05-27-2019, 12:56 PM

Blog Entry: http://edspi31415.blogspot.com/2019/05/f...prime.html

The program CHAOS1 plots a fractal to simulate a Julia Set. Given the complex numbers z_k and c, repeats the equation:

z_n = z_n-1 + c^2

The program will ask for a sample graphing space with the a border B. The viewing window will be set up as:

Xmin = -B

Xmax = B

Ymin = -B

Ymax = B

The program CHAOS1 will also as for the number of points (G), this will determine the number of points plotted in the graph. The higher G is, the more detailed the fractal is; however, the plot will take longer to complete.

We start with each point on the graph z_0 = a + b*i, where i = √-1. Then we calculate:

z_1 = z_0^2 + c

z_2 = z_1^2 + c

z_3 = z_2^2 + c

and so on.

For each z_0, we have two possibilities for the repeated calculations:

1. |z_n| = abs(z_n) eventually goes towards infinity or

2. |z_n| = abs(z_n) eventually settles (or converges) to a specific point.

A color is assigned to each point z_0. The color is determined by the amount of iterations it takes to reach |z| > 2. For such points that fit criteria 2, and never reaches |z| >2, the point is colored black. This method creates the Julia set for any given c.

For the program CHAOS1, I set up a rank of colors to plot each point, and arbitrary pick a maximum amount of iterations. For example, I picked 9 colors for the TI-84 Plus CE version. Hence for each point, if |z| ≤ 2 after 9 iterations, color the point black.

Obviously the more levels (colors) we have, the more accurate our fractal is. With the programs listed, you can adjust the number of colors.

I set 18 color levels, starting from white to black. Because of the faster processor and calculating speed, I have increased the number of grid points to 500. 250 makes a picture with lines.

Note: I use the parenthesis notation (x,y) for the complex number x + yi because the imaginary character i unfortunately does not transfer to computer text.

Download the program (hpprgrm) here: https://drive.google.com/open?id=1A06e_p...jbLSUkVUN2 (The zip file also has a version for the TI-84 Plus CE)

The program CHAOS1 plots a fractal to simulate a Julia Set. Given the complex numbers z_k and c, repeats the equation:

z_n = z_n-1 + c^2

The program will ask for a sample graphing space with the a border B. The viewing window will be set up as:

Xmin = -B

Xmax = B

Ymin = -B

Ymax = B

The program CHAOS1 will also as for the number of points (G), this will determine the number of points plotted in the graph. The higher G is, the more detailed the fractal is; however, the plot will take longer to complete.

We start with each point on the graph z_0 = a + b*i, where i = √-1. Then we calculate:

z_1 = z_0^2 + c

z_2 = z_1^2 + c

z_3 = z_2^2 + c

and so on.

For each z_0, we have two possibilities for the repeated calculations:

1. |z_n| = abs(z_n) eventually goes towards infinity or

2. |z_n| = abs(z_n) eventually settles (or converges) to a specific point.

A color is assigned to each point z_0. The color is determined by the amount of iterations it takes to reach |z| > 2. For such points that fit criteria 2, and never reaches |z| >2, the point is colored black. This method creates the Julia set for any given c.

For the program CHAOS1, I set up a rank of colors to plot each point, and arbitrary pick a maximum amount of iterations. For example, I picked 9 colors for the TI-84 Plus CE version. Hence for each point, if |z| ≤ 2 after 9 iterations, color the point black.

Obviously the more levels (colors) we have, the more accurate our fractal is. With the programs listed, you can adjust the number of colors.

I set 18 color levels, starting from white to black. Because of the faster processor and calculating speed, I have increased the number of grid points to 500. 250 makes a picture with lines.

Note: I use the parenthesis notation (x,y) for the complex number x + yi because the imaginary character i unfortunately does not transfer to computer text.

Code:

`EXPORT CHAOS1()`

BEGIN

// 2019-05-19 EWS

LOCAL b,g,c,l6,s;

LOCAL i,k,j,z,l;

STARTAPP("Function");

INPUT({b,g,{c,[[0],[3]]}},"Fractal",

{"Border: ","Grid: ","C: "},

{"Min/Max - X/Y",

"# Grid Pts",

"C: x + yi"});

// size the screen

Xmin:=−b; Xmax:=b;

Ymin:=−b; Ymax:=b;

// clear

RECT();

// list of colors

l6:={#FFFFFFh,#C0C0C0h,

#D0D0D0h,#FFFF00h,

#B0B0B0h,#FF8000h,

#905000h,#400808h,

#00FF00h,#004060h,

#003000h,#00FFFFh,

#80D0FFh,#0000FFh,

#000080h,#400080h,

#404040h,#000000h};

// plotting

s:=SIZE(l6);

FOR i FROM −b TO b STEP (2*b)/g DO

FOR j FROM −b TO b STEP (2*b)/g DO

k:=0;

REPEAT

k:=k+1;

IF k==1 THEN

z:=(i,j);

ELSE

z:=z^2+c;

END;

UNTIL ABS(z)>2 OR k==s;

l:=l6(k);

PIXON(i,j,l);

END;

END;

WAIT(0);

END;

Download the program (hpprgrm) here: https://drive.google.com/open?id=1A06e_p...jbLSUkVUN2 (The zip file also has a version for the TI-84 Plus CE)