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Author Topic: Simonbrot nth  (Read 2571 times)
Description: Does anybody have experience zooming into a Simonbrot with powers not 4 or 6?
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greentexas
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« on: November 06, 2016, 02:00:33 PM »

Hello, I'm wondering whether anyone has any experience zooming into unfamiliar powers of the Simonbrot.

I know the formula for the Simonbrot 4th as z2 * |z|2 + pixel.

The formula for the 6th Simonbrot is z3 * |z|2 + pixel.

Does anybody have experience zooming into fractals like z * |z|2 + pixel (Quadratic Simonbrot) or z4 * |z|2 + pixel? (8th Simonbrot?)

If so, notify me.
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TheRedshiftRider
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« Reply #1 on: November 06, 2016, 02:14:46 PM »

Yes they have been implemented (exept for the 8th) with perturbation into the program Kalles Fraktaler:
http://chillheimer.de/kallesfraktaler/
http://www.fractalforums.com/kalles-fraktaler/kalles-fraktaler-2-11/

I do not have a lot of experience with zooming into it. But there are others here who do.
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greentexas
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« Reply #2 on: November 06, 2016, 07:46:34 PM »

Yes they have been implemented (exept for the 8th) with perturbation into the program Kalles Fraktaler:
http://chillheimer.de/kallesfraktaler/
http://www.fractalforums.com/kalles-fraktaler/kalles-fraktaler-2-11/

I do not have a lot of experience with zooming into it. But there are others here who do.

You are correct. There is a fourth and sixth power Simonbrot in Kalles Fraktaler. But I'm not aware of any 2nd degree Simonbrot in Kalles Fraktaler.
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TheRedshiftRider
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« Reply #3 on: November 06, 2016, 08:01:49 PM »

You are correct. There is a fourth and sixth power Simonbrot in Kalles Fraktaler. But I'm not aware of any 2nd degree Simonbrot in Kalles Fraktaler.
I must have read your post incorrectly. At this point those variants are not implemented at this point. Would indeed be cool to see them implemented.
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greentexas
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« Reply #4 on: November 06, 2016, 10:47:00 PM »

It would also be nice to see Simonbrot 3rd, 5th, and 7th, but the formulas for these fractals are z1.5 * |z|2 + pixel, z2.5 * |z|2 + pixel, and z3.5 * |z|2 + pixel.

Due to the fact that the powers of some of the terms are fractions, this could be difficult to implement. In a roundabout way, there is a power zero Simonbrot on Kalles Fraktaler, because the power zero Simonbrot is actually the Burning Ship.
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simon.snake
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simon.fez SimonSideBurns
« Reply #5 on: November 07, 2016, 10:37:46 PM »

I have to say I find it really weird (in a nice way) that there is all this talk of the simonbrot.

It is a little surreal to have people discussing something I found.

It may well have been found before I discovered it - there were thousands of formulas created for the formula parser in FractInt.

Anyway, do not stop talking about it just because I find it weird.  There probably are a whole load of different variants of the base formula still waiting to be discovered.

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greentexas
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« Reply #6 on: November 08, 2016, 12:48:02 AM »

I actually found the Simonbrot strange when I first saw it (when it was added to Kalles Fraktaler). I have no evidence of anybody seeing it before that time.

Also, I would think of this formula creating something of a Simon-buffalo:

|z2 * |z|2| + pixel

because the formula for the Buffalo is |z2| + pixel.
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simon.snake
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simon.fez SimonSideBurns
« Reply #7 on: November 08, 2016, 02:26:08 PM »

I was playing in FractInt last night and found that (in FractInt formula parser syntax):

z = abs(z) * z + c        followed by
z = z * z + c

(so only slightly different from the base formula by way of a +c on the first line)

Produces another similar fractal but with mandelbrot style minibrots when zoomed in.

I don't know how you'd code that into your convention.

Strange.
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greentexas
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« Reply #8 on: November 08, 2016, 11:59:30 PM »

The formula for that fractal, as far as I can tell, is: z2 * |z| + c. In |z| power, this formula appears to be something of a hybrid between the Mandelbrot and the Simonbrot.

In a round-about way, this is the Mandelbrot formula:

z2 * |z|0 + c.

(z0 is always 1, and don't forget the identity property of multiplication!)

The Simonbrot has |z| power two, and your new fractal has |z| power one.
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LionHeart
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« Reply #9 on: April 17, 2017, 02:47:02 AM »

Hi,

I'm trying to implement the entire suite of SimonBrot fractals. I have success with the even functions:


int   DoSimonSnakeFormula(void)

    {
    Complex   zabs, tempz;

    zabs.x = fabs(z.x);
    zabs.y = fabs(z.y);
    tempz.y = z.y * zabs.x + z.x * zabs.y;
    tempz.x = z.x * zabs.x - z.y * zabs.y;

    z = tempz;
    z = CPolynomial(z, degree) + pixel;

    return (CSumSqr(z) >= rqlim);
    }

I would like to upgrade my complex arithmetic routines to include fractional powers of complex numbers. Has anyone else been able to make progress here?

Thanks to all who contributed smiley
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Paul the LionHeart
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« Reply #10 on: April 17, 2017, 02:20:45 PM »

Hi Guys,

I have built a square root function into my complex library and seem to have some success with Simon Brot. I have nothing to test against, so I don't know if I have a bug smiley

int   DoSimonSnakeFormula(void)

    {
    Complex   zabs, tempz, sqrtz;

    zabs.x = fabs(z.x);
    zabs.y = fabs(z.y);
    tempz.y = z.y * zabs.x + z.x * zabs.y;
    tempz.x = z.x * zabs.x - z.y * zabs.y;
    sqrtz = (degree % 2 == 1) ? CSqrt(z) : 1.0;         // use square root power if degree is odd
    z = CPolynomial(tempz, degree / 2)* sqrtz + q;
    return (CSumSqr(z) >= rqlim);
    }

Please let me know if there are any images for Simonbrot 3rd, 5th, and 7th, using the formulas for these fractals:
z1.5 * |z|2 + pixel, z2.5 * |z|2 + pixel, and z3.5 * |z|2 + pixel

Thanks
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Paul the LionHeart
simon.snake
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simon.fez SimonSideBurns
« Reply #11 on: April 17, 2017, 08:57:08 PM »

Any pictures yet?  I'd love to see what the initial render at standard coordinates looks like.
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LionHeart
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« Reply #12 on: April 18, 2017, 04:52:24 AM »

Hi Simon,

Here are the 8 SimonBrot images I produced so far using the algorithm in my previous post.

Simonbrot 2nd, 3rd, 4th, 5th, 6th, 7th, 8th and 9th: The formulas for these fractals are:
z^1.0 * |z|^2 + pixel, z^1.5 * |z|^2 + pixel, z^2.0 * |z|^2 + pixel, z^2.5 * |z|^2 + pixel,
z^3.0 * |z|^2 + pixel, z^3.5 * |z|^2 + pixel, z^4.0 * |z|^2 + pixel, z^4.5 * |z|^2 + pixel.
 


Can you verify the fractional ones?

Many thanks

Paul the LionHeart
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Paul the LionHeart
simon.snake
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simon.fez SimonSideBurns
« Reply #13 on: April 18, 2017, 11:26:06 PM »

I'm not sure how I can verify anything.  I made up the SimonBrot but I haven't played extensively with the formula to create other powers.

When the formula was being converted to something usable in Fractal eXtreme there were other variants made.

I don't think they looked much like what you have here though.  I'll try to dig out a picture of the different types, if I still have it on my Laptop.

Had a quick look but no joy so far.  Will continue looking.

Meanwhile, here's a formula for one I call Kung Fu Panda:

Code:
kungfupanda {
  ; burning ship variant
  ; SMF abs(z*z) in place of abs(z).
  if (ismand)
    p = 0-pixel
    z = p
  else
    p = 0-p1
    z = 0-pixel
  endif:
  z = abs(z*z)
  z = z * z + p
  |z| < 4
}
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LionHeart
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« Reply #14 on: April 18, 2017, 11:33:24 PM »

Hi Simon,

Using the ManpWIN fractal interpreter, I got:



z=c=pixel
z = abs(z*z) *abs(z*z) + c

Does it look right?
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Paul the LionHeart
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