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Author Topic: Is there anything novel left to do in M-like escape-time fractals in 2d?  (Read 76380 times)
Description: I think there may be. Your opinions are greatly desired.
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fracmonk
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« Reply #150 on: September 20, 2011, 07:23:13 PM »

The recent storm(s) I mentioned in post 146 devastated the region in which I spend most of my time.  Deposition patterns from erosion have made future floods likely to be more severe than the rainfall that causes them would normally suggest.  I'm watching this dynamic closely, and have to get back.  New heavy rains are forecast for the next handful of days.  I'll elaborate on the last post when I can, however.

Try to stay "on top of things" yourselves...

Later!
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fracmonk
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« Reply #151 on: October 11, 2011, 08:57:47 PM »

Well, those interminable rains are finally over for awhile, and in fact, there have been 5 totally cloudless days in a row here, which is even stranger- here, that NEVER happens.  Good weather for digging out stream beds...

Then there's those people who want to drill for gas under us, and ruin our water, air, and landscape.  There's a big push for it, and NYSDEC (Department of Environmental Compromise) is inclined to fast-track it.

Cheers to O.W.S. crowd, making me feel like a few people actually listen!  Hope it's not just another deception, or eventual "object lesson".
I can recall plainclothes cops getting into demonstrations to incite riot and give the uniforms an excuse to bust everyone...ah, the good ol' days...when we weren't 99% spineless!

I'm very busy still, but I have machines making pix for me, the kind that take awhile.

Be good.  Be CAREFUL!

Later.
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fracmonk
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« Reply #152 on: November 15, 2011, 10:50:57 PM »

Too much time has passed, & I just wanted to say Hi.  Been working on index sets with Julia-like features a little more lately, and I will (eventually) get up some interesting pictures soon.

Elsewhere:  What's a better headline?  1. Winter Weather Withers Protest
                                               -or-2. Billionaire Mayor Crushes Dissent
I'm thinking 2...
Anyone remember the story about the goose that laid the golden egg?  So does it lay more eggs when you strangle it?  The answer is yes, for a very short while.  Then, no...

Bloomberg plans on cooking it for Thanksgiving, maybe.

Best laid plans...


Later!
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fracmonk
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« Reply #153 on: December 23, 2011, 04:30:47 PM »

May your holidays be joyful, and every other moment as well.

Lately I've been taking second looks at the family of functions that dominated my entries here, and lets finally call them Multipowerbrots, since the name Multibrot, which may better have been called Powerbrot, has long since been taken.

Anyway, most recently, I've been looking at a kind of circularity in them, that could come only from formula programming.

If you took  ...
                  s=z*z*c-d
                  t=s*s-d
                  z=t*t
                  |z| < 1000000
                  }

and did       ...
                  t=z*z*z*z*c-d
                  z=t*t-d
                  |z| < 1000000
                  }

instead,
then you would get the same results, as long as you started with the right initial z in each case, and they are integer distance values.  d could be either 1 or 2.

The d=2 case is interesting since it's the simplest case of quartic and squared M shapes possible for d=2, that being degree 8.  I have to look to see if the julias are also identical for any given set of c coords.  In any event, the julias will contain both j2 + j4 shapes if one picks coords in a mini.

Note: I use large bailouts for smoothness (and when they get away, they go fast anyway).

I can say more later, and show some pix then, too.

All my best, and thanx again to m'Bunny...
« Last Edit: December 23, 2011, 04:54:55 PM by fracmonk » Logged
Alef
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« Reply #154 on: December 25, 2011, 05:21:48 PM »

Don't think, this is worth for new thread, so put in some threads.

Adding +some number to colour algorithm allows to see, how exact numbers orbits to mandelbrot border. Imaginary numbers have curved trajectories, real have linear.




http://www.fractalforums.com/the-3d-mandelbulb/more-natural-mandelbulb-surfaces/msg39296/

Would be good enought for Chaos pro. http://www.chaospro.de/formulas/display.php?fileid=224
« Last Edit: December 25, 2011, 05:27:01 PM by Asdam » Logged

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David Makin
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« Reply #155 on: December 25, 2011, 05:47:51 PM »

Don't think, this is worth for new thread, so put in some threads.

Adding +some number to colour algorithm allows to see, how exact numbers orbits to mandelbrot border. Imaginary numbers have curved trajectories, real have linear.


smiley Nice to see you're investigating - but even the orbit trap formulas for Fractint let you do that in around 1995 or earlier - if you haven't done so already import all the Ultra Fractal formulas into ChaosPro and in 95% of the orbit-trap style formulas you'll find a parameter called "Trap Center" or something similar which accomplishes the same thing - essentially what your suggestion does is trap to a fixed point other than the origin.
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Alef
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« Reply #156 on: December 27, 2011, 06:39:55 PM »

Thanks. Well, chaos pro native formulas are quind of poor. I allredy have bunch of ultra fractal formulas on Chaos pro, but they not so smoothly implements in its interface embarrass Hadn't looked throught all, tested just some, and then went to exponential smoothing as Kali allways sugests;) Well, FE hadn't colour algorithms with variables, so at least Choaso pro users now could download easy to use algorithm.


Continuing with the main thread. Interesting feature is that you can write some random formula without any square function, linear z, some sin(z)-exp(z)+c whatever, just no cubes. There will be some set, but zooming in at some point still would show classical mandelbrot sets, it's even hard to get rid of them. So maybe mandelbrot set lake is something more fundamental than definition in wikipedia sugests.
« Last Edit: December 27, 2011, 06:51:28 PM by Asdam » Logged

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fracmonk
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« Reply #157 on: December 27, 2011, 07:23:36 PM »

Asdam- It's interesting, and I take it that it's an overlay, AND I can see the original formula coming thru in the third pic, but why cloud the point of it visually when we're talking theory here?
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David Makin
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« Reply #158 on: December 28, 2011, 11:33:54 AM »

Thanks. Well, chaos pro native formulas are quind of poor. I allredy have bunch of ultra fractal formulas on Chaos pro, but they not so smoothly implements in its interface embarrass Hadn't looked throught all, tested just some, and then went to exponential smoothing as Kali allways sugests;) Well, FE hadn't colour algorithms with variables, so at least Choaso pro users now could download easy to use algorithm.

To be honest when investigating new ideas I'd recommend doing what I do - always search the UF formula collection for any keywords that are relevant and often a main formula and/or colouring (or even a transform) will turn up that's already using the idea - not necessarily as you intended but often involving essentially the same calculations - and of course you can do this even without UF or with an unregistered copy - for those who are more Fractint oriented than myself doing the same with the Fractint formula collection would probably also be useful.
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fracmonk
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« Reply #159 on: December 28, 2011, 04:45:27 PM »

When it rains, it pours!  I was in the middle of a lengthy report about my disastrous situation when the library computer I was on timed out and booted me off.  Starting over...

I was preparing pix related to post 153 this AM when I opened too many windows at once & froze up my ancient standalone.  Ctrl-alt-del did zip, so I had to reboot, except it refused under any circumstances.  Was earlier getting advice online, and have to consider options with a pro I know.  Worst case, I'll have to pop the main drive, put in another, reinstall XP, & hope to recover data from current main.  I don't think it affected external USB drive where most fractal-related biz takes place.  My backup is geographically remote and a month out of date.

Isn't that just wonderful?

Later!
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fracmonk
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« Reply #160 on: December 29, 2011, 07:27:24 PM »

Did what I said I would in last post, and with the replacement drive, it does the same.  Put a new battery in the motherboard, too, nothing, so I have to make new arrangements that'll take time.  But what I cannot show you now flashes complete in my mind:

(Continuing comment from post 153) The most fundamental necessity for the effects given by Multipowerbrot functions is that the relationship between z and c is multiplicative, as opposed to additive in standard M.

When d=2 in the post 153 funcs., you have f(z)->(((z^4)c-2)^2)-2, and you get larger filamentation than in M, despite the presence of the quartic shape.  In a quartic multibrot and higher powers, there is more "lake" and less filament (including larger and larger minis as powers increase).  This func. above gives Julias w. a 4-sym, while f(z)->((((z^2)c-2)^2)-2)^2 gives the same index set, but the Julias for it are 2-sym (and therefore the critical points are different).  So exploring them in greater detail is still an interesting enough universe on its own to keep me happy.  There are still new and interesting things to be found there.  You have plenty enough info to DIY.

Fare better!

Later.
« Last Edit: December 29, 2011, 07:31:39 PM by fracmonk » Logged
Alef
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« Reply #161 on: December 30, 2011, 03:50:21 PM »

Main reason reason were, that no colour algorithm had all the features I wanted to use animeting quaternion cheesy But that is way offtopic.
Not the overlay, dots appears where lies orbits with exact numbers. In first iteration there are just one dot, in second - two dots, in third - four rising in geometrical progression. Quind of interesting.
« Last Edit: December 30, 2011, 04:25:03 PM by Asdam » Logged

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« Reply #162 on: December 30, 2011, 03:57:42 PM »

Didn't found this very simple variation in UF formula database. Throught I think, I alredy had seen this. Maybe as error.
Code:
z= z^2+c;
z=z + 0.12*z/cabs(z);
This equation decreased size of set, but increased proportions of outside features. 0.12 produced unkempt mandelbrot, 0.25 creates almoust like bunch of grass, and -0,25 gives 'bloated' mandelbrot with minibrot rings on antenna.

EDITED: variation by Edgars Malinovskis.
Wow, this idea is spreading and evolving.


* Mandelgrass_all2.JPG (25.02 KB, 411x360 - viewed 240 times.)

* Mandelgrass_bush2.JPG (22.45 KB, 406x360 - viewed 239 times.)
« Last Edit: February 01, 2012, 07:33:22 PM by Asdam, Reason: Added authorship » Logged

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lkmitch
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« Reply #163 on: December 30, 2011, 06:48:59 PM »

Didn't found this very simple variation in UF formula database. Throught I think, I alredy had seen this. Maybe as error.
Code:
z= z^2+c;
z=z + 0.12*z/cabs(z);

Nice variation!  Here are a Mandelbrot zoom using -2/0.1 as the factor instead of 0.12, and a Julia using 0.1/0.1.  Both use the color of the year for 2012, Tangerine Tango.


* dec30-b.jpg (91.03 KB, 800x600 - viewed 40 times.)

* dec30-a.jpg (81.33 KB, 800x600 - viewed 41 times.)
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Alef
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« Reply #164 on: January 03, 2012, 03:20:35 PM »

The second picture is marvelous, looks very natural. Hmm, shouldn't colour of 2012 be mayan, end of the  world and that stuff fiery Not connected with net I was searching with -0.3 and - 1.


Looks as bloated Mandelbrot;)


This fractal have rings on antenna and minibrot rings on all stalks. Tried the most basic colouring, showing  just as it is.
Code:
else if (formula=="MandelGrassBrot")
{
z= z^2+c;
z=z -0.3*z/cabs(z);   //by Edgar Malinovsky
{

I think, I had alredy seen this feature, but I think I never had seen julia sets alike. Changing -0.3 with -1 and having julia seed x = 1.3 and y = 0.5 revealed pretty interesting julia set consisting of circles. Julia set goes nicely with -1, but then rings on mandelbrot are too dense.

This equation works well with another powers, for example cube. Looks like respective standart julias, but of circles and angles.
« Last Edit: January 03, 2012, 03:27:37 PM by Asdam » Logged

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