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Author Topic: Is there anything novel left to do in M-like escape-time fractals in 2d?  (Read 32691 times)
Description: I think there may be. Your opinions are greatly desired.
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fracmonk
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« Reply #90 on: March 17, 2011, 01:39:18 PM »

(pix via bunny express)

makc-  I apologize sincerely!  For days, I'd been thinking that I had not been fair with you.  Maybe I've let my obsession with simple and multiple connectedness in M-like sets narrow my notion of "M-like" too much.  Universality of M is very strong in the complex parameter plane.  Too often, when I search for critical points, I'll miss, and YET, still see M all over...what you often see is still M-like even if its not what you're looking for.

I don't know if anyone has ever fully explained why there are discontinuities in objects made from functions with non-integer exponents of z.  It would be nice to know.

Also, is there a simple way to summarize what Perlin noise is, how it works, and what it can tell us?



The storyboard series continues below with d=-1.75, then d=-1.5, another pinch point, and d=-1.25, which corresponds to the center of the main bulb in M.  More next time...

Later!




* Ls4.GIF (6.9 KB, 320x200 - viewed 347 times.)

* Ls5.GIF (6.82 KB, 320x200 - viewed 340 times.)

* Ls6.GIF (6.47 KB, 320x200 - viewed 347 times.)
« Last Edit: March 17, 2011, 06:22:43 PM by fracmonk » Logged
fracmonk
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« Reply #91 on: March 18, 2011, 01:54:52 PM »

(pix via bunny express)

The progression continues below with d=-1, another pinch point, d=-.75, and d=-.5, the last 2 corresponding to the cardioid interior.


Got potassium iodide?

I'm going with that old standby, the tin-foil hat.  I can make a spiffy one in less than a minute, so...

I'm good.


Have a non-ionizing weekend, if you can manage it.

Later!



* Ls7.GIF (6.4 KB, 320x200 - viewed 328 times.)

* Ls8.GIF (6.11 KB, 320x200 - viewed 327 times.)

* Ls9.GIF (5.84 KB, 320x200 - viewed 329 times.)
« Last Edit: March 18, 2011, 03:14:49 PM by fracmonk » Logged
makc
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« Reply #92 on: March 20, 2011, 10:45:08 PM »

Also, is there a simple way to summarize what Perlin noise is, how it works, and what it can tell us?
it's just to make random but continuous function. Perlin was the 1st guy who published an algorithm for that, IIRC. check out wikipedia if you're interested.
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fracmonk
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« Reply #93 on: March 21, 2011, 01:56:33 PM »

(pix via bunny express)

makc-  I did, recently.  Originally though, my first reaction was instinctual, that a CG overlay of pseudorandom noise on M would only serve to OBSCURE discussion of theory here.  M-like sets are exacting and deterministic, and they're about how numbers behave.  Why smear up the view?

Perlin noise is a different KIND of fractal phenomena, and I don't think that imposing it on M will enhance understanding of either.

Further, if one defines "M-like" too broadly, the concept becomes meaningless, especially for our purposes here, which is to discuss how they work.

CG visual effects imposed on M, if they make you happy, can always be discussed in their own thread.  I don't think they help here, really, but that's just my opinion, of course.

If you want to figure out what qualifies as M-like, however, this is the place.  THAT may need consensus...


The exploration into real d in the series concludes in the pix below.  The 1st, d=-.25, still corresponds to the interior of the cardioid.  When d=0, as in the 2nd pic, you should recognize the lambda function.  As d increases from zero, as in the last pic, d=.25, the split widens, w. the resulting halves becoming ever smaller.

Next time, a small peek into imaginary d.

Later.



* Ls10.GIF (5.72 KB, 320x200 - viewed 304 times.)

* Ls11.GIF (5.81 KB, 320x200 - viewed 304 times.)

* Ls12.GIF (5.68 KB, 320x200 - viewed 303 times.)
« Last Edit: April 08, 2011, 04:09:06 PM by fracmonk, Reason: typo » Logged
makc
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« Reply #94 on: March 21, 2011, 03:55:21 PM »

Future posts will continue the series in .25 increments, while a "smooth" animation would require significantly smaller ones, say, .01 per frame.
Why not calculate at .01 and make a movie? It's fairly easy to do with a tool like virtualdub once you get your images numbered (000.jpg, 001.jpg, 002.jpg and so on). Would save you a lot of posts smiley Also, what's with putting text some time after the post? This way only people who come at right time are able to read it.
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fracmonk
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« Reply #95 on: March 21, 2011, 06:32:54 PM »

makc-  My bunny posts in advance because I cannot where I am.  It was explained earlier.  Uploads and downloads are restricted too.
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makc
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« Reply #96 on: March 21, 2011, 07:35:36 PM »

that a CG overlay of psuedorandom noise on M would only serve to OBSCURE discussion of theory here.  M-like sets are exacting and deterministic, and they're about how numbers behave.  Why smear up the view?
the idea was to be able to see images of multiple exponents M-sets simultaneously, but it did not work out like I expected. so consider that as a dead-end xperiment branch.
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fracmonk
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« Reply #97 on: March 22, 2011, 01:59:01 PM »

(pix via bunny express)

The lambda 3d series shown recently can be extended to 4d when complex d is used.  Below are pure imaginary values for d=.25i, d=.5i, and d=.75i, respectively.  Separation of continental features takes place at approx. |d imag|>0+,-.569946625...i.  There may be a greater imaginary value for d that connects dendritically, but I'll never find it...


M has distinct and well-documented properties.  I propose that M-like sets, to be considered that, must have these, in approximate order of importance:

1. well-defined prisoner and escape sets
2. connectedness
3. geometric symmetry
4. infinite perimeter
5. finite area (in 2d)
6. repeated fractal shapes
7. a hierarchical structure
8. scale-dependent features (so-called theme & variation)

I think all of those may be necessary.  Any to add or subtract?  Your critique very welcome.


makc- Dead ends- I'm not sticking my neck out too far to say WE get a lot of those.  I know I do.  But you keep looking.  There's a lot of dry times, but I gotta say I've been hot lately- at least in terms of what I'd been looking for, -for AGES...

Happy hunting!

Later!


* Ls13.GIF (5.99 KB, 320x200 - viewed 295 times.)

* Ls14.GIF (6.39 KB, 320x200 - viewed 294 times.)

* Ls15.GIF (6.28 KB, 320x200 - viewed 286 times.)
« Last Edit: March 23, 2011, 03:06:31 PM by fracmonk, Reason: adjust value... » Logged
fracmonk
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« Reply #98 on: March 23, 2011, 03:17:18 PM »

I've been working on that overview of nesting formula permutations, and it's been a very fruitful exercise, as some interesting behavioral patterns have revealed themselves.  I think it promises better insights, so that's what I'm doing.  There is also a bunch of documentation I have to catch up with.  This stuff seriously needs its own website, or maybe a few papers or a book, but none of that promises to happen very soon.  As the season here warms, other more personal responsibilities will allow me less time for this work.  "Time.", he moans sadly...

But I'll be in touch, and report things that seemingly just can't wait.

Later.
« Last Edit: March 28, 2011, 07:17:04 PM by fracmonk, Reason: \"is\" » Logged
fracmonk
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« Reply #99 on: March 31, 2011, 07:14:37 PM »

The work is going well, but unfortunately, not fast enough for the hordes of eager, brilliant fellow researchers clamoring to discuss the theories behind it...

I forgot, it's naptime.  It's always naptime now.

Later.
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fracmonk
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« Reply #100 on: April 01, 2011, 04:26:18 PM »

I research many things besides fractal math, as you might have guessed.  Some are even of a serious and sensitive nature, but I digress...

While incredibly off-topic, the following is loosely based on a true story (and somehow connected):

In 1960, it was decided that men would be sent to the moon and returned safely to earth within the decade.  Assuming a high probability that it actually happened as reported, one expert on space exploration more recently stated that if we chose to repeat this exercise today, it would cost much, much more and take about 20 years to accomplish.  (...with more advanced technology...)

When the 1st Earth Day was declared, it immediately occurred to me that every day was Earth Day, unless, of course, you were doing lots and lots of drugs.  I often feel the same way about April Fools' Day (today), especially lately, when the late departure of winter where I live was attributed to global warming.

So today is as good as any to make predictions:
The international standard for "normal background radiation" will have to be increased to take into account pollution from recently damaged reactors in Japan, by which all additional fallout from future nuclear disasters will be measured.  Monitoring of new cancer cases close to such sites will end sooner than the 10 years after the 3 Mile Island accident, after which it gets a little embarrassing for plant operators.

And let's hear it for international banking, always a day late and 60 trillion dollars short.

Oil commodity speculators attempting to falsify Mandelbrot's price curve theory are reminded to send Col. Khaddaffi a thank-you card.

It's only 54 days until Towel Day is celebrated again.  Don't panic.

Thank you for your indulgence.

Later.
« Last Edit: April 01, 2011, 04:43:29 PM by fracmonk, Reason: sloppiness » Logged
makc
Strange Attractor
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Posts: 272



« Reply #101 on: April 03, 2011, 12:50:53 AM »

have you seen this video btw

<a href="http://www.youtube.com/v/9mTSdAtlhRY&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/9mTSdAtlhRY&rel=1&fs=1&hd=1</a>

it's along the lines of what you were doing here
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fracmonk
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Posts: 356


« Reply #102 on: April 04, 2011, 07:17:09 PM »

In the aftermath of Fools' Day, my "Violent Torpedo of Publicity Tour" continues, in truth, with due deference to C.S.

Wish You Were Here.


makc- Saw the animation.  I figure that the exponent changes incrementally, but did stigomaster specify whether init. z does too?
Do you know the formula?  Also, did you notice how the disconnected (dendritic) bits at some point can connect in its 3rd dimension? 
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fracmonk
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Posts: 356


« Reply #103 on: April 05, 2011, 07:12:22 PM »

makc-  Thought I'd remind you that I'm almost always accessing via library computer.  So there's no sound, if the animation had any.  To discuss theory, explanation to accompany pix, etc., is very desirable, and best as text.
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makc
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Posts: 272



« Reply #104 on: April 05, 2011, 08:06:27 PM »

from video description:
Quote
Based on the formula:
z = a * z^3 + b * z^2 + c * z + seed
This animation changes nothing but a, b and c from -1 to 1, back and forth.

p.s. "I'm almost always accessing via library computer" - that's too bad. Obama has his own computer, and you should, too

<a href="http://www.youtube.com/v/miaAC3d4RDU&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/miaAC3d4RDU&rel=1&fs=1&hd=1</a>
« Last Edit: April 05, 2011, 08:09:04 PM by makc » Logged
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