HPDZ Buffalo

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Offline LionHeart

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« on: May 25, 2019, 05:44:59 AM »
Hi Friends,

Does anyone have the equations and any other information on HPDZ Buffalo? I can see it in Kalles Fraktaler, but it is hard to find specific information.

I would like to add this fractal to my program ManpWIN.

Any help greatly appreciated.

Thanks.
Paul the LionHeart

Offline mclarekin

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« Reply #1 on: May 25, 2019, 06:10:54 AM »
maybe   
Quote
The formula for the HPDZ Buffalo is abs(z2 - z + z0.)

http://www.fractalforums.com/other-types/i-figured-out-why-there-are-no-minibrots-in-the-hpdz-buffalo!/

Offline LionHeart

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« Reply #2 on: May 25, 2019, 08:05:42 AM »
Many thanks mclarekin.

Pardon my ignorance, but isn't the absolute value of a complex number a scalar and not complex? https://www.varsitytutors.com/hotmath/hotmath_help/topics/absolute-value-complex-number which says:

The absolute value of a complex number , a+bi is defined as the distance between the origin (0,0) and the point (a,b) in the complex plane. | a+bi |=sqrt(a*a+b*b).

So what would Z.y be if Z.x =  abs(z2 - z + z0) ? I would assume it would be 0, but doesn't seem to make sense.

Are you able to express it as real and imaginary components?

Many thanks.

Offline mclarekin

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« Reply #3 on: May 25, 2019, 09:45:36 AM »
Quote
Pardon my ignorance
  ;D It is likely you know more than me. I avoid complex number mathematics  I simply work in a 3D world where sqrt(-1) exists.

Someone else can probably answer your question :)

Offline marcm200

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« Reply #4 on: May 25, 2019, 10:08:09 AM »
I think the abs is just a non-mathematical way of stating, that after calculating zn+1 in a non-abs way you take the absolute value of each component instead, according to https://theory.org/fracdyn/buffalo/.

Offline LionHeart

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« Reply #5 on: May 25, 2019, 02:26:41 PM »
Thanks anyway  mclarekin.

You tried ;)

Hi marcm200,

I tried your idea on the website https://theory.org/fracdyn/buffalo/

I interpreted their code:

Code: [Select]
A derivative of the Burning Ship, the Buffalo fractal was composed as follows:

    Let,
     z0 = 0
     zn+1 = zn2 - zn + c

    Take,
     z = x + i*y,

    Then,
     z2 - z + c = (x + i*y)2 - (x + i*y) + c

    Lastly, take the absolute values of x and y:
     (|x| + i*|y|)2 - (|x| + i*|y|) + c

this way:

Code: [Select]
int HPDZ_Buffalo(void)
    {
    z.x = fabs(z.x);
    z.y = fabs(z.y);
    z = z * z - z;
    z += q;
    return (CSumSqr(z) >= rqlim);
    }

The result was close, yet not quite right.

My result:



And the website you recommended had the correct image:



Notice the left side of the image. It is quite different. What am I doing wrong? Is my code accurate?

Thanks for any suggestions.

Offline claude

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« Reply #6 on: May 25, 2019, 02:35:16 PM »
https://code.mathr.co.uk/et/blob/kf:/kf/formulas.et#l136
https://mathr.co.uk/kf/manual.html#formulas
Code: [Select]
z := (((x^2 - y^2) - |x|) + i (|2xy| - |y|)) + cthough may be clearer written as
Code: [Select]
w := |x| + i |y|
z := w^2 - w + c

Seed value should be 0.5+0.0i instead of 0.0+0.0i.  There may be other valid seed values, I haven't checked the full derivative matrices.

Offline claude

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« Reply #7 on: May 25, 2019, 02:37:17 PM »
The result was close, yet not quite right.
Looks like your rqlim is too low, perhaps 2 instead of 2^2 = 4? Not sure...

Offline LionHeart

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« Reply #8 on: May 25, 2019, 02:57:09 PM »
Hi Claude,

Yep, that did it. I upped rqlim from 4 to 16 and up she came.

Thanks for the full list of fractals in your last post too. You have been most helpful.

Just as an aside referring to link https://fractalforums.org/programming/11/perturbation-success-story/2826/new#new, I have coded many fractals in Perturbation. Once I understood the theory, it was much easier.

Thanks for all your help with my program  ;)


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