5 ms·

How to Fold a Julia Fractal

171 points by rosstex 11y ago

gus_massa 11y ago
The article is interesting, but it's very difficult to read with that background. Also, my slow netbook becomes very slow rendering it.
- elektromekatron 11y ago
  It is definitely written for a faster computer and larger screen, however the animations are worth viewing properly.
- kzrdude 11y ago
  The Julia set slideshow/animation (36 steps) is really the best part, the animation of the set "folding" (square every point) is great.
kzrdude 11y ago
Previous discussion: https://news.ycombinator.com/item?id=7898883 https://news.ycombinator.com/item?id=7898883
- tricolon 11y ago
  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 11y ago
WOW! The masthead alone is worth a scroll, you even get an achievement badge. LOL
- corysama 11y ago
  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 11y ago
    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 11y ago
      Thanks for adding more information, but that's a perfectly cromulent use of the word they.
      - skeuomorf 11y ago
        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.
        pavel_lishin 11y ago
        It's become a real word by virtue of its use.
        drdeca 11y ago
        "Correctness" of word choice is determined by usage. I prefer a consensus where "they" is accepted as being possibly singular, so when I have no reason not to, I choose to treat it as being a correct usage. I also tend to use it sometimes despite knowing the gender of the person I am referring to. This seems reasonable to me, because I don't see any reason why it is important to always specify the person's gender, when I don't need to specify who the person is, and doing so allows me to refer to someone without specifying a gender, should any situation occur where that would be useful, without anyone commenting on my word choice, because I would already be in the habit of sometimes leaving it out.
jfarmer 11y ago
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
divs1210 11y ago
What brilliant article and website design! I feel stupid for not being able to visualize complex numbers before.
bittercynic 11y ago
Reading this gave me a moderate ASMR response. Anyone else have that occasionally from reading about math?
lasryaric 11y ago
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 :)
skybrian 11y ago
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-LieSGnZEwNpceG8KTIvbnus9olu3KqnM5bg/edit#slide=id.i0 https://docs.google.com/presentation/d/1oMNjkDp-LieSGnZEwNpc...
- rosstex 11y ago
  That's awesome, thank you! It really did "connect the dots" in my head :)
deleted 11y ago
[deleted]
jimmyspencerjr 11y ago
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.
echoneptune 11y ago
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.
dbalan 11y ago
[Slightly OT] Crashes my firefox everytime (FF 40 on OSX).
- tripzilch 11y ago
  I know. My low-powered laptop can barely handle it either. But this particular page, is worth trying on a slightly more powerful computer :-)
codeshaman 11y ago
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.
mrcactu5 11y ago
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
tripzilch 11y ago
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!)

171 points by rosstex 11y ago

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

elektromekatron 11y ago
It is definitely written for a faster computer and larger screen, however the animations are worth viewing properly.
kzrdude 11y ago
The Julia set slideshow/animation (36 steps) is really the best part, the animation of the set "folding" (square every point) is great.

kzrdude 11y ago

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

tricolon 11y ago
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 11y ago

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

corysama 11y ago
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 11y ago
  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 11y ago
    Thanks for adding more information, but that's a perfectly cromulent use of the word they.
    - skeuomorf 11y ago
      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.
      - pavel_lishin 11y ago
        It's become a real word by virtue of its use.
      - drdeca 11y ago
        "Correctness" of word choice is determined by usage. I prefer a consensus where "they" is accepted as being possibly singular, so when I have no reason not to, I choose to treat it as being a correct usage. I also tend to use it sometimes despite knowing the gender of the person I am referring to. This seems reasonable to me, because I don't see any reason why it is important to always specify the person's gender, when I don't need to specify who the person is, and doing so allows me to refer to someone without specifying a gender, should any situation occur where that would be useful, without anyone commenting on my word choice, because I would already be in the habit of sometimes leaving it out.

jfarmer 11y ago

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

divs1210 11y ago

What brilliant article and website design! I feel stupid for not being able to visualize complex numbers before.

bittercynic 11y ago

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

lasryaric 11y ago

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 :)

skybrian 11y ago

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-LieSGnZEwNpceG8KTIvbnus9olu3KqnM5bg/edit#slide=id.i0 https://docs.google.com/presentation/d/1oMNjkDp-LieSGnZEwNpc...

rosstex 11y ago
That's awesome, thank you! It really did "connect the dots" in my head :)

deleted 11y ago

[deleted]

jimmyspencerjr 11y ago

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.

echoneptune 11y ago

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.

dbalan 11y ago

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

tripzilch 11y ago
I know. My low-powered laptop can barely handle it either. But this particular page, is worth trying on a slightly more powerful computer :-)

codeshaman 11y ago

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.

mrcactu5 11y ago

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

tripzilch 11y ago

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!)