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 Author Topic: Is there anything novel left to do in M-like escape-time fractals in 2d?  (Read 25595 times) Description: I think there may be. Your opinions are greatly desired. 0 Members and 1 Guest are viewing this topic.
fracmonk
Fractal Fertilizer

Posts: 356

 « on: December 08, 2010, 04:19:59 PM »

This, for now, is a personal survey.  I'll be making some posts of a few peculiar items I've found over my journeys on the parameter plane in the future here, where theory is properly discussed.  But feel free to beat me to the punch...

Later...

>>>>>>>>>>>>>>>>>>>>A LATE NOTE FOR NEWCOMERS TO THIS THREAD, 2/3/2012:

I encourage you to examine the relatively SIMPLE mathematics of my own offerings here, which chronicle the discovery and early evolution of a new and significant fractal type, now called Multipowerbrot.

There is, I sense, a battle for the heart and soul of fractal geometry, with those who favor graphic sophistication and polish on one side, and simplicity of math and presentation on the other.  I am on the latter side.  For the mathematically oriented, studying my entries in detail, in a forensic sort of way, should be a fairly rewarding experience for those interested in the workings of the Mandelbrot set and its closest relatives.

Enjoy!
 « Last Edit: February 06, 2012, 07:30:42 PM by fracmonk, Reason: perspective » Logged
lkmitch
Fractal Lover

Posts: 204

 « Reply #1 on: December 09, 2010, 04:08:24 PM »

I remember when Cliff Pickover asked essentially the same thing back in the late '80s/early '90s.  To me, yes, there's not much big novel stuff going on or left to do, but, being that fractals are math, there are always new math ideas to incorporate.  So, I hope that there will always be novel things going on with even the "lowly" 2D M-set.
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fracmonk
Fractal Fertilizer

Posts: 356

 « Reply #2 on: December 09, 2010, 07:40:15 PM »

Lowly?  It still impresses ME.  Last year, I went on this journey that BEGAN w. f(z)->zc+1/zc.  Did index sets for f(z)->(za+b) + (1/(za+b)) where b is a nonzero constant.  On the complex *a* param. plane, interesting M-like things emerge outside the unit circle.  Stuck for pix right now, on a public computer w. disabled drives & time limits.  *b* plane for fixed *a* is kinda neat, too.  Will post pix under better circumstances.
 « Last Edit: December 20, 2010, 07:05:12 PM by fracmonk, Reason: clarity » Logged
reesej2
Guest
 « Reply #3 on: December 10, 2010, 05:56:27 AM »

There's definitely a lot more to be done with M-type 2D fractals. Even the good old Mandelbrot has more to be coaxed out of it! For one thing, there's the matter of optimization--I for one desperately want a faster algorithm for computing the Mandelbrot. From a more picturesque point of view, there's always a new way of coloring the Mandelbrot that reveals new details.

One big question I've been pondering lately is strongly applicable to M-type 2D fractals, though also to 3D ones--how does the shape arise from the formula? How well can we "steer" the shape?

That was a very long-winded way to say: YES, there is definitely something novel to do.
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fracmonk
Fractal Fertilizer

Posts: 356

 « Reply #4 on: December 10, 2010, 04:26:24 PM »

reesej2- On "steering" the shape, oh yeah, there's lots of things I've called Mandelbrot gymnastics, involving alternative funcs., but I could never really predict what I'd get. Later, I'd see the results and say: "yeah, that makes sense..."

On 3d, I don't know if you're familiar with my* (let's call it) "notquat" commutative 4d space.  (under 3d on this site).  I'm starting to get a clearer idea of the shape there myself, despite some horrendous errors on the way...

>I need to add about my last post that the params a and b in that formula are both COMPLEX ones.<

Maybe they will look familiar.  I thought I've seen such things out on the web in the past.  I'll show examples when I can.

Happy Computing!

*--As if it doesn't belong to everyone.  I just FOUND it...
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lkmitch
Fractal Lover

Posts: 204

 « Reply #5 on: December 10, 2010, 06:21:01 PM »

One big question I've been pondering lately is strongly applicable to M-type 2D fractals, though also to 3D ones--how does the shape arise from the formula? How well can we "steer" the shape?

The shape of the main cardioid arises from describing the region containing orbits that settle down to a final fixed point:  Require z2 + c = z to get the interior of the region and also require |2z| = 1 to get the boundary.  The second condition is the neutral stability criterion: |df/dz| < 1 for stable orbits (interior of the cardioid) > 1 for unstable orbits (Misiurewicz points on dendrites) and = 1 for the boundary of the cardioid.

How the other shapes arise is left as an exercise for the reader.
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reesej2
Guest
 « Reply #6 on: December 10, 2010, 08:57:50 PM »

@Ikmitch: Yes, I'd found that information, but it's not as helpful as it seems. That allows us to, given a formula, compute the general shape it yields. What it most certainly doesn't allow us to do is backsolve--given a certain general shape, how can we construct a formula that will produce it? I've been working on finding a way to control the relative positions and shapes of the period-1 and period-2 bulbs independently, but that's much easier said than done. The results I'm getting involve solving nonlinear systems with dozens of variables, under the modest assumption that the formula is quadratic in both coordinates.
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Thunderwave
Guest
 « Reply #7 on: December 11, 2010, 07:04:36 AM »

If you look at the M-set with a simple point of view, yes, there's nothing new...

But! As I have been steadfastly studying the M-set, I am finding value in prediction and analysis; patterns that exist within patterns of zooming.  I find predictions in what is to come, based on what I zoom in on.  I find new ways to create patterns using different zooming patterns.  I have been logging all my work privately since not many care about the process of personal research of a probably over-researched fractal algorithm; yet nonetheless, I go on and spurt out a little thought once and a while about a fascinating bug called the M-set.
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Tater
Explorer

Posts: 55

 « Reply #8 on: December 11, 2010, 06:12:12 PM »

This, for now, is a personal survey.  I'll be making some posts of a few peculiar items I've found over my journeys on the parameter plane in the future here, where theory is properly discussed.  But feel free to beat me to the punch...

Later...

I have not been here long enough to know if this has been discussed already, but there is probably a real 2d mandelbrot analog to the mandelbulb of the form $v_{n}= \rightarrow v_{n+1}=+$ where p and q are real or integer multipliers of the rotation angle.

Here are some examples: http://kernsholler.net/KHJS/RealMandel/RealMandelbrot.html

Tater
 « Last Edit: December 11, 2010, 10:07:27 PM by Tater, Reason: add url » Logged
Tater
Explorer

Posts: 55

 « Reply #9 on: December 11, 2010, 06:18:27 PM »

I have not been here long enough to know if this has been discussed already, but there is probably a real 2d mandelbrot analog to the mandelbulb of the form <Quoted Image Removed> where p and q are real or integer multipliers of the rotation angle.

And it would have a three and four dimensional version by extending the parameters p and q as new dimensions.

Tater
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Tater
Explorer

Posts: 55

 « Reply #10 on: December 12, 2010, 08:59:42 PM »

I have not been here long enough to know if this has been discussed already, but there is probably a real 2d mandelbrot analog to the mandelbulb of the form <Quoted Image Removed> where p and q are real or integer multipliers of the rotation angle.

Here is a gallery of images made by this method

http://www.aurapiercing.com/gallery/main.php?g2_itemId=143

Tater
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fracmonk
Fractal Fertilizer

Posts: 356

 « Reply #11 on: December 14, 2010, 07:33:43 PM »

Reesej2-  Years ago, Barnsley claimed to have a program that could backsolve a formula, but I had my doubts about it being able to come up with more than an approximation of the original one.  It was related to image compression in general.

Have to take other responses back for a read.  Later!
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fracmonk
Fractal Fertilizer

Posts: 356

 « Reply #12 on: December 15, 2010, 02:32:01 PM »

(via bunneemail) -pix were posted by my "agent" in advance on a machine that allows uploads, unlike the one I use now...

1st is of f(z)->(za+b)+(1/(za+b)), where b=0 and the view is of a param. plane.  Critical pts are z=2, z=-2.

2nd has -2 @ its ctr. for b=-7. b<-7 results in separation into 2 objects.

3rd is of b param. plane, where a=1, ctr. is b=-1, & starting z=2.

4th: a=-1, & ctr. is b=2  This pic borders on multiple connectedness.
Since division often involves wild gyrations, I got in the habit of setting hi escape values, which FractInt often seems to reject arbitrarily.  Here, is was 127.9.  Still there is "glare" of quickly escaping points about what would normally be terminating Misiurewicz (sp?) pts.

Many of you HAD to have seen this stuff before.

Thought of posting a teaser suggesting where this story leads, but I don't want to give away the ending.
In this tale, it's important that the journey is retraced.
 A0.GIF (29.95 KB, 320x200 - viewed 1520 times.)  Abn7.GIF (9.49 KB, 320x200 - viewed 1518 times.)  Ba1.GIF (14.81 KB, 320x200 - viewed 1638 times.)  Ban1.GIF (15.11 KB, 320x200 - viewed 1516 times.) « Last Edit: December 20, 2010, 07:06:31 PM by fracmonk, Reason: clarity » Logged
jwm-art
Conqueror

Posts: 132

 « Reply #13 on: December 15, 2010, 03:42:28 PM »

If you look at the M-set with a simple point of view, yes, there's nothing new...

But! As I have been steadfastly studying the M-set, I am finding value in prediction and analysis; patterns that exist within patterns of zooming.  I find predictions in what is to come, based on what I zoom in on.  I find new ways to create patterns using different zooming patterns.  I have been logging all my work privately since not many care about the process of personal research of a probably over-researched fractal algorithm; yet nonetheless, I go on and spurt out a little thought once and a while about a fascinating bug called the M-set.

I agree with you Thunderwave. We have barely scratched the surface of the Mandelbrot set. We will only ever be able to scratch about on it's surface, no matter how powerful our computers are and the ever expanding depths they allow us to zoom into. Infinity is infinite!

If you take this view, that we have barely scratched the surface, then you can imagine that beyond the depths we don't have patience to compute, there could very well be new patterns of zooming to explore. Take for instance zooming into embedded Julia Sets. There are certain steps you must take to be able to perform certain zooms.

At this moment in time, I've reached another plateau in my knowledge of methods to zoom into the Mandelbrot Set.

At the same time, still regarding ways of zooming, it's not that I'm finding more ways of zooming into the Mandelbrot Set, it's more like I'm integrating the many into the few.
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fracmonk
Fractal Fertilizer

Posts: 356

 « Reply #14 on: December 16, 2010, 02:38:24 PM »

(via bunneemail)

With the previous pix, you get a squaring effect without squaring due to the difference between highest (1)and lowest (-1) exponents.

After a zoom into the a param. plane while b=-2, you arrive at the 1st pic below.
The next is of the julia set for the ctr. of the last.  It's ctrd. on the critical point z=2, and this point is not visually obvious in the assymmetrical julia set.  The critical point is magnified greatly in the last 2 pix.  V. noisy, this spot...

->Please excuse the amount of time it may take to get to the next page of this thread, as it runs rather long, with many pictures.
(Really good stuff though, I think...)   (1-11-2011)
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