Print Page - Omnibrot: Getting m-sets of all powers

Title: Omnibrot: Getting m-sets of all powers
Post by: kjknohw on May 04, 2014, 06:03:29 AM

Wouldn't it be fun if every (integer 2 or larger) mset power was included at once?

There is a formula that has contains m-sets of every power: z = (sin(z))^(z/pi)+c, this formula has roots of every order. But it is disgusting since it has branch lines and large inside regions going off to infinity.

You can alternate iterations, like the Stutterbrot fractal. You do 10 at z^2, 20 at z^3, 30 at z^2 again, 40 at z^3, 50 at z^4, and so on. This allows you to explore any Mandelbrot power up to any depth as long as you zoom in. It gives strange results:
(http://www.developmentserver.com/kevin/2014_mandelPower/region/stutter.png)

So what next? Simply divide up the complex plane into regions (an infinite number in a finite space) and put an mset of each power there. This sounds unnatrual, and it is, but it works surprisingly well:
(http://www.developmentserver.com/kevin/2014_mandelPower/region/region%20main.png)

You don't just get isolated msets, they interact because an orbit can jump to another region on the 100th iteration. More images:
http://www.developmentserver.com/kevin/2014_mandelPower/region/omnibrot.html (http://www.developmentserver.com/kevin/2014_mandelPower/region/omnibrot.html)

Finally, why settle for finite power when you can have infinite? Simply put a large positive power zero next to a slightly smaller pole and take the limit (courtasy wolfram alpha) as the power goes to infinity (the pole and zero close in on eachother for the function to converge to a finite non-zero value). The formula we use is z = z*z*exp(1/z)+c
(http://www.developmentserver.com/kevin/2014_mandelPower/region/infinite.png)

Formula (with julias) attached.

Title: Re: Omnibrot: Getting m-sets of all powers
Post by: DarkBeam on May 04, 2014, 09:28:10 AM

This is super interesting, especially last two approaches!

Title: Re: Omnibrot: Getting m-sets of all powers
Post by: kram1032 on May 04, 2014, 10:49:35 AM

Great stuff :D

I like the infinity-brot and its weird appendages.
Can you somehow do this in the opposite order? Starting with infinity-brot, going all the way down to 2-brot?

Title: Re: Omnibrot: Getting m-sets of all powers
Post by: DarkBeam on May 04, 2014, 11:29:14 AM

Message for the formula author;

When I visit the link for the ufm file it looks all messed; without line-feeds.

Please fix :)

Title: Re: Omnibrot: Getting m-sets of all powers
Post by: kjknohw on May 04, 2014, 11:55:46 AM

Quote from: DarkBeam on May 04, 2014, 11:29:14 AM

Message for the formula author;

When I visit the link for the ufm file it looks all messed; without line-feeds.

Please fix :)

I attached a .txt file of the formula, which works on my (mac) machine if you turn text wrap off. For some reason the forums doesn't allow attaching .ufm :hmh: so I renamed the file as .txt.

Edit: Ok now it does (see post below), and I didn't notice that I could have just sent a .zip

Title: Re: Omnibrot: Getting m-sets of all powers
Post by: DarkBeam on May 04, 2014, 12:10:51 PM

Now, the big trouble is that arbitrary powers of Mandelbulb are non-trivial expressions of x,y,z (except if you code them using trig expressions, damn slow!) so I cannot code it in general.

As for exp(z) for now nobody gave a convincing 3D Mandelbulb-like expression so far.

I am so sorry. :'(

Title: Re: Omnibrot: Getting m-sets of all powers
Post by: Sockratease on May 04, 2014, 12:12:56 PM

Quote from: kjknohw on May 04, 2014, 11:55:46 AM

...For some reason the forums doesn't allow attaching .ufm so I renamed the file as .txt :hmh: .

It does now!

The reason was simply that it was overlooked. As new programs and situations arise, our list of allowed extensions for attachments continues to grow.

We do allow the .zip extension so if you ever find another file type you wish to attach, but is not on our list, just compress it to a zip file and use that O0

Title: Re: Omnibrot: Getting m-sets of all powers
Post by: Dinkydau on May 04, 2014, 04:03:16 PM

Very nice! This works exceptionally well. I have imagined a single fractal with all mandelbrot set powers and even suggested something here on the forum about it before, but never saw something like it.

Title: Re: Omnibrot: Getting m-sets of all powers
Post by: laser blaster on May 04, 2014, 09:30:07 PM

Wow, this is great! I imagine the second method could be used to make a fusion of any two fractal sets. The third one is interesting, too. Does it contain the multibrot for every power?

Title: Re: Omnibrot: Getting m-sets of all powers
Post by: kjknohw on May 05, 2014, 01:45:54 AM

The second image has every power.

The third image is actually an infinite power mini-mset, which is "mapped" in such a way that it isn't a circle. If you look closely, you can see period "infinitying" just like you have period doublinc for z=z^2+c or tripling for z=z^3+c. There is only one power (infinity).

Title: Re: Omnibrot: Getting m-sets of all powers
Post by: cKleinhuis on May 05, 2014, 03:08:14 AM

congrats, nice approach! i like how they actually do interact on every combination!
gotta go and take a look at them, perhaps i am looking into it with my dotty visualisation, quite chaotic though ;)

Title: Re: Omnibrot: Getting m-sets of all powers
Post by: Endemyon on May 07, 2014, 05:33:28 PM

Beautiful Discover ! :dink:

Title: Re: Omnibrot: Getting m-sets of all powers
Post by: kjknohw on May 18, 2014, 03:29:21 AM

Thank you all for your feedback. Here is a simple pixel-cenered orbit trap that traps all mini-mandelbrots. It also works on any fractals that have "coexistence" (different powers in one).

Code:

pixelcenterorbittrap {
;pixel-centered orbit trap

init:
     float err = 1e38;
     int dontStartRightAway = 0
loop:
     if dontStartRightAway>0
     z = #z-#pixel;#center
     if real(z)*real(z)+imag(z)*imag(z) < err
          err = real(z)*real(z)+imag(z)*imag(z)
     endif
     else
     dontStartRightAway = 1
     endif
final:
  #index = 200-log(err);
default:
  title = "#pixel-centerd orbit trap";
}

Title: Re: Omnibrot: Getting m-sets of all powers
Post by: Alef on July 14, 2014, 06:36:37 PM

You could register as UF formula author and upload the file in UF site;) There are lots of formulas but this could be nice new adition.
Alsou this http://www.fractalforums.com/new-theories-and-research/complex-not-so-complex/ (http://www.fractalforums.com/new-theories-and-research/complex-not-so-complex/) never entered UF database :-\

Title: Re: Omnibrot: Getting m-sets of all powers
Post by: M Benesi on July 16, 2014, 04:21:03 AM

You try the exponential function version?

$z_{n+1}= e^z -z -1 + e^c -c - 1$

  Just calculate e^c -c -1 once.... so:

$c= e^c -c -1$

  You subtract 1 and the first term from each exponential function.

  If you do it, just take the absolute value of the real portion for your bailout value. Not too interesting. z^2 dominates.

However... something interesting.

While for abs(z) <1 doesn't really allow too much (although one COULD alter the magnitude of the pixel??):

$\sum_{n=2}^\infty z^n = \frac{-z^2}{z-1}$

One can do a partial sum, which does not diverge:

$\sum_{n=2}^k z^n = \frac{-z\times(z-z^n)}{z-1}$

Set n to whatever, and you have z^2 +z^3 +z^4+....+ z^(n-2) + z^(n-1) +z^n.
Slightly interesting.

Title: Re: Omnibrot: Getting m-sets of all powers
Post by: Alef on July 18, 2014, 04:49:16 PM

Pictures :D

Title: Re: Omnibrot: Getting m-sets of all powers
Post by: M Benesi on July 20, 2014, 04:52:24 AM

Yes, ohh master. Just zoomed into where the stalk would be. There are other patterns in other locations. Makes various patterns.

(https://lh6.googleusercontent.com/-AXAmwOwbxiQ/U8ssEnCU7RI/AAAAAAAACRs/XJZOloHrjn4/s288/multibrot%2520n66.jpg) (https://lh5.googleusercontent.com/-mHC-7NVCgVg/U8ssEoTLpzI/AAAAAAAACRM/eIT7FUlLBGg/s288/n66%2520zoom%25201.jpg) (https://lh3.googleusercontent.com/-yCk1UwNeqUA/U8ssE3RUSoI/AAAAAAAACPg/LobavB6adTA/s288/n66%2520zoom%25202.jpg) (https://lh3.googleusercontent.com/-LR3CTgU2u6A/U8ssE60cH8I/AAAAAAAACQY/_WBOCDedgBI/s288/n66%2520zoom%25203.jpg) (https://lh6.googleusercontent.com/-bg-uN-qMgc8/U8ssFPP6_LI/AAAAAAAACQI/PXl9ZV4DozA/s288/n66%2520zoom%25204.jpg) (https://lh5.googleusercontent.com/-IM1hfHlgQGo/U8ssFbOA1oI/AAAAAAAACPs/cS0tAAKHllg/s288/n66%2520zoom%25205.jpg) (https://lh6.googleusercontent.com/-JaqC2tGQVPw/U8ssF4Jrl7I/AAAAAAAACP8/GfbdQb0y3MI/s288/n66%2520zoom%25206.jpg) (https://lh5.googleusercontent.com/--cACWOs_B7E/U8ssGGP97UI/AAAAAAAACP4/kSEgryT_eL4/s288/n66%2520zoom%25207.jpg) (https://lh3.googleusercontent.com/-b92FcqUDH0Q/U8ssGpzpWJI/AAAAAAAACQE/hnm5akiMh7I/s288/n66%2520zoom%25208.jpg) (https://lh6.googleusercontent.com/-xFwFCv55X7k/U8ssGhs8qcI/AAAAAAAACQA/XJ9A0ikTUa0/s288/n66%2520zoom%25209.jpg) (https://lh3.googleusercontent.com/-iPbcwIFulpM/U8ssEdyaQlI/AAAAAAAACQU/ODAolk8sM5g/s288/n66%2520zoom%252010.jpg)

Title: Re: Omnibrot: Getting m-sets of all powers
Post by: kjknohw on July 20, 2014, 04:13:50 PM

Interesting pattern, however it will only have z^2 minibrots. I do like how the powers go to infinity toward the left and converge to a circle.

Title: Re: Omnibrot: Getting m-sets of all powers
Post by: M Benesi on July 27, 2014, 05:25:42 AM

It's pretty smooth if you just do (z-z^n)/(z-1) and set n high and odd (like 111 or 217 or whatever). Patterns change depending on z. If you want symmetry, set z to an odd number and travel along the stalk.

Circles and circles... pretty normal stuff really.

(https://lh6.googleusercontent.com/-PDu7aCzwk7g/U9RwjVWk85I/AAAAAAAACSg/T6PRmQ_0huU/s2048/z111%252520pn0_1.jpg)

Title: Re: Omnibrot: Getting m-sets of all powers
Post by: kram1032 on July 27, 2014, 12:12:51 PM

normal? yes.
pretty? no.
What the heck is up with that JPEG compression? :/

Title: Re: Omnibrot: Getting m-sets of all powers
Post by: M Benesi on July 28, 2014, 09:14:46 PM

Dunno. I should check the google image settings- I think picasa chomped on the jpeg a bit too hard.

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Fractal Math, Chaos Theory & Research => (new) Theories & Research => Topic started by: kjknohw on May 04, 2014, 06:03:29 AM