Calcyman's Idea (2D)

[LesPaul]

Tangent is a strange beast!

The first image here is what appears to be the whole set.  I had to zoom out quite a bit to see it -- the top-left is (-1000, 750) and the bottom right is (1000, -750).  This image is so unusual that it almost seems like there is some calculation error happening.  I did see some strange behavior (like the image changing when zooming) until I turned off "guessing" and "periodicity checking."

The two zooms here are not deep at all.  There is no layering or any other post-processing of the images.  "Function" is set to "tan" and "Power" is set to one.  It's just a really strange set!  The different areas show almost no resemblance to each other...

[kram1032]

nice O.o

Did you also try different bailout values?

[LesPaul]

I'm not sure if you know, but the shape featured in z_2_sin.png is a multibrot; the shape that appears from iterating z=z^3 + z0, instead of z^2. I find it interesting that this appears in your z^2*sin(z) version.

It is very similar to the z³ multibrot, but different in a lot of ways.  The standard multibrot doesn't have that "stem" along the real axis.  The overall shape is slightly different, too.

You can toggle between the two by just switching the formula from "sin" to "ident."  The identity formula just returns z, so you get

   z_n+1 = z_n² * z_n + z₀ = z_n³ + z₀.

That would suggest that the z_n² term dominates the shape on the large scale, so that multiplying by sin(z_n) isn't much different from multiplying by just z_n.  But they are quite different when you zoom in.

[LesPaul]

nice O.o

Thanks

Did you also try different bailout values?

Good idea...  It looks like changing the bailout does have a dramatic effect.  The black semicircle on the +real half just gets larger and larger if you increase the bailout.  I guess that suggests that the entire +real half of the graph is in the set?  Or that it isn't in the set but tends to infinity very slowly?  I'm not sure.  Are there coloring algorithms that don't rely on a bailout value?  It would probably make rendering unacceptably slow if there was no bailout.

[kram1032]

that's what I thought...
With functions like the tan, going to infinity in a cyclic pattern, I guess you can't define a true bailout value.
You could have a value close to infinity which then returns a value close to zero, if you plot it in again.

However it might help a bit (for bringing slightly more order in this) to use , rather than , as then the intervals will be at lenght 2, having infinity at odd and zero at even numbers...

[Timeroot]

It seemed to me it would be interesting to find the locus of points (I believe they would be Cantor Dust) that do escape to infinity. Although that isn't very easy to do, I created an image of the locus of points that monotonically increase. Here's the code I used:

Code:

TangentEscape {
init:
 c=#pixel
 z=@Perturbation
loop:
 zOld=z
 z=z*tan(z) + c
bailout:
 |z| >= |zOld|
}

It created a Cantor Set along the positive real axis (with the exception of an narrow, boring, error-ridden shape from 0 to 0.23) with no other points in the positive half-plane. In the other half plane, some interesting figures were produced. I'll eventually post some pics, but I'm doing some other stuff right now... 

[bib]

a video showing the effect of bailout on z*tan(z)+c

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

[bib]

Another one that dezooms from near the origin and zooms far from the origin:

<a href="http://www.youtube.com/v/-wK6tMPR_Nc&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/-wK6tMPR_Nc&rel=1&fs=1&hd=1</a>

[Calcyman]

Wow! Some truly stunning images from such simple formulae. The z*tan(z) fractal looks totally pathological until you zoom into its core, where it is dominated with minibrots and multiply connected lacy structures. By comparison, ordinary z² Mandelbrot-Julia fractals have simply connected dendritic structures.

The diversity of phenomena in the tangent fractal is exciting. The fractal detail seems to lie on radial lines emanating from the origin, and the Manhattan-style lines that bridge them. It's quite unexpected to see Mandelbrot Sets hiding in the details.

The Maclaurin Series of the natural logarithm function also rapidly converges. z*ln(z) should be interesting -- the real line terminates at the base of the logarithm (e in this case).

My 'idea' was to investigate fractals with Maclaurin series (integer powers of z). I wonder whether Fourier series conceal the same detail, considering the fractal complexity of the sinusoidal fractals. This would yeild a new family of fractals, all composed of sinusoidal components.

In fact, the Weierstrass function, a particular Fourier series, is a fractal on the first iteration!

[David Makin]

I think you'll find formulas already built for Fractint, Ultra Fractal and ChaosPro investigating formulas along the lines you're describing

There are also some where a continued series is extended with iteration.

[LesPaul]

Another one that dezooms from near the origin and zooms far from the origin:

Wow, I watched this one multiple times! 

Did you find that the calculation time becomes unbearably slow as soon as you zoom to a level where arbitrary precision is required?  In UltraFractal, the "pixels per second" drops from several thousand down to like 9!  The algorithm they use for tan(z) must not be very friendly to arbitrary-precision math.

I took a shot at animating what happens as the formula varies smoothly from  z²*tan(z)  to just  z*tan(z)  and the result is interesting.  The set kind of "folds over" on itself and all kinds of strange details emerge right before the power reaches 1.  It looks as if the set "explodes" in the -real direction.

I'd post it but I used an evaluation version of UltraFractal and the whole AVI is stamped with "Evaluation Copy."  I guess it's time to shell out the hundred dollars. 

Those videos are really great, bib.

[LesPaul]

The Maclaurin Series of the natural logarithm function also rapidly converges. z*ln(z) should be interesting ...

I'm not able to get z*ln(z) to do anything interesting, so far.  The entire complex plane appears to be in the set.  I'll have to check a few points by hand and figure out what's going on there.

[Timeroot]

I'd post it but I used an evaluation version of UltraFractal and the whole AVI is stamped with "Evaluation Copy."  I guess it's time to shell out the hundred dollars. 

Those videos are really great, bib.

Just a tip: While UltraFractal is generally more user-friendly and what not and has a better GUI and stuff, ChaosPro can load UF formulas, animate them, and it's free. I just use it when I have a good fractal form UF and want to render it. :-P

Can't wait to see vid!

[LesPaul]

Just a tip: While UltraFractal is generally more user-friendly and what not and has a better GUI and stuff, ChaosPro can load UF formulas, animate them, and it's free. I just use it when I have a good fractal form UF and want to render it. :-P

Can't wait to see vid!

Nice, "free" is my favorite price. 

Is it also open source?  I'd be more than willing to contribute to the project, if so.

[LesPaul]

Can't wait to see vid!

Here it is:
<a href="http://www.youtube.com/v/dwHUkbhI4CA&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/dwHUkbhI4CA&rel=1&fs=1&hd=1</a>

Sorry the compression is a bit severe.  I'm just getting the hang of uploading to YouTube.

Many of the small black islands that you see swirling around are minibrots.