fractalforums.com - Calcyman's Idea (2D)

LesPaul

Safarist

Posts: 47

Calcyman's Idea (2D)

« 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.

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 8 times.)
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LesPaul

Safarist

Posts: 47

Re: Calcyman's Idea (2D)

« Reply #1 on: January 23, 2010, 12:36:39 AM »

and cos(z) + z₀ looks like this:

cos.png (97.04 KB, 640x480 - viewed 5 times.)
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LesPaul

Safarist

Posts: 47

Re: Calcyman's Idea (2D)

« 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 10 times.)
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LesPaul

Safarist

Posts: 47

Re: Calcyman's Idea (2D)

« 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 6 times.) z_1_sin_zoom_1.png (235.35 KB, 1280x320 - viewed 9 times.) z_1_sin_zoom_2.png (281.28 KB, 1280x960 - viewed 11 times.)
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LesPaul

Safarist

Posts: 47

Re: Calcyman's Idea (2D)

« 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 7 times.) z_2_sin_zoom_1.png (285.61 KB, 640x480 - viewed 7 times.) z_2_sin_zoom_2.png (46.66 KB, 640x480 - viewed 7 times.)
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LesPaul

Safarist

Posts: 47

Re: Calcyman's Idea (2D)

« 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.

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 5 times.) z_8_cos_zoom_1.png (58.8 KB, 640x480 - viewed 5 times.)
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Timeroot

Fractal Fanatic

Posts: 258

The pwnge.

Re: Calcyman's Idea (2D)

« 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

I really like your picture z_1_sin_zoom_2.png, by the way!


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Paolo Bonzini

Conqueror

Posts: 68

Re: Calcyman's Idea (2D)

« Reply #7 on: January 23, 2010, 09:34:34 AM »

Quote from: Timeroot on January 23, 2010, 03:32:54 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).

Quote from: Timeroot on January 23, 2010, 03:32:54 AM

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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Posts: 45

Re: Calcyman's Idea (2D)

« 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

Fractal Bachius

Posts: 542

Re: Calcyman's Idea (2D)

« Reply #9 on: January 23, 2010, 01:51:46 PM »

very nice experiments cheesy

They look great

Just an other thought:

what's about doing $z^n*cos(z)+i*z^k*sin(z)+z0$ ?


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bib

Fractal Phenom

Posts: 437

Re: Calcyman's Idea (2D)

« Reply #10 on: January 23, 2010, 02:13:22 PM »

here is a test using z*tan(z)

ztanz.jpg (306.09 KB, 1024x768 - viewed 26 times.)
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kram1032

Fractal Bachius

Posts: 542

Re: Calcyman's Idea (2D)

« Reply #11 on: January 23, 2010, 02:19:01 PM »

O.o How's that coloured?
Really nice!


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bib

Fractal Phenom

Posts: 437

Re: Calcyman's Idea (2D)

« 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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Nahee_Enterprises

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Global Moderator
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email contact is best

Re: Calcyman's Idea (2D)

« Reply #13 on: January 23, 2010, 03:58:09 PM »

Quote from: LesPaul 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

<Quoted Image Removed>

This produces the attached images.

These "z_1_sin" images are quite interesting.


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LesPaul

Safarist

Posts: 47

Re: Calcyman's Idea (2D)

« Reply #14 on: January 23, 2010, 06:17:21 PM »

Quote from: bib on January 23, 2010, 02:13:22 PM

here is a test using z*tan(z)

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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