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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.

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.

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.

It would also be nice to see Simonbrot 3rd, 5th, and 7th, but the formulas for these fractals are z^1.5 * |z|² + pixel, z^2.5 * |z|² + pixel, and z^3.5 * |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.

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.

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.

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.

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.

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 :)

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 :)

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

Any pictures yet?  I'd love to see what the initial render at standard coordinates looks like.

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.
 
(http://deleeuw.com.au/misc/SimonBrot.jpg)

Can you verify the fractional ones?

Many thanks

Paul the LionHeart

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:

Hi Simon,

Using the ManpWIN fractal interpreter, I got:

(http://deleeuw.com.au/misc/Panda.png)

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

Does it look right?

He's upside down but apart from that, yes he's right.

Goody Simon,

I'll add him to ManpWIN :)

Thanks for the tip.

If you change the initial condition to z=c=0-pixel then he goes the right way up.

Yep, thanks Simon.

I did indeed. I'll add it to ManpWIN that way.

In April of this year, I discovered something:

My original formula for the Simonbrot 6th was wrong. The symmetry for the formula resembled the Simonbrot 6th (which is why I tricked myself), but it was so stupid of me for considering this to be the formula for the Simonbrot 6th:

z = (z^3 * |z|^2) +z₀

This formula is obviously a 5th order fractal. z * z * z * abs(z) * abs(z) obviously has 5 zs. However, I will take credit for this formula as the 5th GT Quadratic Simonbrot. (GT stands for greentexas.) As an even bigger bonus, here is the code for the fractal (for you programmers):

ozx = zx;
zx = zx*zx*zx*zx*zx - 4*zx*zx*zx*zy*zy - (6*abs(zx*zx*zx))*zy*abs(zy) + 2*abs(zx)*abs(zy)*zy*zy*zy + 3*zx*zy*zy*zy*zy + cx;
zy = 3*ozx*ozx*ozx*ozx*zy - 4*ozx*ozx*zy*zy*zy - 6*ozx*abs(ozx)*abs(zy*zy*zy) + 2*ozx*ozx*ozx*abs(ozx)*abs(zy) + zy*zy*zy*zy*zy + cy;

The nth GT Quadratic Simonbrot has a formula of:

(z^{(n - 2)} * abs(z)²) + z₀

However, I used to think the formula for this fractal (and the normal Simonbrot) was:

(z^{(n / 2)} * abs(z)²) + z₀

It's difficult to explain when you have to clean up an ambiguous mess like this.

For an nth order Simonbrot, I discovered the formula is:

z = (z^n/2 * abs(z)^n/2) + z₀

This is the true formula.

I was also mistaken by claiming the 0th power Simonbrot looked exactly like the 2nd power Burning Ship. It is just a boring circle.

Thank you, LionHeart, for showing us what the nth order Simonbrot looked like!

how are these?  questioning the non-symmetry in some of the regions (actually normal if you look at evolution of multibrot from z^2, the transitions are smooth but non-integer exponents introduce non-symmetries)

my understanding is that |z|^2 is a purely real quantity and has little effect on the original mandelbrot fractal other than the slight 'scaling'

1,1.5,2,2.5,3,3.5,10 not all that different from the multibrot

That quote I made was a mistake. I made it a while back, but the formulas are all wrong! (Now I know the correct Simonbrot formulas.)