Fractal Find : Explore Fractal and Quantum Variations

CHAPTER 8
State Changes for Mandelbrot Quantum Fractal Two Spin-1 Device

Two Spin-1 Systems

Two Spin-1 systems are extracted from a single Mandelbrot construction.
The first spin, zⱼ, reflects state changes in the y-direction.
The second spin, zᵢ, reflects state changes in the x-direction.
Each pixel has a state change of {-, 0, +} associated wth it for each spin.
The spins and associated pseudocode are shown below.

Quantum notation is shown using a modified form of Feynman's Dirac notations:
|+X>, |0X>, |-X> denotes x-direction where real values are on the x-axis.
|+Y>, |0Y>, |-Y> denotes y-direction where imaginary values are on the y-axis.
|+Z>, |0Z>, |-Z> denotes z-direction where changes in state are on the z-axis.

zᵢ=△kᵢ
zⱼ=△kⱼ
The state change is {-, 0, +}
If zⱼ≠0, the state has changed in the y-direction.
If zᵢ≠0, the state has changed in the x-direction.
Since the current graphs show only two dimensions, the state change is reflected in colors.
zⱼ≠0 and zᵢ≠0 are plotted separately in the graphs below.
zⱼ and zᵢ are also plotted to show magnitude.

Resource for Dirac Notation:
Feynman, Richard et al. Feynman Lectures on Physics Volume 3 Quantum Mechanics. NY: Basic Books, 2010, New Millenium Edition.

State Changes

Each pixel, p, requires two values to calculate the state change.
No pixel can have two states or two state changes in a direction.
There is a single state and a single state change per pixel in each direction.
For a complete state change map (-m to +m, -n to +n), pixels range from (-m-1 to m+1, -n-1 to n+1).
Pixels have width = 1.0 units and height = 1.0 units.
State changes are step functions.

State Change where i = -m-1 or j = -n-1
For the y-direction, p_{(x, -n-1)} and zⱼ = z_{(x, -n-1)} = k_{(x, -n-1)}
For the x-direction, p_{(-m-1, y)} and zᵢ = z_{(-m-1, y)} = k_{(-m-1, y)}

State Change where -m ≤ i ≤ m+1 or -n ≤ j ≤ n+1
For the y-direction, p_{(x, j)} and zⱼ = z_{(x, j)} = k_{(x, j)} - k_{(x, j-1)}
For the x-direction, p_{(i, y)} and zᵢ = z_{(i, y)} = k_{(i, y)} - k_{(i-1, y)}

State Change Totals for {-, 0, +} where -m-1 ≤ i ≤ m+1 or -n-1 ≤ j ≤ n+1
Each time the state changes, a total state change count {-, 0, +} is also incremented:
For the y-direction, zⱼ total count for {-, 0, +} increases over |-Y>, |0Y>, |+Y> respectively.
For the x-direction, zᵢ total count for {-, 0, +} increases over |-X>, |0X>, |+X> respectively.

Mandelbrot (zⱼ)
Here is pseudocode and the resulting graph for the Mandelbrot set where (x² - y², 2.0*x*y) is the build. x remains constant while △kⱼ = zⱼ in the y-direction is calculated. Each pixel referenced by the start point(i, j) is plotted when zⱼ ≠ 0.

for (int i = -251; i ≤ 251; i++)
{
	oldk = 0;
	for (int j = -251; j ≤ 251; 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 State Change Totals (zⱼ)
Section	{-}	{0}	{+}	Sum
y1	382	59508	3111	63001	24.90
y2	0	195	56	251	0.10
y3	2916	59703	382	63001	24.90
y4	8	217	26	251	0.10
y5	0	1	0	1	0.00
y6	25	218	8	251	0.10
y7	233	61468	1300	63001	24.90
y8	3	247	1	251	0.10
y9	1050	61715	236	63001	24.90
Sum	4617	243272	5120	253009	100.00
%	1.82	96.15	2.02	100.00

Mandelbrot (zᵢ)
Here is pseudocode and the resulting graph for the Mandelbrot set where (x² - y², 2.0*x*y) is the build. y remains constant while △kᵢ = zᵢ in the x-direction is calculated. Each pixel referenced by the start point(i, j) is plotted when zᵢ ≠ 0.

for (int j = -251; j ≤ 251; j++)
{
	oldk = 0;
	for (int i = -251; i ≤ 251; 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;
	}
}

Mandelbrot State Change Totals (zᵢ)
Section	{-}	{0}	{+}	Sum
x1	653	59933	2415	63001	24.90
x2	25	218	8	251	0.10
x3	1416	61444	141	63001	24.90
x4	0	245	6	251	0.10
x5	0	1	0	1	0.00
x6	16	235	0	251	0.10
x7	653	59933	2415	63001	24.90
x8	25	218	8	251	0.10
x9	1416	61444	141	63001	24.90
Sum	4204	243671	5134	253009	100.00
%	1.66	96.31	2.03	100.00