Complex powers

[Kali]

I don't know how complex powers are calculated, but this are some formulas I've tried in UF5 with interesting results.

z = c ^ z * @m
z = c ^ z * @m + c
z = 1 / c ^ z * @m + 1 / c
(bailout above 1000)
Note: '@m' is a multiply factor entered as a parameter to modify the drawing, for example 1, 0.9, -0.7...


z = z ^ (0,2)  + c
z = z ^ (0,-2)  + c
(bailout above 10000)

They don't generate something as wonderful and with the great diversity of the original mandelbrot, but it's worth seeing, the images are very non-conventional fractals.

A few samples... I don't remember the exact formulas for each one, but they derivates from the above:








Pretty strange patterns, don't you think?

[Fractal Ken]

Kali, those pictures are very interesting, and I think I've done similar things, but I have a question about the notation everyone here seems to use.

What is the meaning of c? Is it a constant across all pixels, or does it vary by pixel? In other words, If I write z = z^2 + c, am I describing the Mandelbrot set or a Julia set?

Ken

[Kali]

I was talking about varying c by pixel like the M-Set, but I also tried the formulas like Julia sets with some good results.

[Fractal Ken]

Thanks for the clarification!

[cKleinhuis]

@ken, you describe a formula, if initial values come along you have the julia/mandelbrot distinguation
alone it can not be distinguished, usually people mean the mandelbrot method ( varying per pixel )
and the julia method is implied ( constant per pixel )

regards

[Fractal Ken]

Thanks, Christian.

Here are two simple Julia-type examples using complex exponents, colored with escape time.


z = z^(0, 2) + (1.08, -0.50); Iterations = 75; Bailout = 10; The red is the non-escape region


z = z^(0, -2) + (0, 0); Iterations = 1000; Bailout = 100

[Kali]

Nice! Here's another:



z=z^(0,2)+(0.2330725,0.228134) - Iterations:100 - Bailout: 15000 - Location center: 0,0 - Standard zoom

[Fractal Ken]

Cool pic!

Here's a Mandelbrot-type image where I've zoomed into the interior, i.e, non-escape region. I believe the coloring method I've used is called an orbit trap. (I can provide details if anyone's interested.)


z = z^(0.0, 0.5) + c; Iterations = 110; Rotated -90 degrees

I struggle with the notation and terminology folks use. The unrotated image is centered at (-0.9065, -0.2870), but I don't have a frame of reference to compute a zoom magnification. If it helps: x varies from -0.951 to -0.862, and y varies from -0.350 to -0.224.

[Kali]

As I mentioned I was using UF5, I mean with "standard zoom", the zoom the program has for default, without touching anything.
I guess the notation you used it's the right one...

[Fractal Ken]

Kali, I use software I wrote myself, and I'm unfamiliar with UF5's terminology. I certainly didn't mean to imply there's anything wrong with it. Perhaps someone else can provide a translation between the two ways of describing zoom.

Regards,
Ken

[Kali]

Anyway, besides UF5 terminology, just wanted to say "no zoom", or the zoom level that shows the whole mandelbrot set if using the standard formula... -2 to 2 on the real axis (x), I guess... I'm also don't know much about notations