How to Fold a Julia Fractal

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

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, 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.

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...

Previous discussion: https://news.ycombinator.com/item?id=7898883

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

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 -- 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!)

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

What brilliant article and website design!

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

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.

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 from this paper: http://arxiv.org/abs/1201.4225v1

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 :)

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.

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

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