just thought it was convenient to refer to the author in a more
specific manner since his details were obvious. I had no idea whether _their_
-relative OP- :) intention was to use "their" in a singular or plural manner.
Heck, one of my favorite Stephen Fry videos [0] talks exactly about this!

Also, the videos you referenced are great :)

[0]
[https://www.youtube.com/watch?v=J7E-aoXLZGY](https://www.youtube.com/watch?v=J7E-aoXLZGY)

------
tripzilch
This page was very useful when I was teaching a (particularly clever) 12 year
old kid about complex numbers. He really wanted to render Mandelbrot/Julia
fractals (using Processing), had done some googling on the subject of complex
numbers, but most of the articles he found were ever-so-slightly above the
level of math he had learned in school (turned out he hadn't yet learned about
the distributive rule for multiplication, that (a+b)*(c+d) = ac + ad + bc +
bd, which is kinda important if you want to work out (x + iy)^2 given that i^2
= -1).

I was lucky that someone explained me complex numbers when I was 15 (I also
had wanted to plot Mandelbrot fractals for a long time, but back then I didn't
even have the Internet to help me), using a very visual approach similar to
the featured article. That is, multiplication by -1 is the same as a 180
degree rotation around the zero ... so what would happen if we decided we
could rotate by 90 degrees?

So I took a similar approach. Then I remembered this article about "folding
Julia fractals", the visualizations in this article were a great supplement to
the graphs and scribbles we made on paper, exploring the weird world of
complex numbers.

I did a little video interview with him to show off his work (cause, you know,
I was kinda proud):
[https://www.youtube.com/watch?v=rR6klRdtjsg](https://www.youtube.com/watch?v=rR6klRdtjsg)
\-- It's in Dutch and I'm not a very good interviewer, also no editing (and
yes I should've held my phone horizontally, sorry).

But the best part was two weeks later, I kinda feared I had dumped too much
information onto him at once (especially given I also had to explain the
distributive rule), I asked him if he had made any improvements or additions;
"Yeah, I had to wait at the dentist's this week, and I had Processing for
Android (APDE) on my phone, so I wrote the Julia version of the Mandelbrot
zoomer" ... Oh, if only I had have a powerful pocket computer when I was 12!!!
(so jealous!)

------
bittercynic
Reading this gave me a moderate ASMR response. Anyone else have that
occasionally from reading about math?

------
divs1210
What brilliant article and website design!

I feel stupid for not being able to visualize complex numbers before.

------
echoneptune
This would have helped me tremendously when I was studying complex numbers for
signal processing. I guess I'm a visual learner. I find that to be helpful in
some field in math, but held me back when I was studying statistics.

------
mrcactu5
There's not a whole lot of material about the Julia set itself.

When he talks about "folding" the Julia set, I immediately thought of this
picture of some fractions
[http://i.imgur.com/wxz2a3t.png](http://i.imgur.com/wxz2a3t.png) from this
paper: [http://arxiv.org/abs/1201.4225v1](http://arxiv.org/abs/1201.4225v1)

------
lasryaric
OWAW OWAW! I am starting to understand complex number, which I had to "flirt
with" a few times working on signal processing.

Thanks, amazing blogpost / webapp :)

------
codeshaman
Thank you ! I've been so looking for this article, I was totally amazed when I
first read it (back in 2013, I guess). I think it's one of the best math
explanations (or rather, visualisations) that I've ever seen.

------
dbalan
[Slightly OT] Crashes my firefox everytime (FF 40 on OSX).

~~~
tripzilch
I know. My low-powered laptop can barely handle it either. But this particular
page, is worth trying on a slightly more powerful computer :-)

------
gus_massa
The article is interesting, but it's very difficult to read with that
background. Also, my slow netbook becomes very slow rendering it.

~~~
kzrdude
The Julia set slideshow/animation (36 steps) is really the best part, the
animation of the set "folding" (square every point) is great.

