Alternative ways of creating the Mandelbrot set

[twinbee]

I didn't realise there was another way of creating the set until I found this:

I quote:

You don't need imaginary numbers to define the mandelbrot set. Start with point (x[1],y[1]). When you do the iterations, x[next] = x[previous]^2 - y^2 + x[1] and y[next] = -y[previous]^2 + y[1]. And the image shown is most definitely a rendering of the mandelbrot set.

Unfortunately, he lost me at the y^2 bit. Perhaps he meant y[1] (which doesn't work btw)?

My attempt at reproducing the method has failed so far:

Code:

double nx;		double ny;
double ox=0.4;		double oy=0.5;  // Initialize to any numbers? (these chosen at random)
double px=ox;		double py=oy;

for (int n=0; n<100000; n++) {
	nx = pow(px,2.0) - pow(oy,2.0) + ox;
	ny = -pow(py,2.0)              + oy;

	px=nx;		py=ny;

	int drawx=(int)(nx*100.0 + 400);   // Enlarging tiny numbers to big so we can see...
	int drawy=(int)(ny*100.0 + 300);  // ...the results properly. Plus centering to middle.
		
	plot(drawx, drawy);
}

Any ideas of what needs to be changed?

And to to keep on topic generally, are there any other simple ways of creating the Mandelbrot set?

[juharkonen]

Calculations with complex numbers can always be translated to calculations in real numbers (using the definition of complex numbers). For the logistic map z(n+1) = z(n)^2 + C used to calculate the Mandelbrot set, this translates to
x(n+1) = x(n)^2 - y(n)^2 + xC
y(n+1) = 2*x(n)+y(n) + yC
where
C = (xC, yC) = xC + i*yC and
z(n) = (x(n), y(n)) = x(n) + i*y(n).

Comparing to your reference, it seems that the calculation of the y coordinate is incorrect. I didn't test this one neither but I hope you got the idea. I don't know about other ways to calculate the Mandelbrot set and I wouldn't say this essentially differs from the calculations using complex numbers - it's just a different representation of the same calculations.

[GFWorld]

Twinbee - thank´s for your visit and comments - you picked up here some of my own favorites 
***
FeSt 8 - I lost it, but I am sometimes trying to return back , knowing about the impossibility at last
:-)))
It was created with Fe Sterlingware / Twister Weed , playing around here with CAbs , CAbs 2 & CSin Functions ...
***
>These 3 also come very close:
Mb1/mb4.html  / Formula by Paul W.Carlson
Fr1/fe126.html / Fe JT_Rosegarden
Fr1/fe116.html / Fe Sierpinskie Formula by Rico Wack
***
>These aren't as good as the above, but still really nice:
Fr1/fe100.html / Fe JT_Dahlia
UF/7.html / UF Sorensen Cubic - one of *my* places starting for Mini´s :-)
UF/120.htm / UF Julia Tricorn
Fr1/fe162.html / Fe JT_Venice
Fr1/fe181.html / Fe Newton Var by Girvan
Mut/Mut23.htm / MutatorKammer Alternating Fractals CM & Orbit Traps

Margit

[cKleinhuis]

as far as i know the above method works for julia sets, did you paint mandelbrot fractals with that rendering loop ?

[lycium]

Comparing to your reference, it seems that the calculation of the y coordinate is incorrect. I didn't test this one neither but I hope you got the idea. I don't know about other ways to calculate the Mandelbrot set and I wouldn't say this essentially differs from the calculations using complex numbers - it's just a different representation of the same calculations.

different and less meaningful representations, agreed.