Rotational Symmetry of Order 4

Making Patterns with Rotational Symmetry of Order 4 Using a n by n Grid of Random Shaded Squares

(1) n is even

(n = 12)

gnuplot script:

n = 12
m = n/2
max(x,y) = x>y?x:y
a = 0.875
f(t) = a*0.5*cos(2*pi*t) / max(abs(cos(2*pi*t)), abs(sin(2*pi*t)))
g(t) = a*0.5*sin(2*pi*t) / max(abs(cos(2*pi*t)), abs(sin(2*pi*t)))
funX(i,t) = f(t) + 0.5 + i
funY(j,t) = g(t) + 0.5 + j
set xrange [-m-1:m+1]
set yrange [-m-1:m+1]
unset key
unset tics
unset border
set margins 0,0,0,0
set size ratio -1
set samples 1000
set parametric
set trange [0:1]
Color = "#00cc00"
set style fill solid noborder
set terminal wxt 0 size 600,600 position 0,0 title "n = ".n
seed = ceil(2147483646*rand(time(0)))
random_test = rand((time(0),seed))
set multiplot
do for [i=0:m-1]{
    do for [j=0:m-1]{
        b = rand(0)>0.5?1:0
        if (!b) {continue}
        plot funX(i,t),  funY(j,t) w filledc closed lw 1 lc rgb Color, \
            -funY(j,t),  funX(i,t) w filledc closed lw 1 lc rgb Color, \
            -funX(i,t), -funY(j,t) w filledc closed lw 1 lc rgb Color, \
             funY(j,t), -funX(i,t) w filledc closed lw 1 lc rgb Color
    }
}
unset multiplot

(2) n is odd

(n = 11)

gnuplot script:

n = 11
m = (n+1)/2.
max(x,y) = x>y?x:y
a = 0.875
f(t) = a*0.5*cos(2*pi*t) / max(abs(cos(2*pi*t)), abs(sin(2*pi*t)))
g(t) = a*0.5*sin(2*pi*t) / max(abs(cos(2*pi*t)), abs(sin(2*pi*t)))
funX(i,t) = f(t) + i
funY(j,t) = g(t) + j
set xrange [-n/2.-1:n/2.+1]
set yrange [-n/2.-1:n/2.+1]
unset key
unset tics
unset border
set margins 0,0,0,0
set size ratio -1
set samples 1000
set parametric
set trange [0:1]
Color = "#00cc00"
set style fill solid noborder
set terminal wxt 0 size 600,600 position 0,0 title "n = ".n
seed = ceil(2147483646*rand(time(0)))
random_test = rand((time(0),seed))
set multiplot
b = rand(0)>0.5?1:0
if (b) {plot f(t), g(t) w filledc closed lw 1 lc rgb Color}
do for [i=1:m-1]{
    do for [j=0:m-1]{
        b = rand(0)>0.5?1:0
        if (!b) {continue}
        plot funX(i,t),  funY(j,t) w filledc closed lw 1 lc rgb Color, \
            -funY(j,t),  funX(i,t) w filledc closed lw 1 lc rgb Color, \
            -funX(i,t), -funY(j,t) w filledc closed lw 1 lc rgb Color, \
             funY(j,t), -funX(i,t) w filledc closed lw 1 lc rgb Color
    }
}
unset multiplot

(3) n is even (Non-Reflectional Symmetry)

(n = 12)

gnuplot script:

n = 12
m = n/2
max(x,y) = x>y?x:y
f(t) = 0.5*cos(2*pi*t) / max(abs(cos(2*pi*t)), abs(sin(2*pi*t)))
g(t) = 0.5*sin(2*pi*t) / max(abs(cos(2*pi*t)), abs(sin(2*pi*t)))
funX(i,t) = f(t) + 0.5 + i
funY(j,t) = g(t) + 0.5 + j
set xrange [-m-1:m+1]
set yrange [-m-1:m+1]
unset key
unset tics
unset border
set margins 0,0,0,0
set size ratio -1
set samples 1000
set parametric
set trange [0:1]
Color = "#00cc00"
set terminal wxt 0 size 600,600 position 0,0 title "n = ".n
seed = ceil(2147483646*rand(time(0)))
random_test = rand((time(0),seed))
set multiplot
do for [i=0:m-1]{
    do for [j=0:i]{
        b = rand(0)>0.5?1:0
        if (b) {
           plot funX(i,t),  funY(j,t) w filledc closed lw 1 lc rgb Color, \
               -funY(j,t),  funX(i,t) w filledc closed lw 1 lc rgb Color, \
               -funX(i,t), -funY(j,t) w filledc closed lw 1 lc rgb Color, \
                funY(j,t), -funX(i,t) w filledc closed lw 1 lc rgb Color
        }
        if (j==i) {continue}
        if (!b) {
           plot funX(j,t),  funY(i,t) w filledc closed lw 1 lc rgb Color, \
               -funY(i,t),  funX(j,t) w filledc closed lw 1 lc rgb Color, \
               -funX(j,t), -funY(i,t) w filledc closed lw 1 lc rgb Color, \
                funY(i,t), -funX(j,t) w filledc closed lw 1 lc rgb Color
        }
    }
}
unset multiplot

(4) n is odd (Non-Reflectional Symmetry)

(n = 11)

gnuplot script:

n = 11
m = (n+1)/2.
max(x,y) = x>y?x:y
f(t) = 0.5*cos(2*pi*t) / max(abs(cos(2*pi*t)), abs(sin(2*pi*t)))
g(t) = 0.5*sin(2*pi*t) / max(abs(cos(2*pi*t)), abs(sin(2*pi*t)))
funX(i,t) = f(t) + i
funY(j,t) = g(t) + j
set xrange [-n/2.-1:n/2.+1]
set yrange [-n/2.-1:n/2.+1]
unset key
unset tics
unset border
set margins 0,0,0,0
set size ratio -1
set samples 1000
set parametric
set trange [0:1]
Color = "#00cc00"
set terminal wxt 0 size 600,600 position 0,0 title "n = ".n
seed = ceil(2147483646*rand(time(0)))
random_test = rand((time(0),seed))
set multiplot
do for [i=0:m-1]{
    do for [j=0:i]{
        b = rand(0)>0.5?1:0
        if (b) {
           plot funX(i,t),  funY(j,t) w filledc closed lw 1 lc rgb Color, \
               -funY(j,t),  funX(i,t) w filledc closed lw 1 lc rgb Color, \
               -funX(i,t), -funY(j,t) w filledc closed lw 1 lc rgb Color, \
                funY(j,t), -funX(i,t) w filledc closed lw 1 lc rgb Color
        }
        if (j==0 || j==i) {continue}
        if (!b) {
           plot funX(j,t),  funY(i,t) w filledc closed lw 1 lc rgb Color, \
               -funY(i,t),  funX(j,t) w filledc closed lw 1 lc rgb Color, \
               -funX(j,t), -funY(i,t) w filledc closed lw 1 lc rgb Color, \
                funY(i,t), -funX(j,t) w filledc closed lw 1 lc rgb Color
        }
    }
}
unset multiplot

Graph files:

Download page

( Mathematical softwares used: Graph, gnuplot )

Swastika Tiling

An implicit curve: swastika(x,y) = 0

The swastika here is an ancient Buddhist symbol, not the Nazi one.

Composition of swastika(x,y), arctan(x) and tan(x)

(k = 0.25)

Composition of swastika(x,y), arctan(x) and tan(x)

(k = 0.125)

Composition of swastika(x,y), arctan(x) and tan(x)

(k = 0.4)

Composition of swastika(x,y), arcsin(x) and sin(x)

(k = 0.25)

Composition of swastika(x,y), arcsin(x) and sin(x)

(k = 0.4)

Swastika of Swastikas

(k = 1/4, 5/24)

Swastika of Swastikas of Swastikas

(k = 1/4, 1/4, 5/24)

Swastika of Swastikas of Swastikas

(Unicode of 卍: 0x534D)

A Maze of Swastikas

(k = 0.1)

( Mathematical software used: Graph )


Related posts:

• Swastika (卍)

Funny Scalar Fields (Heat Maps) on a Surface

Heat Map on a Torus

Heat Map on a Torus

Scalar Field on a Torus

Scalar Field on a Möbius Strip

Scalar Field on a Möbius Strip

( Mathematical software used: gnuplot )

Funny Inequalities on a Torus

(1)

(2-1)

(2-2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

( Mathematical software used: gnuplot )

Plotting Slope Fields using Graph

y' = x² + y² - r²  (r = 0, 1, 2, ..., 7)

y' = (x + y)/(x - y)  (method 1)

y' = (x + y)/(x - y)  (method 2)

y' = sin(x)/y + sin(y)/x

y' = cos(x) + cos(y)

y' = cos(x) sqrt(1+x²) + cos(y) sqrt(1+y²)

Download Examples:

• slope_field.zip

( Mathematical software used: Graph )

A Monster Equation

x² + y² - a[exp(sin³(x-y)) + exp(sin³(-x-y))]² = c

• x and y are variables.
• a and c are parameters.

(a = 3, c = 0.5)

Scilab script:

clear;
clf();
xset("wdim",400,400);
xset("fpf"," ");
xset("thickness", 2);
square(-8.2,-8.2,8.2,8.2);

a = 3;
c = 0.5;
function z = fun(x,y)
   z = x^2 + y^2 - a*(exp(sin(x-y)^3) + exp(sin(-x-y)^3))^2 - c;
endfunction
data = linspace(-8.2, 8.2, 2000);

contour2d(data, data, fun, [0,0], style=color("red"), axesflag=0);

(a = -1, c = 24)

(a = -2, c = 50)

( Mathematical software used: Scilab )