Calcyman's Idea (2D)

[kram1032]

There you have your escaping orbits:

http://kram1032.deviantart.com/art/A-Nice-Tan-II-157962594

and a try on combining the two:

http://kram1032.deviantart.com/art/A-Nice-Tan-III-157962821

in the combination, blue is rather from the escaping point, green from the non-escaping ones.

I personally think, the non-escaping points are more interesting in case of tan.

Note, that the two have different bailout values.
The radius is 2*pi for both of them but as the results wheren't clear enough on the escaping one, I set the bailout to 1000 while it was 200 for the inner one.
My guess is, that you'd need WAY higher bailouts for clear images

[Timeroot]

Woah! The Buddha has a bunch of other monks with him!

In the third picture, I like the two little globs towards the bottom that are so twisted... and now I see that the corresponding monks have two "Jewels" on their heads, instead of three! They look like ears! I think that, undoubtedly, one of the coolest features of the tan function is how it can behave locally like x^2, x^3, x^4, etc, which results in the various multibrots. I would be curious to see what happens for atanh, instead of tan, because they have somewhat similar shapes and expansions. If that doesn't look interesting, then maybe atanh(pi*x/2).   

[kram1032]

I can try but I don't really think, it will be that interesting^^
(Interesting, yes. THAT interesting, no...)

Also worth a try might be 1/tan (cot)

[Calcyman]

Whoa! Impressive nebula-esque renderings!

As for the co-tangent idea, the set will have similar features to the z*tan(z) set, since, after all, cot(z) = tan((pi/2)-z).

The main difference is that tan(z) resembles z for small values of z, whereas cot(z) resembles 1/z. This means that the linear term will actually cancel out the dominant term in cot(z), resulting in small terms being mapped towards 1.

[kram1032]

Well, I see an important difference:

While tan(0) is 0, so tan(tan(0)+0)+ .... is also 0, cot 0 might be totally different. cot 0 = infinity...

[bib]

z*cotan(z)+c gives interesting results (if zstart<>0) with a lot of similarities and also a lot of differences compared to z*tan(z)+c

[kram1032]

tried already?

I noticed that z=0 seemingly doesn't converge or diverge at all.
All the points seem to get undefined or so. The screen stays black
(Thinking about it, they probably get undefined.)
So I'll use c0=z0 as starting condition...

EDIT: Or actually, I'll first try random conditions between +/-2*pi - probably not the most interesting way to do it but the shape 'till now looks interesting anyway. Just not overly fractal

[kram1032]

Silly me:
I forgot the multiplication part and just did cot(z), rather than cot(z)*z

cot(z)*z actually also gives results for x=0 and y=0
It's 1 in the limit.
So, the "correct" starting conditions could either be  or, skipping the first step
 [1,0]
it seems to be nice but it's late now and I'll have to start the render over tomorrow....
It looks promissing, though.

Bib: Would you bother reendering propper high-iteration and high-bail-radius renders of tan and cot?

In case of such a complicated function, it'd probably be neccesary to skip bailout alltogeher.. Too bad,  that this makes things so extremely slow

[bib]

A zoom into an outer region of z=z*tan(z)+C


http://www.fractalforums.com/index.php?action=gallery;sa=view;id=2102

I will try to produce some more images soon, but to answer kram's request for a global kind of "template" view of these fractals, calculation is extremely long with high iteration count AND high bailout, and the result is not very beautiful, lots of chaotic places (/artifacts?)

[kram1032]

yup...

Due to the way, tan works, mapping + and - infinity to every pi/2 steps periodically, to REALLY know the shape of this, you'd have to do a no-bailout render with insanely high iterations...

Beautiful render

I looked at cot and it looked nice but rather boring, at least to me.
Maybe, an actual Mbrot render of this would work better...

[kram1032]

I tried something different now. While it's not as beautiful as ztanz, it's not too far away of that in my opinion:

First, I tried to do a logistic growth Mset. Not too interesting. The graph already made that to be expected.
Then I found out, that certain logistic growths simplify to 1+tanh(z) - still not interesting, but the connection to tanh(z) made me even more curious about the results. - It now seems like tanh(z) is in a way the base of any logistic growth.

Then I thought: Hmmm: this thing actually looks like an integral. It's how many individuals exist per timestep.
So, my next idea was, to take the differential of tanh(z) which is sech²(z). - It's basically how many individuals become added in that certain time-step.
So that's what I'm rendering right now. - The anti-buddhabrot version for now.

It looks very nice and promising. Still quite noisy though. 1/sin(iz)² - I just renormed it a little bit so that the integral over the total set gives 1 but that shouldn't change too much. In theory, that curve now could be used as a probabillity function, at least over the reals

So, the actual formula I'm using is with a being any positive real - they will always produce 1 as an integral. The specific value of a, I chose, was , as that made it more symmetric... (2/sqrt(2) = sqrt(2))

Probably, you could directly use sech²(z) for nice results, too. The image shouldn't vary too much...



Oh, btw, I also tried the exiting orbit version. It looked nice as well but cleared up very slowly...

There surely are good reasons for using the gaussean normal distribution but the sech(x)² shares the property of the simple integral and on the imaginaryaxis it's pretty much like a cyclic version of the normal distribution... Looking at the general normal distrubution with sigma and µ, I think it would be very trivial to add that to the sech distribution aswell...

EDIT: Ok, found. It's actually called Logistic distribution (d'uh)
http://en.wikipedia.org/wiki/Logistic_distribution

EDIT:
Argh! As usual: Errors make beauty. I missed an h, turning sinh to sin. Now I have to check, wether it still looks as nice, without the error...

EDIT:
Lucky: It changed shape quite a bit, but it still looks interesting (Hyper short rendertime 'till now - just a few secs... But it already looks reasonable) - certain features still are more or less unaltered, actually...

If you want to look at the error, just use the above mentioned formulae but change the line

to

[kram1032]

Ok, here you go. It's still very noisy but you get the idea...



Here, I used a bailout value of 200 and a bailout radius of sqrt(20) (so x²+y²=20)
However for this it's probably similar to ztanz: It would need a bailout-free render with insanely high iterations to truely know how this one looks.

You can clearly see a lot of places where details only begin to show up. This would seriously need a way longer render.

If the image isn't there, try it here:
http://kram1032.deviantart.com/art/Sech-Mask-162296856
There you can also click the download button to see a higher resolution but due to all the noise, it's not quite worth it.

[kram1032]

Ok, I did an overnight render with roughly 18h in total...

http://kram1032.deviantart.com/art/Mask-of-Sechsqua-162544028
Certain details still aren't there, most likely. It litterally takes for ever, especially for the smaller details, to show up. for instance, inside all the holes of the bright area, structures emerged but they converged so slowly, I didn't get the chance to ever see, how they actually looked...

[stigomaster]

I was going to complain about your inappropriate use of the term “Literally”, but then I thought that because fractals have detail on an arbitrarily small level, the smallest details do literally take forever to show up. 
How does the orbit density map of the escaped orbits look?

[kram1032]

To show you that, I'll have to render for ever yet again xD
I did a short test on that and what I saw by that time was nice

I'd love to see an Mset of it, too, like the last two ztanz renders

Yup, I actually DID mean it litterally


Oh, I forgot to mention: My renders where with bailout at xxyy=20 and iteration=2000
To fasten things up, I now do 200 iterations