
Clifford Attractors - d00r
http://paulbourke.net/fractals/clifford/
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jheitmann
Julien C. Sprott has a very fun, approachable, and free book on generating
various kinds of strange attractor visualizations:

[http://sprott.physics.wisc.edu/sa.htm](http://sprott.physics.wisc.edu/sa.htm)

It has 1993-era code, which paradoxically might make it good for a coding
beginner since you'd have to work a bit to figure out a basic syntactic
translation, but the high level idea is already there. You get immediate
visual results, and there is a lot of room to experiment and play.

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bhouston
I wrote a renderer (eventually became Krakatoa) for these and similar systems
developed by "Dr" Baily:

[http://www.imagesavant.com/](http://www.imagesavant.com/)

Was used (and still used) on a bunch of films for Krytonite, dream sequences,
plasma, etc. The longest render time I remember for getting rid of noise was
over 24Hrs and involved +1B computed samples of the underlying functions.

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qubex
Wow, these bring back deep memories.

I was basically weaned on a slew of Clifford Pickover's books — first his
Black Holes: A Traveller's Guide and later Mazes For The Mind, The Loom Of
God, Keys To Infinity, Fractal Horizons and others.

I distinctly remember a vacation spent in Portugal... I can date it to summer
1996 because I remember the TV droning on about the campaign between Bill
Clinton and Bon Dole (!) furiously typing in Pickover's BASIC code to QBASIC
(or was it GWBASIC?) in DOS mode on my first laptop, a Cyrix 586 ACER thing
made of cheap, slightly springy plastic. I was fascinated by those patterns. I
remember how at first the code all seemed like incantations but how gradually
I got a feel for what the various parts did, and the strange pride when I
eventually located (and modified) the parts that actually represented the
formula being iterated, which I then proceeded to modify, and I'd sit there
staring at the slowly-emerging patterns.

I'd forgotten all that. Thanks for bringing it back.

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purpled_haze
Neat site.

Someone should make a screensaver like the last two random attractors on:
[http://paulbourke.net/fractals/lyapunov/](http://paulbourke.net/fractals/lyapunov/)

[http://paulbourke.net/fractals/lyapunov/explore1.jpg](http://paulbourke.net/fractals/lyapunov/explore1.jpg)

[http://paulbourke.net/fractals/lyapunov/explore2.jpg](http://paulbourke.net/fractals/lyapunov/explore2.jpg)

Those look badass.

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klodolph
There's a lot of noise in those images. You could reduce that by increasing
the sample count significantly, which should be easy on modern GPUs. (I
recognize the noise levels because I wrote a renderer for Clifford Attractors
circa 2000, but back then, it took all night on a nice Mac G4 to get images
with that kind of quality.) Maybe I'll post my OpenGL-based version.

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tonetheman
What determines alpha on those images? On any attractors really? You end up
with an infinitely large set of points between say -1.5 and 2.0. How does that
end up being graphed. Perhaps that is magic/beauty of it.

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notthegov
Could you elaborate or suggest any references? It sounds like you are saying
something that appears chaotic forms some coherent beautiful image?

I know alpha refers to many things but one of those is the fine-structure
constant. There's some remarkable beauty in it because it unites
electromagnetism, relativity and quantum mechanics. And this is why many
thought the reciprocal of alpha (137) would become as important as 0 and Pi.

I am assuming though none of that relates to fractals and that alpha in this
context refers to the angles?

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Twisol
"Alpha" in this context probably refers to opacity, the A in RGBA. GP is
making a comment on the apparent translucency of the filaments that comprise
the attractor.

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spitfire
I loved Clifford pick over's books as a teenager. Implemented a lot of his
ideas as graphics effects on a 486. They're still fresh.

Computers, Pattern, Chaos and Beauty in particular was good, and cover
attractors quite a bit.

His website is a monster of 1990's kitsch - Over 50 books.
[http://www.pickover.com](http://www.pickover.com)

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IIAOPSW
Imma let you finish, but the Lorenz attractor is the best strange attractor of
all time.

[https://en.wikipedia.org/wiki/Lorenz_system](https://en.wikipedia.org/wiki/Lorenz_system)

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peapicker
I spent some time rendering these sorts of attractors back between 1995 and
2004, influenced by Pickover and Sprott...

[http://technocosm.org/chaos/gallery.html](http://technocosm.org/chaos/gallery.html)

Pardon me for not updating the site to modern standards, the html is very late
90s.

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Retr0spectrum
Here's a quick JS implementation:
[https://jsfiddle.net/7hx2mq03/](https://jsfiddle.net/7hx2mq03/)

