Wow. I "understood" complex numbers, in that I could use them in specific contexts in the past and got that they represented rotation, but this really drove it home in a way that I could've used a decade ago. Great article.
I am blown away by this intuitive explanation of why Julia sets look the way they do.
I was obsessed with the Mandelbrot and Julia sets waaaay back in high school, with writing programs to produce them in real time, back on my 386. But I never had the slightest understanding of why they looked the way they do. So to discover that now, almost 20 years later, is just amazing.
I could have easily written this comment! My favourite programming memory of all time was finally succeeding in turning the pseudo-code at the back of a book[1] that my mom had bought me into a Turbo Pascal program that would generate a Mandlebrot set. It was exhilarating!
Me too - among my first, and best, programming experiences was turning pseudo-code into QBASIC and marvel at the fractals and strange attractors growing on the screen.
I did so as well. I was fascinated that such a simple formula could produce an infinitely complex image. To build it, I used Lazarus/Freepascal and Wikipedia. It’s actually still hosted on Google code.
Me too - I wrote a Mandlebrot generator on the Amstrad CPC 6128 (3.7MHz Z80 based computer with 128kb RAM) after reading a copy of "Chaos" by James Gleick that I found in the school library. It took 3-4 hours to generate a 16 colour image of 160x120 (Mode 0?) and used some undocumented calls into the BASIC interpreter ROM for the floating point calculations. I'm sure I could have made it faster but I was producing it as a static image for a demo (demo scene).
Wow, simply incredible work. Stuff like this offers a glimpse into the future of what the web can be and how it can be used to pass knowledge on. I knew some of the concepts presented here, but I never knew what it all meant. Nice touch in adding the equations as well. Beautiful presentation.
For some reason the Mandelbrot fractal (slides 30 and 31 in the second part) didn't render in Firefox but they worked in Chrome. Everything else seemed to work fine.
Oddly enough, it works fine if you open the iframe in a new tab. I think it may just be Firefox denying my admittedly greedy requests for video memory.
I'm impressed with the explanations and everything... but what really got my panties in a bunch is the visualization technology. That is one of the prime examples I've seen of actually leveraging current web tech to communicate actual content instead of just doing fancy stuff just because. I was initially annoyed by the 3D header movement effect, but the content, graphs, and animations of the actual content made me completely forget that and allowed me to be bewildered by what we can do this day and age on the web. Congrats to the maker!
Consider the fancy header to be the first answer to: what happens when you put a 3D engine inside a webpage. This article is what happens 12 months later...
Hah I can relate to that! I will clarify something though, the effect is really cool and I actually like it. What annoyed me is that I had initially opened it and moved it to another pane (dwm here) and then when I moved it the main panel I was using a smaller window than usual and when I scrolled I couldn't hit any links, animation looked weird, and then I couldn't scroll past the header.
I browsed it again in a floating window in a different tab (or space if you wanna call it that way) where I keep a maximized chromium and I was no longer annoyed :) Good job on being on the cutting edge though, that is something I completely can appreciate.
Sigh, every time my site hits hacker news or reddit I have to mention this... it's a Webkit bug. I reported it 5 releases ago, it sits unloved and unfixed. I imagine i'll just redo the site before they address it.
Ohh that makes sense, I also wondered if it was Chromium in Linux thing so I opened it up in Chrome in my work iMac and I noticed the same thing, albeit less pronounced. It's very pretty when it does work so that kinda balances it out. In any case your content is worth it, thanks again for the awesome sauce.
For me, it fails in both Chrome Canary (26.0.1378.2) and Firefox says I do not have a compatible graphics card. (Possibly just hates my Optimus setup) And, of course, IE9 lacks webGl in the first place, so it doesn't even get to the point of causing a process hang like Canary did. All Windows 7.
Once I did find a browser that worked with the site, though- that's the best explanation of both fractals and complex numbers I've ever seen!
wow! I heard many times that great mathematician can picture theories in their mind... and that for some theories only few people on earth are able to picture them.
I always wondered if that's true and for people using math on daily basis (mathematician or others) it's very important to picture what you are trying to achieve or only numbers matter? Thanks!
It's more the case that the numbers and the mental picture are the same thing. When I manage that kind of insight (rarely) that's what it's like anyway.
The square root animation, slide 29-30, is incorrect. It should not tear at -i, it should tear at the direction of 1 instead. Other than that everything else looks fine. Amazing work.
I loved this so much. The visualizations were beautiful, and I'm appreciative of the effort that must have gone into such clear illustrations. The only thing I would suggest would be to make the slide progression arrows somewhat more eye-catching--I almost missed their presence entirely when I was quickly scrolling down the page trying to parse whether or not I was going to read the whole thing. As a result, I almost missed out on the defining part of the experience.
I would absolutely love to see more visualizations like this in e-textbooks. I think it would assist with the readers' understanding, particularly in secondary school math or physics where students may have difficulty grasping new concepts in geometry, algebra, and the like. As a math tutor, having something to show my students really helps things. Well done!
Based on the comments, this sounds like a great article that I'll have to check out later... for the moment, however, all it does is crash Safari on my ipad.
Yes, I suspect the mobile devices will not have the ability to display this content. However, if you enable WebGL on Safari via the Developer menu, the page will render really well on a Mac.
I was reading interestedly until my laptop fan got so loud I couldn't concentrate anymore and had to shut it down. The room is now a degree or two warmer.
