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Author Topic: Calcyman's Idea (2D)  (Read 13292 times)
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LesPaul
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« on: January 23, 2010, 12:35:16 AM »

Hi all,

Calcyman had an interesting idea regarding the 3D Mandelbulb.  His original post is here: http://www.fractalforums.com/index.php?topic=2354.msg10836#msg10836

His idea was to include all the "higher orders" when calculating the Mandelbrot (Mandelbulb) set, and he came up with a clever way to do it that can be calculated efficiently.  His example started with:

   z_{n+1} \: = \: {z_n}^8 \: + \: z_0

... which is the basic form of the Mandelbulb.  Then he added in higher orders, like so:

   q_{n+1} \: = \: z_0 \: + \: {z_n}^{8} \: - \: \frac{{z_n}^{10}}{2!} \: + \: \frac{{z_n}^{12}}{4!} \: - \: \frac{{z_n}^{14}}{6!} \: + \: \frac{{z_n}^{16}}{8!} \: + \: \cdots

The idea is for each term to contribute less as the exponent gets higher.  So, each term is divided by a factorial, which gets large very quickly, and the really high-order terms become negligible.  But the clever thing about this is that this series is actually just the MacLaurin series expansion for {z^8}\cdot{cos(z)}.  So, his formula can be calculated just like the original with one additional multiplication by a cosine term.

When I saw his post, the question that came to my mind was whether this had been studied in the 2D realm of the original Mandelbrot.  I started some very basic research and found some interesting things lurking within that magical complex plane.  smiley

I decided to generalize his idea to be "z raised to any power, multiplied by any function of z, plus z0."

First, let me paste the UltraFractal formula that I used, in case anyone else is interested in experimenting further.  The formula is so simple that I'm nearly certain this has been done before, to some extent.

Code:
MandelTimesFunc {
init:
  z = 0
loop:
  z = z^@power * @Function(z) + #pixel
bailout:
  |z| < @bail
default:
  title = "Complex Power Times Complex Function"
  float param power
    caption = "Power"
    hint = "The power to which Z is raised.  Set to zero to reduce calculation to just Function(Z)."
    default = 2.0
  endparam
  float param bail
    caption = "Bailout"
    default = 1e6
  endparam
}

That will allow you to choose any function you like (as long as it's supported by UltraFractal) and multiply it by z raised to any power, in other words:

   z_{n+1} \: = \: {{z_n}^p} \: \cdot \: {function(z)} \: + \: z_0

As the "hint" suggests, you can also set the power to zero, reducing the calculation to simply:

   z_{n+1} \: = \: {function(z)} \: + \: z_0

So, I began by looking at just sin(z) and cos(z), without multiplication by a power of z.

sin(z) + z0 looks like this:


* sin.png (85.08 KB, 640x480 - viewed 364 times.)
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LesPaul
Guest
« Reply #1 on: January 23, 2010, 12:36:39 AM »

and cos(z) + z0 looks like this:


* cos.png (97.04 KB, 640x480 - viewed 350 times.)
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LesPaul
Guest
« Reply #2 on: January 23, 2010, 12:38:38 AM »

These images are interesting and definitely have some fractal-like qualities.  In fact, if you zoom in on the cosine version, you can find some familiar things!


* cos_zooms.png (446.03 KB, 1280x960 - viewed 547 times.)
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LesPaul
Guest
« Reply #3 on: January 23, 2010, 12:47:55 AM »

I continued by getting back to the original idea and including a power of z in the calculation.  Things become interesting even with a power of one, giving

   z_{n+1} \: = \: z_n \: \cdot \: {sin({z_n})} \: + \: z_0

This produces the attached images.  "z_1_sin.png" shows the bulk of the set, although it appears to continue quite far in both directions.  "z_1_sin_zoom_1.png" is a zoom into the flat-ish area between the two main "bulbs" shown in the original.  Surprise!  Another minibrot is hiding there, along with some other shapes that are distinctly not similar to the Mandelbrot.  "z_1_sin_zoom_2.png" is a further zoom into the same area showing some really surprising features.


* z_1_sin.png (62.1 KB, 640x480 - viewed 343 times.)

* z_1_sin_zoom_1.png (235.35 KB, 1280x320 - viewed 374 times.)

* z_1_sin_zoom_2.png (281.28 KB, 1280x960 - viewed 667 times.)
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LesPaul
Guest
« Reply #4 on: January 23, 2010, 12:56:19 AM »

The next experiment was to increase the power of z to 2:

   z_{n+1} \: = \: {z_n}^2 \: \cdot \: {sin({z_n})} \: + \: z_0

This gives the attached images and zooms.  This set is interesting because it has a lot of characteristics of the basic Mandelbrot.  There are many, many instances of smaller, rotated/skewed versions of the original (what we would call "minibrots" or "midgets" in the the Mandelbrot).  These midgets appear to be far more dense in this set than in the Mandelbrot.  But interestingly, there are also some parts of the set that are not self-similar.  I don't believe I've ever seen a part of the Mandelbrot set that was actually inside the set that didn't look similar to the original set.

The "stem" area of this set looks rather non-interesting at first, but there are some really bizarre shapes hiding there.


* z_2_sin.png (44.79 KB, 640x480 - viewed 372 times.)

* z_2_sin_zoom_1.png (285.61 KB, 640x480 - viewed 397 times.)

* z_2_sin_zoom_2.png (46.66 KB, 640x480 - viewed 373 times.)
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LesPaul
Guest
« Reply #5 on: January 23, 2010, 01:01:26 AM »

There is a lot more experimentation that I'd like to do, but there's only so much time in a day.  smiley  One last thing I tried was Calcyman's original suggestion, which was the eigth-order cosine version:

   z_{n+1} \: = \: {z_n}^8 \: \cdot \: {cos({z_n})} \: + \: z_0

That gives the attached image (and zoom), which really does have the "feel" of the Mandelbulb.  Here again, many parts are self-similar and many parts are not.  It's all interesting, to me, anyway!


* z_8_cos.png (41.95 KB, 640x480 - viewed 429 times.)

* z_8_cos_zoom_1.png (58.8 KB, 640x480 - viewed 368 times.)
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Timeroot
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« Reply #6 on: January 23, 2010, 03:32:54 AM »

I think that, yes, much experimentation has been done with other functions such as sin(z) or cos(z), but I'd never seen z^n * fn(z). Nice images! I'm not sure if you know, but the shape featured in z_2_sin.png is a multibrot; the shape that appears from iterating z=z^3 + z0, instead of z^2. I find it interesting that this appears in your z^2*sin(z) version. The object to the right in  z_2_sin_zoom_2.png is Julia set. Nothing new, really, to see that hiding there, but the interesting thing is that it's filled in - as far as I know, that never happens in the regular M-Set. For the record, it can be easily shown that any formula of the form z=z^n * sin(z)  +  z0    or z=z^n * cos(z)   +   z0 extends out to infinity in both directions.  tongue stuck out I really like your picture z_1_sin_zoom_2.png, by the way!
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Someday, man will understand primary theory; how every aspect of our universe has come about. Then we will describe all of physics, build a complete understanding of genetic engineering, catalog all planets, and find intelligent life. And then we'll just puzzle over fractals for eternity.
Paolo Bonzini
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« Reply #7 on: January 23, 2010, 09:34:34 AM »

I'm not sure if you know, but the shape featured in z_2_sin.png is a multibrot; the shape that appears from iterating z=z^3 + z0, instead of z^2. I find it interesting that this appears in your z^2*sin(z) version.

That's because z^2*sin(z) = z^3+O(z^5).

The object to the right in  z_2_sin_zoom_2.png is Julia set. Nothing new, really, to see that hiding there, but the interesting thing is that it's filled in - as far as I know, that never happens in the regular M-Set.
Yes, that's very interesting!
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matsoljare
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« Reply #8 on: January 23, 2010, 12:39:54 PM »

Don't forget to try the "julibrot" projections as well!
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kram1032
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« Reply #9 on: January 23, 2010, 01:51:46 PM »

very nice experiments cheesy

They look great smiley

Just an other thought:

what's about doing z^n*cos(z)+i*z^k*sin(z)+z0?
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bib
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« Reply #10 on: January 23, 2010, 02:13:22 PM »

here is a test using z*tan(z) smiley


* ztanz.jpg (306.09 KB, 1024x768 - viewed 432 times.)
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kram1032
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« Reply #11 on: January 23, 2010, 02:19:01 PM »

O.o How's that coloured?
Really nice!
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bib
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« Reply #12 on: January 23, 2010, 02:40:38 PM »

There are 2 layers : one with smooth iterations in yellow and blue, with a steep variation in gradient to make the patterns inside the big bud appear, and another with orbit traps to highlight the minibrots.
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« Reply #13 on: January 23, 2010, 03:58:09 PM »

I continued by getting back to the original idea and including a power of z in the calculation. 
Things become interesting even with a power of one, giving

   <Quoted Image Removed>

This produces the attached images.

These "z_1_sin" images are quite interesting.   smiley
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LesPaul
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« Reply #14 on: January 23, 2010, 06:17:21 PM »

here is a test using z*tan(z) smiley

WOW!!  I will definitely experiment with tan(z) having seen that image!

The MacLaurin series for tan(x) is interesting.  It is identical to sin(x) except for two things:
1) the tan(x) series has no negative terms, and
2) the coefficients of the tan(x) series are larger.

   sin(x) \: = \: x \: - \: \frac{x^3}{6} \: + \: \frac{x^5}{120} \: - \: \frac{x^7}{5040} \: + \: \cdots

   tan(x) \: = \: x \: + \: \frac{x^3}{3} \: + \: \frac{2x^5}{15} \: + \: \frac{17x^7}{315} \: + \: \cdots

This makes me think that inverse-tangent is probably also interesting:

   tan^{-1}(x) \: = \: x \: - \: \frac{x^3}{3} \: + \: \frac{x^5}{5} \: - \: \frac{x^7}{7} \: + \: \cdots
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