APPENDIX A.B
Mandelbar Variation Snowflakes
Mandelbar Variation Snowflakes use Mandelbrot variations, actually Mandelbar variations.
Mandelbrot is (x+iy)² and Mandelbar is (x-iy)² which produces a three-pronged figure.
The expression (x-iy)⁵ produces a six-pronged figure with the figure not quite radially symmetric, but with a snowflake shape.
Six-Pronged Mandelbar Expanded Center
Six-Pronged Mandelbar Snowflake
for (i = 0; i ≤ 800; i++) {
for (j = 0; j ≤ 800; j++) {
x = 0.0;
y = 0.0;
xs = -4.0+(i / 100.0);
ys = -4.0+(j / 100.0);
k = 0;
do {
k = k+1;
xnew = x⁵-10.0*x³*y²+5.0*x*y⁴+xs;
ynew = -y⁵+10.0*x²*y³-5.0*x⁴*y+ys;
x = xnew;
y = ynew;
} while ((k ≤ 255) && (x*x+y*y ≤ 16.0));
PlotPixel(i, j, color);
}
}
Six-Pronged Mandelbar Expanded Center with PseudocodeThe center of the Six-Pronged Mandelbar is expanded to reveal the prongs.
for (i = 0; i ≤ 1200; i++) {
for (j = 0; j ≤ 1200; j++) {
x = 0.0;
y = 0.0;
xs = -1.5+(i / 400.0);
ys = -1.5+(j / 400.0);
k = 0;
do {
k = k+1;
xnew = x⁵-10.0*x³*y²+5.0*x*y⁴+xs;
ynew = -y⁵+10.0*x²*y³-5.0*x⁴*y+ys;
x = xnew;
y = ynew;
} while ((k ≤ 255) && (x*x+y*y ≤ 16.0));
PlotPixel(i, j, color);
}
}
Six-Pronged Mandelbar Snowflake with PseudocodeTo obtain this figure, pixels are plotted at iteration count increment.
for (i = 0; i ≤ 1200; i++) {
oldk = 0;
for (j = 0; j ≤ 1200; j++) {
x = 0.0;
y = 0.0;
xs = -1.5+(i / 400.0);
ys = -1.5+(j / 400.0);
k = 0;
do {
k = k+1;
xnew = x⁵-10.0*x³*y²+5.0*x*y⁴+xs;
ynew = -y⁵+10.0*x²*y³-5.0*x⁴*y+ys;
x = xnew;
y = ynew;
} while ((k ≤ 255) && (x*x+y*y ≤ 16.0));
if ((oldk != k) && (k > 3) && (oldk > 3)) PlotPixel(i, j, color);
oldk = k;
}
}
| Snowflake | Map (m,n) | Build (f(x,y), g(x,y)) | Escape h(x,y)>value | Plot |
|---|---|---|---|---|
| Example | (1201, 1201) | (x⁵-10.0*x³*y²+5.0*x*y⁴, -y⁵+10.0*x²*y³-5.0*x⁴*y) | x²+y²>16.0 | Pixel |
| Snowflake #25 | (1201, 1201) | (x⁵-10.0*x³*y²+5.0*x*y⁴, -y⁵+10.0*x²*y³-5.0*x⁴*y) | x²+y²+|x|+|y|>16.0 | Pixel |
| Snowflake #26 | (1201, 1201) | (x⁵*k-10.0*x³*y²*k+5.0*x*y⁴*k-0.5*x, -y⁵*k+10.0*x²*y³*k-5.0*x⁴*y*k-0.5*y) | x²+y²>16.0 | Pixel |
| Snowflake #27 | (1201, 1201) | (x⁵-10.0*x³*y²+5.0*x*y⁴, -y⁵+10.0*x²*y³-5.0*x⁴*y) | x²+y²+|x|>16.0/k | Pixel |
| Snowflake #28 | (1201, 1201) | (x⁵-10.0*x³*y²+5.0*x*y⁴, -y⁵+10.0*x²*y³-5.0*x⁴*y) | x*x+|y|+|x|>16.0/k | Pixel |
| Snowflake #29 | (1201, 1201) | (x⁵-10.0*x³*y²+5.0*x*y⁴-0.5*x, -y⁵+10.0*x²*y³-5.0*x⁴*y-0.5*y) | |x|+|y|>16.0/k | Pixel |
| Snowflake #30 | (1201, 1201) | (x⁵-10.0*x³*y²+5.0*x*y⁴+x/k, -y⁵+10.0*x²*y³-5.0*x⁴*y+y/k) | y*y+0.25*|x| >16.0/k | Pixel |
| Snowflake #31 | (1201, 1201) | (x⁵-10.0*x³*y²+5.0*x*y⁴+0.25*x, -y⁵+10.0*x²*y³-5.0*x⁴*y+0.25*y) | y*y+0.25*|x| >16.0/k | Pixel |
| Snowflake #32 | (1201, 1201) | (x⁵-10.0*x³*y²+5.0*x*y⁴+x/3.0, -y⁵+10.0*x²*y³-5.0*x⁴*y+y/3.0) | y*y+0.25*|x| >16.0/k | Pixel |
| Snowflake #33 | (1201, 1201) | (x⁵-10.0*x³*y²+5.0*x*y⁴-x*0.25, -y⁵+10.0*x²*y³-5.0*x⁴*y-y*0.25) | y*y+0.25*|x| >16.0/k | Pixel |
| Snowflake #34 | (1201, 1201) | (x⁵-10.0*x³*y²+5.0*x*y⁴-x/1.2, -y⁵+10.0*x²*y³-5.0*x⁴*y-y/1.2) | x²+y²>16.0/k | Pixel |
| Snowflake #35 | (1201, 1201) | (x⁵-10.0*x³*y²+5.0*x*y⁴-x/10.0, -y⁵+10.0*x²*y³-5.0*x⁴*y-y/10.0) | 0.125*|y|+0.25*|x|+y*y>16.0/k | Pixel |
| Snowflake #36 | (1201, 1201) | (x⁵-10.0*x³*y²+5.0*x*y⁴+x/20.0, -y⁵+10.0*x²*y³-5.0*x⁴*y+y/20.0) | x²+y²+|x|+|y|>16.0/k | Pixel |
| Snowflake #37 | (1201, 1201) | (x⁵-10.0*x³*y²+5.0*x*y⁴-x/20.0, -y⁵+10.0*x²*y³-5.0*x⁴*y+y/4.0) | x²+y²-0.25*|y|>16.0/k | Pixel |
| Snowflake #38 | (1201, 1201) | (x⁵-10.0*x³*y²+5.0*x*y⁴+x/k, -y⁵+10.0*x²*y³-5.0*x⁴*y)+y/k) | x*x+y*y-0.2*|y|>16.0/k | Pixel |
| Snowflake #39 | (1201, 1201) | (x⁵-10.0*x³*y²+5.0*x*y⁴+x/k, -y⁵+10.0*x²*y³-5.0*x⁴*y)+y/k) | x*x-0.8*|y|>16.0/k | Pixel |
| Snowflake #40 | (1201, 1201) | (x⁵-10.0*x³*y²+5.0*x*y⁴+x/k, -y⁵+10.0*x²*y³-5.0*x⁴*y)+y/k) | 0.5*|x|>16.0/k | Pixel |
| Snowflake #41 | (1201, 1201) | (x⁵-10.0*x³*y²+5.0*x*y⁴+x/2.0, -y⁵+10.0*x²*y³-5.0*x⁴*y+y/2.0) | |x|>16.0/k | Pixel |
| Snowflake #42 | (1201, 1201) | (x⁵-10.0*x³*y²+5.0*x*y⁴+x*k/12.0, -y⁵+10.0*x²*y³-5.0*x⁴*y+y*k/12.0) | 0.75*|x|>16.0/k | Pixel |
| Snowflake #43 | (1201, 1201) | (x⁵-10.0*x³*y²+5.0*x*y⁴-x*k/12.0, -y⁵+10.0*x²*y³-5.0*x⁴*y-y*k/12.0) | 0.5*|x|>16.0/k | Pixel |
| Snowflake #44 | (1201, 1201) | (x⁵*k-10.0*x³*y²*k+5.0*x*y⁴*k-x/k, -y⁵*k+10.0*x²*y³*k-5.0*x⁴*y*k-y/k) | 2.0*|x|>16.0/k | Pixel |
| Snowflake #45 | (1201, 1201) | (x⁵*k-10.0*x³*y²*k+5.0*x*y⁴*k-x/k³, -y⁵*k+10.0*x²*y³*k-5.0*x⁴*y*k-y/k³) | 0.5*x*x+|y*y*x|>16.0/k | Pixel |
| Snowflake #46 | (1201, 1201) | (x⁵-10.0*x³*y²+5.0*x*y⁴-x/k, -y⁵+10.0*x²*y³-5.0*x⁴*y -y/k) | x²+0.5*y²>16.0/k | Pixel |
| Snowflake #47 | (1201, 1201) | (x⁵-10.0*x³*y²+5.0*x*y⁴-x/k², -y⁵+10.0*x²*y³-5.0*x⁴*y-y/k²) | 0.4*x²+0.3*y²+|x|+|y|>16.0/k | Pixel |
| Snowflake #48 | (1201, 1201) | (2.0*(x⁵/k-10.0*x³*y²/k+5.0*x*y⁴)/k, 2.0*(-y⁵/k+10.0*x²*y³/k-5.0*x⁴*y)/k) | x²+y²>16.0/k | Pixel |
FRACTAL FIND