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
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.
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
