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How to Fold a Julia Fractal
21–30 of 31 posts
Re: How to Fold a Julia Fractal
#22What 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.
Re: How to Fold a Julia Fractal
#23Earlier quoted context omitted.
Thanks for adding more information, but that's a perfectly cromulent use of the word they.
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.
If singular they was good enough for Shakespeare, it ought to be good enough for us.
Re: How to Fold a Julia Fractal
#24This 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.
Re: How to Fold a Julia Fractal
#25Earlier quoted context omitted.
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.
"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, wh…
It is also happens to be consistent with the way most academics refer to work, right? A single author will use "we" and others will use "they" to refer their work.
Re: How to Fold a Julia Fractal
#26[Slightly OT] Crashes my firefox everytime (FF 40 on OSX).
Re: How to Fold a Julia Fractal
#27Earlier quoted context omitted.
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.
> 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 pro…
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 :)
Re: How to Fold a Julia Fractal
#28Thank 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.
Re: How to Fold a Julia Fractal
#29There'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
Re: How to Fold a Julia Fractal
#30This 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!)