“Burning Ship” fractal(paulbourke.net) |
“Burning Ship” fractal(paulbourke.net) |
When you violate this sacred notion, of treating a complex number as a singular entity, indivisible, the operations or functions you get, no longer have nice properties, like being analytic [2]. You are no longer doing algebra, you have gone into the realm of chaotics over R^2 with an equivalence relation through rotations, which is a far different beast.
That's why other fractals that treat complex numbers as first class citizens, singular entities, and don't use hacks like Re or Im, are much prettier and have less branch cuts.
The operation * (meaning conjugation) isn't totally ugly. The reason for that is because it shows up in matrix theory. A lot of families of matrices (like the unitary and the self-adjoint matrices) are defined using *. Therefore, while the function theory of the split-complex numbers might be boring, their matrix theory is still somewhat interesting because of its dependence on *.
The relevance to generating fractals using the split-complex numbers is that you need to use non-holomorphic functions to get interesting results. Some results here: https://news.ycombinator.com/item?id=32211495
There are holomorphic split-complex functions and they aren't uninteresting at all - they are quite beautiful and complex, relating to wave equations: https://en.wikipedia.org/wiki/Motor_variable#D-holomorphic_f...
They're not something often looked into in the fractal scene but if anyone has spare time, I'd love to see a D-holomorphic fractal analog of the Mandelbrot :-)
powi(rabs(z)+i*rabs(im(z)),2)+c
;Position file automatically generated by XaoS 4.2.1
; - a realtime interactive fractal zoomer
;Use xaos -loadpos <filename> to display it
(initstate)
(filter 'anti #t)
(palette 2 73629707 0)
(formula 'user)
(usrform "powi(rabs(z)+i*rabs(im(z)),2)+c")
(usrformInit "0")
(maxiter 5000)
(bailout 5)
(view -1.6924 -0.02769 0.1137 0.1137)
[1] https://github.com/xaos-project/XaoS [2] https://i.imgur.com/mJ0uZG7.png
edit: Wait a minute, it's right on the same site! http://www.paulbourke.net/fractals
Burning Ship Fractal - https://news.ycombinator.com/item?id=12581569 - Sept 2016 (65 comments)
Edit: or, to dereference all the way: https://news.ycombinator.com/item?id=26158300.
A year ago, I built a tool to explore the Mandelbrot set fractal on the browser using vanilla JS.
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.
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
The result isn't completely trivial, but isn't particularly impressive either.
Anybody want to try Burning Ship over the split-complex numbers? It looks like you only need to replace the complex "i" with the split-complex "j".
This is my first time using that program. It doesn't look too bad.
Same thing, but with the dual numbers: https://imgur.com/a/kTU5ztn
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 least that you barely could predict where a repeating part could be. Sort of a chaotic fractal.
I manually searched the space by zooming and panning into various areas from 2x all the way to about 10^13 zoom ans rendered them in high intensity color schemes
Spend enough time applying human interpretation to non-human processes, and something will eventually come up that tickles your senses.
What I'm saying is, asking this question isn't a whole lot different than cutting open the liver of an animal sacrifice and trying to divine the will of the gods from the interior structure. Or cutting the throat of a prisoner of war and reading the splatter. Or watching the flight of birds. It doesn't say much.
I guess it would be called the running pig fractal.
This might be a message from god: don't eat pork.
It's very good and by modern standards it's quite short.
As you read it you will discover why it's relevant to your question.
Also check Library of Babel, there are some there too.
/s
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.
edit: I guess if they find an interesting thing very zoomed-in, then zoom out from that, the whole video will be interesting.
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.
On top of that I think that locations on (or near) the boundary, tend to stay on the boundary (and stay in the center of the image too) when zooming out.
While purely zooming (not translating) to a known boundary (or near) point, you won't ever see a move to another section-of-boundary, so if there are both 'inside' and 'outside' regions, corresponding to attractors at 0 and infinity, (the 2 main ones in these types of fractals) in the most-zoomed in state, then there will always be regions of both states contained in the final image when zoomed out (until you get to the 'top').
Maybe it would be possible for there to be formulae that don't hold to this? If the fractal had an incredibly sparse structure, say? To be honest I'm more interested in the opposite myself: Structures where the boundary (between N regions or behaviors) is so wiggly, it's almost 2 dimensional itself!. (If anyone wants to read more, I've called one particular interesting example of this: 'mandelfield' on UltraIterator)
You will be very disappointed once you see the split-complex Mandelbrot set. It will be an instructive exercise for you to see why it looks the way it does.
My point about complex conjugation is illustrated in [1]. Linear algebra over the complex numbers does not limit itself to the four arithmetic operations {+,-,*,/}, but uses complex conjugation as the 5th and final operation. Look at the definition of a unitary matrix, or a self-adjoint matrix, or a normal matrix. Also, look at the definitions of the QR decomposition and SVD. Complex conjugation is everywhere in linear algebra. Consequently, matrix theory over the split-complex numbers is actually somewhat interesting.
[1] - https://en.wikipedia.org/wiki/Matrix_decomposition
To some extent, this seems like a weird discussion about mathematical aesthetics.
Linear algebra does a lot of things, ie. transpose, that aren't algebraic. I think it's important we understand what is computational versus algebraic. In that vain, linear algebra proves to be immensely useful, but we will trip ourselves if we start thinking certain aspects of it are analytic.
Anyhow, point taken, that image hardly looks interesting.
I think we agree and like you said, are just talking about aesthetics. I agree that the utility of math doesn't end at algebra or analyticity.
Read my other reply.
(I had to click a lot of "here's a post where I explained it before" links that had "here's a post where I explained it before" before I found this one)
