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“Burning Ship” fractal

paulbourke.net

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Re: “Burning Ship” fractal

#41

Post by ithinkso »

messe wrote: post #34
kenshoen wrote: post #31

These kind of fractal have actually 4 dimentional structure since c_x and c_y can also be parameters. I'd love to see their slice as 3D, but have yet to find a good way to visualize them... https://i.imgur.com/JRfLy6R.mp4

> These kind of fractal have actually 4 dimentional structure since c_x and c_y can also be parameters They're not parameters in that sense. The fractal is computed by taking each point on the plane as coordinates (c_x, c_y), and then iteratively applying the recursion relation. Then, with luminosity depending on how quickly that sequence escapes to infinity, we color in that point (c_x, c_y) in our image.

That's not what kenshoen meant. For example for the Mandelbrot set we have a function f(z) = z^2 + c for complex z,c and each pixel in the image represents c_x,c_y in c = c_x + i*c_y and then you iterate f(0), f(f(0))... On the other hand, if you have c constant and each pixel represents z_x, z_y in z=z_x + i*z_y instead, then iterating f(z),f(f(z))... gives you a Julia set.

But you can think of f as a function of two complex arguments f(z,c)=z^2 + c and iterate it on the whole domain (two complex = four real dimensions) and then have a picture being a slice through any 2D or (even 3D, which is what parent is talking about) plane you like. In other words, the famous Mandelbrot fractal picture is a slice of f(z,c) through a plane z=0, and Julia set pictures are slices through planes c=constant but there is no reason one cannot make other pictures of f(z,c) (just be careful what you meant by iterating a function f: C^2 -> C).

The burning ship fractal in the article is the same but the function f(z,c) is a bit weirder

Re: “Burning Ship” fractal

#42

Post by ogogmad »

Question: Do fractals have projective duals? In the projective plane RP^2, the dual of a point is a line, and vice versa. Do fractals like "Burning Ship" and "Mandelbrot" have projective duals, where each point in the set is replaced with a line? What would these look like?

Re: “Burning Ship” fractal

#43

Post by ronald_raygun »

ogogmad wrote: post #42

Question: Do fractals have projective duals? In the projective plane RP^2, the dual of a point is a line, and vice versa. Do fractals like "Burning Ship" and "Mandelbrot" have projective duals, where each point in the set is replaced with a line? What would these look like?

Mmm. I don't think that's a super useful way to think about it. The way I've always conceptualized it was that if you look at fractals, they are determined by some complex polynomial. So you could look at the space of polynomials, and each point in there , would be its own fractal. For the right space of polys thats a Hilbert space, and so by Riesz representation theorem it would have a dual space, but I dont think thats super useful here

Re: “Burning Ship” fractal

#44

Post by ronald_raygun »

wruza wrote: post #30

Oh, fractals thread and I have a couple of questions. Are fractal “features” enumerable? Finite? By feature I mean in fractal zoom videos they pick some zoom path and it generates repeating shapes again and again until they switch to different path. How many repeating patterns there are? Do fractals exist with an infinite number of features? Do fractals exist where features cannot repeat in future zoom levels? Or at…

Oh they fractal features are definitely infinite. I worked on a genetic algo a to breed fractals together a number of years ago, and one thing I noticed was that the highest non-zero poly power would determine the shape of the fractal - so a c^3 fractal would have "three petals", a c^4 would have "four petals". So clearly these things can have infinite features

https://github.com/zoso95/genetic-algorithm-fractals

Re: “Burning Ship” fractal

#48

Post by layer8 »

dr_kiszonka wrote: post #26
dang wrote: post #9

Related: Burning Ship Fractal - https://news.ycombinator.com/item?id=12581569 - Sept 2016 (65 comments)

Just curious - do you have a script that finds related posts for each submission or do you rely on your memory, Dang?

https://hn.algolia.com/?query=burning%20ship%20fractal&sort=...

Re: “Burning Ship” fractal

#49

Post by simias »

rubatuga wrote: post #5

Great YouTube video of a deep zoom: https://www.youtube.com/watch?v=2S3lc2G3rWs

What's the trick to implement a zoom like this in software? I remember many years ago implementing a trivial program to display the mandelbrot fractal, but as you zoomed in you quickly ran out of precision even if you used 64bit double floats.

Is it just using "bignums" behind the scenes or is there a trick to "reset" the exponent due to the fractal nature of the display? I always wondered if, thanks to the self-similar nature of fractals, one could convert a set of coordinates to another at a different scale and yield the same results.

My intuition tells me that it wouldn't work for all fractals though, and probably not for Mandelbrot because while it's self-similar it never seems to look exactly the same at different scales.

Re: “Burning Ship” fractal

#50

Post by rob74 »

mike_hock wrote: post #36
dmurray wrote: post #20

Earlier quoted context omitted.

I'm not sure your edit is right. No reason why there can't be a tiny but very complex thing which looks single-colour at the next 1 or 10 or 100 orders of magnitude. But there are an infinite number of very interesting things in this fractal. If you constantly zoom in on interesting looking areas, you will find this kind of complexity with minimal need for backtracking.

I'm guessing most fractals don't have a smooth boundary anywhere, so if you know any boundary point, you can zoom in on it and it'll be "interesting."

...except when you run out of precision in an "unsophisticated" implementation as simias mentioned, then you'll eventually get smooth boundaries - which are just an artifact however.
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