## Thursday, December 20, 2012

### Fractal Star - Star fractal using GeoGebra

(GIF animation)

(1275x1275 pixels)

(GIF animation)

(1240x1240 pixels)

(GIF animation)

(1240x1240 pixels)

(GIF animation)

(1240x1240 pixels)

(GIF animation)

(1240x1240 pixels)

(GIF animation)

(1179x1179 pixels)

(GIF animation)

(1240x1240 pixels)

( Mathematical software used: GeoGebra )

### Five Crescent Moons Fractal (Fractal Pinwheel)

(GIF animation)

(1240x1240 pixels)

(1240x1240 pixels)

Fractal Pinwheel (GIF animation)

( Mathematical softwares used: GeoGebra, gnuplot )

## Tuesday, December 4, 2012

### Sierpinski Triangle Using Inequalities

(Sierpinski triangle with hearts)

Example 1:
(iterations: 6)

(GIF animation)

gnuplot script 1:

triangle(x,y) = sqrt(3)*abs(x) - 1.5 + abs(sqrt(3)*abs(x) + 2*y - 0.5)
S0(x,y) = sqrt(3)*abs(x) - 0.75 + abs(sqrt(3)*abs(x) - 2*y - 0.25)
S1(x,y) = S0(2*x, 2*y-1) * S0(2*abs(x)-0.5*sqrt(3), 2*y+0.5)
S2(x,y) = S1(2*x, 2*y-1) * S1(2*abs(x)-0.5*sqrt(3), 2*y+0.5)
S3(x,y) = S2(2*x, 2*y-1) * S2(2*abs(x)-0.5*sqrt(3), 2*y+0.5)
S4(x,y) = S3(2*x, 2*y-1) * S3(2*abs(x)-0.5*sqrt(3), 2*y+0.5)
S5(x,y) = S4(2*x, 2*y-1) * S4(2*abs(x)-0.5*sqrt(3), 2*y+0.5)
fun(x,y) = triangle(x,y)>0 || S0(x,y)<0 || S1(x,y)<0 || S2(x,y)<0 || \
S3(x,y)<0 || S4(x,y)<0 || S5(x,y)<0 ? 0:1
set xrange [-0.5*sqrt(3):0.5*sqrt(3)]
set yrange [-0.5:1]
set pm3d map
unset colorbox
set palette model RGB defined (0 'white', 1 'red')
set size ratio -1
set samples 640
set isosamples 640
unset key
unset xtics
unset ytics
unset border
splot fun(x,y)

Example 2:
(iterations: 6)

(GIF animation)

gnuplot script 2:

S0(x,y) = sqrt(3)*abs(x) - 0.75 + abs(sqrt(3)*abs(x) - 2*y - 0.25)
S1(x,y) = S0(2*x, 2*y - 1) * S0(2*abs(x) - 0.5*sqrt(3), 2*y + 0.5)
S2(x,y) = S1(2*x, 2*y - 1) * S1(2*abs(x) - 0.5*sqrt(3), 2*y + 0.5)
S3(x,y) = S2(2*x, 2*y - 1) * S2(2*abs(x) - 0.5*sqrt(3), 2*y + 0.5)
S4(x,y) = S3(2*x, 2*y - 1) * S3(2*abs(x) - 0.5*sqrt(3), 2*y + 0.5)
S5(x,y) = S4(2*x, 2*y - 1) * S4(2*abs(x) - 0.5*sqrt(3), 2*y + 0.5)
fun(x,y) = (S0(x,y)<0 || S2(x,y)<0 || S4(x,y)<0) ? 2 : \
( (S1(x,y)<0 || S3(x,y)<0 || S5(x,y)<0) ? 1 : 0 )
set xrange [-0.5*sqrt(3):0.5*sqrt(3)]
set yrange [-0.5:1]
set pm3d map
unset colorbox
set palette model RGB defined (0 'white', 1 'green', 2 'magenta')
set size ratio -1
set samples 640
set isosamples 640
unset key
unset xtics
unset ytics
unset border
splot fun(x,y)

Graph examples:

( Mathematical softwares used: Graph, gnuplot )

## Monday, December 3, 2012

### Sierpinski Carpet Using Inequalities

gnuplot script:

square(x,y) = abs(x + y) + abs(x - y) - 3
S0(x,y) = abs(x + y) + abs(x - y) - 1
f(x) = 3*abs(x) - 3
S1(x,y) = S0(3*x, f(y)) * S0(f(x), 3*y) * S0(f(x), f(y))
S2(x,y) = S1(3*x, f(y)) * S1(f(x), 3*y) * S1(f(x), f(y))
S3(x,y) = S2(3*x, f(y)) * S2(f(x), 3*y) * S2(f(x), f(y))
S4(x,y) = S3(3*x, f(y)) * S3(f(x), 3*y) * S3(f(x), f(y))
fun(x,y) = square(x,y)>0 || S0(x,y)<0 || S1(x,y)<0 || \
S2(x,y)<0 || S3(x,y)<0 || S4(x,y)<0 ? 0:1
set xrange [-1.5:1.5]
set yrange [-1.5:1.5]
set pm3d map
unset colorbox
set palette model RGB defined (0 'white', 1 'red')
set size ratio -1
set samples 640
set isosamples 640
unset key
unset xtics
unset ytics
unset border
splot fun(x,y)

Graph example:

Click here to download the example file (sierpinski_carpet.grf).

( Mathematical softwares used: Graph, gnuplot )