Mandelbrot set for e^x *c

[laser blaster]

I certainly didn't discover this fractal, but I haven't seen it mentioned on this site and I think it's really interesting. While the mandelbrot sets for generalized rational functions still typically have the same overall look and feel as the classic Mandelbrot, the m-set for the exponential function looks pretty unique. And it's still just a simple escape-time fractal based on a conformal map! I see a lot of potential for deep-zooming in this. There's as much variety as there is in the standard m-set, and shape stacking seems to apply here as well- only the shapes are much denser and you never have to zoom in very far to reach a "minibrot"- they don't seem to decrease in size with depth, unlike the standard m-set.

You'll notice that there are regions of solid colors with sharp edges- this is an artifact of the bailout value used. A higher bailout reduces the blobs, but then the interior of the set becomes dense in the plane, and the intricate fractal structures are impossible to see! So this is the only way that I know of to visualize this fractal. Exponential smoothing could be another way, but the values would quickly overflow past the limits of single and even double precision floats.

Here's an overall picture of the set- it continues infinitely in both vertical directions.

And here's a close-up of some details.

I've also experimented with z^z formulas. The results are even more interesting, but unfortunately the function e^x is discontinuous, as it introduces discontinuous cuts into the fractal like the fractional power mandelbrots.

And here is a high-res gallery image: http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16658

[Ryan D]

Here's something sort of the same - e^z*lambda, where lambda is continuously modified (following a Lissajous curve over the complex plane).  Same sort of toothy look, those shapes are common in e^... formulas.

<a href="http://vimeo.com/moogaloop.swf?clip_id=50422809&amp;amp;server=vimeo.com&amp;amp;fullscreen=1" target="_blank">http://vimeo.com/moogaloop.swf?clip_id=50422809&amp;amp;server=vimeo.com&amp;amp;fullscreen=1</a>

Ryan

[youhn]

Reminds me of http://www.fractalforums.com/new-theories-and-research/negative-multibrots/

Made a video in which the bailout was increased :

<a href="https://www.youtube.com/v/2IvEctTlPU8&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/2IvEctTlPU8&rel=1&fs=1&hd=1</a>

I expect this fractal to have the same behavior. Does it get more space-filling when you increase the bailout ... ?

[youhn]

That video! I don't really like the color-flashing, but the swinging red arms are great! 

Another one in this broad family:

x = 2^(x * c + c)

Results in images like:

1.


2.


3.


4.


Made with Gnofract4D. Bailout was varied between 800 and 3000. Max iterations around 7000.

[kram1032]

If I recall correctly, iterating z->z²-c just gives the same as iterating z->z²+c, right?
If so (and even if not), I wonder how e^z/c compares to e^z*c.
That particular family does make "more sense" as far as different formulae even /can/ make different amount of sense, because exponentiation essentially turns addition into multiplication
().

[hermann]

I made some experiments with exponetial and trigommetric functions.
The Java-Applet should still work. This two pages give an overview.
http://www.wackerart.de/trigonometric-functions.html
http://www.wackerart.de/trigonometric-functions_2.html

Java-Applets are know very much restricted. May be you have to allow the execution of the applet.
http://www.wackerart.de/java.html#probleme

Hermann

[lkmitch]

The nice thing about polynomials functions, like z -> z^2+c, is that you can effectively use a circular bailout criterion (|z|>4, e.g.).  But with transcendental functions, having a large |z| at one iteration doesn't guarantee divergence.  For example, the mandelbrot set for c*exp(z) has interior points on the entire left half plane (real(c)<0) and spikes that extend to real(z) = +infinity.  This isn't important for capturing the overall nature of the set, but is something to be considered if you're doing anything more sophisticated.