Mandelbrot (△kⱼ ≠ 0)
Here is pseudocode for the basic Mandelbrot set.

for (int i = -250; i ≤ 250; i++)
{
	oldk = 0;
	for (int j = -250; j ≤ 250; j++)
	{
		x = 0.0;
		y = 0.0;
		xs = i / 100.0;
		ys = j / 100.0;
		k = 0;
		do
		{
			k = k + 1;
			xnew = x * x - y * y + xs;
			ynew = 2.0 * x * y + ys;
			x = xnew;
			y = ynew;
		} while ((k ≤ 24) && (x * x + y * y ≤ 6.25));
		if (k != oldk) PlotPixel(i, j, color));
		oldk = k;
	}
}

Mandelbrot (△kᵢ ≠ 0)
Here is pseudocode for the basic Mandelbrot set.

for (int j = -250; j ≤ 250; j++)
{
	oldk = 0;
	for (int i = -250; i ≤ 250; i++)
	{
		x = 0.0;
		y = 0.0;
		xs = i / 100.0;
		ys = j / 100.0;
		k = 0;
		do
		{
			k = k + 1;
			xnew = x * x - y * y + xs;
			ynew = 2.0 * x * y + ys;
			x = xnew;
			y = ynew;
		} while ((k ≤ 24) && (x * x + y * y ≤ 6.25));
		if (k != oldk) PlotPixel(i, j, color);
		oldk = k;
	}
}

In quantum fractal theory, the entire Mandelbrot plane is mapped by calculating the state change (△k) in both the x-direction and the y-direction.
The state change for each direction is calculated separately.

For any Mandelbrot in Cartesian coordinates, a count, k is incremented when f(x, y) > escape.
To obtain the complete map for each pixel (i, j):
i is held constant while j is incremented for kⱼ, j is held constant while i is incremented for kᵢ
or
j is held constant while i is incremented for kᵢ, i is held constant while j is incremented for kⱼ
For every point, (x,y) at pixel (i,j), there are two associated △k: kⱼ and kᵢ.
△kⱼ = kⱼ - k_j-1 for the y-direction.
△kᵢ = kᵢ - k_i-1 for the x-direction.
If △kⱼ≠0, the pixel is plotted in the y-direction.
If △kᵢ≠0, the pixel is plotted in the x-direction.
k is always positive; △k is positive, negative, or zero.

The first graph, Mandelbrot (△kⱼ≠0), shows △kⱼ≠0 where x remains constant while calculating △k=△kⱼ in the y-direction.
Each start pixel(i, j) is plotted when △kⱼ ≠ 0.

The second graph, Mandelbrot (△kᵢ≠0), shows △kᵢ≠0 where y remains constant while calculating △k=△kᵢ in the x-direction.
Each start pixel(i, j) is plotted when △kᵢ ≠ 0.

Mandelbrot (△kⱼ≠0) and Mandelbrot (△kᵢ≠0) contain some pixels (i, j) in common.