Mandelbrot set

Defination of Mandelbort set

According to wiki, the Mandelbrot set is the set of complex numbers c for which the function f_{c}(z)=z^{2}+c does not diverge when iterated from  z=0 .

What magic does it has

If we try to find the set of c in a complex plane. Here is the magic:

where x axis is the real part of c, y axis is the imaginary of c. A point c is colored black if it belongs to the set.

How to find set

here is the pseducode from wiki:

For each pixel (Px, Py) on the screen, do:
{
  x0 = scaled x coordinate of pixel (scaled to lie in the Mandelbrot X scale (-2.5, 1))
  y0 = scaled y coordinate of pixel (scaled to lie in the Mandelbrot Y scale (-1, 1))
  x = 0.0
  y = 0.0
  iteration = 0
  max_iteration = 1000

  //when the sum of the squares of the real and imaginary parts exceed 4, the point has reached escape.
  while (x*x + y*y < 2*2  AND  iteration < max_iteration) {
    xtemp = x*x - y*y + x0
    y = 2*x*y + y0
    x = xtemp
    iteration = iteration + 1
  }
  color = palette[iteration]
  plot(Px, Py, color)
}

for complex numbers:

z=x+iy

 z^{2}=x^{2}+i2xy-y^{2}

 c=x_{0}+iy_{0}

Hence: x is sum of real parts in z and c, y is sum of imaginary parts of z and c. x={Re}(z^{2}+c)=x^{2}-y^{2}+x_{0} and  y={Im}(z^{2}+c)=2xy+y_{0}

How about 3D

I am going to try my 3D later, but please see the detail from skytopia firstly, I cited the formula directly.

3D formula is defined by:

z -> z^n + c where z and c are defined by {x,y,z}


{x,y,z}^n = r^n { sin(theta*n) * cos(phi*n) , sin(theta*n) * sin(phi*n) , cos(theta*n) }

where
r = sqrt(x^2 + y^2 + z^2)
theta = atan2( sqrt(x^2+y^2), z )
phi = atan2(y,x)

// z^n + c is similar to standard complex addition

{x,y,z}+{a,b,c} = {x+a, y+b, z+c}

//The rest of the algorithm is similar to the 2D Mandelbrot!

//Here is some pseudo code of the above:

r = sqrt(x*x + y*y + z*z )
theta = atan2(sqrt(x*x + y*y) , z)
phi = atan2(y,x)

newx = r^n * sin(theta*n) * cos(phi*n)
newy = r^n * sin(theta*n) * sin(phi*n)
newz = r^n * cos(theta*n)

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