Fun math art (pictures) - benice equation

Nested Ellipses (Ellipse Whirl)

2019-01-18T08:27:00.000-08:00

These animations are constructed by shrinking and rotating

a sequence of concentric and similar ellipses,

so that each ellipse lies inside the previous ellipse and is tangent to it.

這些動畫是藉由收縮和旋轉

一序列同心且相似的橢圓來構造的，

使得每個橢圓都位於前一個橢圓內並與之相切。

All the angles of rotations are equal to 5°.

All the angles of rotations are equal to 10°.

All the angles of rotations are equal to 15°.

The angles of rotations are 0.5°, 1°, 1.5°, 2°, .....

All the angles of rotations are equal to 5°.

All the angles of rotations are equal to the value of the animation parameter.

Related posts:

( Mathematical software used: GeoGebra )

Rotational Symmetry of Order 4

2017-12-12T08:18:00.000-08:00

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

2017-11-12T21:53:00.000-08:00

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

2017-09-08T17:32:00.000-07:00

Heat Map on a Torus

Scalar Field on a Torus

Scalar Field on a Möbius Strip

( Mathematical software used: gnuplot )

Funny Inequalities on a Torus

2017-09-08T17:10:00.000-07:00

(1)

(2-1)

(2-2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

( Mathematical software used: gnuplot )

Plotting Slope Fields using Graph

2017-01-27T06:04:00.000-08:00

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

2017-01-07T00:10:00.000-08:00

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 )

Five Interlaced Moons

2016-12-14T05:02:00.000-08:00

Five Interlaced Crescents

Generalization:

Let A and B be two disks with radii a and b.

The set difference A\B is used to plot the moons

in the following animations.

Animation parameter: the position of B

Animation parameter: b (the radius of B)

Animation parameter: the position of A

Animation parameter: a (the radius of A)

( Mathematical softwares used: gnuplot, GeoGebra )

Four Interlaced Crescents

2016-12-14T03:37:00.000-08:00

A cross pattée!

( Mathematical softwares used: Graph, gnuplot )

Related posts:

Cross Patty (cross pattée)

Triple Crescent Moon

2016-12-14T02:30:00.000-08:00

Triple Crescent Moon Symbol

(Boundary curves: circles)

Boundary curves: superellipses with exponent 2.5

Boundary curves: superellipses with exponent 4

A monster face!

Hi!

Exponents: 1.5, 2, 2.5, 3, 3.5, 4

( Mathematical software used: gnuplot )

Cross Patty (cross pattée)

2016-11-28T02:31:00.000-08:00

max( f(abs(x), abs(y)), f(abs(y), abs(x)) ) > 0

(1) f(x,y) = ax^2 - y + b (parabola)

a = 0.035, b = 0.85

(2) f(x,y) = x^2 + (y - a)^2 - b^2 (circle)

a = 15.6, b = 14.75

(3) f(x,y) = abs(x)^p + abs(y - a)^p - b^p (superellipse)

a = 15.6, b = 14.75

p = 2.5, 2.4, 2.3, ...,1.5

p = 4, a = 10, b = 8

(4) f(x,y) = a[exp(-abs(x+b)^p) + exp(-abs(x-b)^p)] - y + c

p = 1.5, a = 1, b = 7, c = 2

p = 3, a = 1, b = 7, c = 2

(5) f(x,y) = a cos(bx)^p - y + c

p = 1, a = -1, b = π/7, c = 2

p = 2, a = 1, b = π/7, c = 2

p = 3, a = -1, b = π/7, c = 2

p = 4, a = 1, b = π/7, c = 2

( Mathematical software used: Graph )

Rounded Cuboid with Superquadric Corners

2016-10-21T02:58:00.000-07:00

The corners of the rounded cuboids on my previous post are spherical corners.

We can generalize the spherical corners (power = 2) to superquadric corners (power > 0).

Equation 1.

The equation for a rounded cuboid with superquadric corners can be given by:

max(|x| - a, 0)^p + max(|y| - b, 0)^p + max(|z| - c, 0)^p = d^p,

where a, b, c, d ＞ 0, and p ≧ 1.

p = 1

p = 1.5

p = 2

p = 4

p = 10

Equation 2.

Furthermore, the above equation can be generalized as follows:

max(|x| - a, 0)^p + max(|y| - b, 0)^q + max(|z| - c, 0)^r = d,

where a, b, c ≧ 0, d ＞ 0, and p, q, r ＞ 0.

Equation of a Rounded Rectangle

Equation of a Rounded Rectangle

2016-10-17T11:57:00.000-07:00

Equations:

The equation of a rounded rectangle centered at (0,0) with width 2(a+c), height 2(b+c)
and corner radius c can be expressed as

max(|x| - a, 0)² + max(|y| - b, 0)² = c² ...... (I)

or

(|x| - a + │|x| - a│)² + (|y| - b + │|y| - b│)² = (2c)² ...... (II)

, where a, b, and c are positive numbers.

Proof:

The rounded rectangle can be broken down into four parts (see the image below):
╭
│ (|x| - a)² + (|y| - b)² = c², if |x|≧a and |y|≧b ...... (1)
│ |x| = a + c , if |x|≧a and |y|＜b ...... (2)
│ |y| = b + c , if |x|＜a and |y|≧b ...... (3)
│ no graph , if |x|＜a and |y|＜b ...... (4)
╰

Since |x|≧a in part (2) and |y|≧b in part (3),
the above expression is equivalent to
╭
│ (|x| - a)² + (|y| - b)² = c², if |x|≧a and |y|≧b;
│ (|x| - a)² = c² , if |x|≧a and |y|＜b;
│ (|y| - b)² = c² , if |x|＜a and |y|≧b;
│ 0 = c² , if |x|＜a and |y|＜b.
╰

Now make the equations of the four parts look similar:
╭
│ (|x| - a)² + (|y| - b)² = c², if |x|≧a and |y|≧b;
│ (|x| - a)² + (b - b)² = c², if |x|≧a and |y|＜b;
│ (a - a)² + (|y| - b)² = c², if |x|＜a and |y|≧b;
│ (a - a)² + (b - b)² = c², if |x|＜a and |y|＜b.
╰

Since max(|x|, a) =
╭
│ |x|, if |x|≧a
│ a, if |x|＜a
╰
and max(|y|, b) =
╭
│ |y|, if |y|≧b
│ b, if |y|＜b,
╰
the four equations can be combined into a single one as follows:

(max(|x|, a) - a)² + (max(|y|, b) - b)² = c²,

or equivalently,

max(|x| - a, 0)² + max(|y| - b, 0)² = c² ...... (I)

Using the formula max(z, 0) = (z + |z|) / 2,
the above equation can be expanded to

(|x| - a + │|x| - a│)² + (|y| - b + │|y| - b│)² = (2c)² ...... (II)

Note:

When a or b is zero (but not both), the equation (I) describes a stadium (curve).
When c is zero (a and b are positive), the equation (I) describes a rectangular region rather than a rectangle.
The equation max(|x| - a, |y| - b) = 0 describes a rectangle.
The equation max(|x| - a, 0)² + max(|y| - b, 0)² + max(|z| - c, 0)² = d² describes a rounded cuboid.

max(|x|, 0)² + max(|y| - 1, 0)² = 2²

(a = 0)

max(|x| - 3, 0)² + max(|y|, 0)² = (1.5)²

(b = 0)

max(|x| - 3, 0)² + max(|y| - 2, 0)² = 0

(c = 0)

max(|x| - 4, 0)² + max(|y| - 3, 0)² + max(|z| - 1, 0)² = 1

Funny Boxing Champion (2)

2016-10-11T19:01:00.001-07:00

( Mathematical software used: gnuplot )

Funny Boxing Champion (1)

2016-10-05T06:16:00.000-07:00

Equations (方程式):

(1) body (身体):

x² + y² - 72[exp(1/(sin(y+x) - 1)) ± exp(1/(sin(y-x) - 1))]² = 5/2

(2) eye sockets (眼眶):

(abs(x) - π/4)² + (y - 3π/2)² = (π/8)²

(3) eyes (眼睛):

arcsin²(k(abs(x) - π/4)) = arcsin²(k(y - 3π/2))

(4) mouth (嘴巴):

arcsin²(kx) = arcsin²(k(y - 19π/16))

(5) navel (肚臍):

arcsin²(kx) = arcsin²(k(y + π/3))

(8 ≦ k ≦ 10)

( Mathematical software used: gnuplot )

Tiling using Rounded Crosses

2016-08-21T21:41:00.001-07:00

( Mathematical software used: Graph )

Related posts:

Rounded Cross - cross with rounded corners (圓角十字形)

2016-08-17T22:53:00.000-07:00

gnuplot script:

a = 2
b = 1
c = 1
w = 8
max(x,y) = x < y ? y : x
min(x,y) = x < y ? x : y
fun(x,y) = \
min(min(max(abs(x)-b,0)**2 + max(abs(y)-(a+b+2*c),0)**2 - c**2, \
max(abs(x)-(a+b+2*c),0)**2 + max(abs(y)-b,0)**2 - c**2), \
max((abs(x)-(b+c))**2 + (abs(y)-(b+c))**2 - c**2, \
c**2 - (abs(x)-(b+2*c))**2 - (abs(y)-(b+2*c))**2))
set xrange [-w:w]
set yrange [-w:w]
set size ratio -1
set samples 512
set isosamples 512
set contour base
set cntrparam levels discrete 0.0
unset key
unset surface
set table 'fun.dat'
splot fun(x,y)
unset table
plot 'fun.dat' w l lw 1.5 lc rgb "red"

(a = 2, b = 1, c = 1)

(a = 4, b = 0.5, c = 0.5)

(a = 0, b = 0, c = 2)

(a = 0, b = 2, c = 4/3)

(a = 2, b = 0, c = 4/3)

( Mathematical softwares used: Graph, gnuplot )

Related posts:

How to Draw the Olympic Rings Mathematically?

2016-08-07T00:42:00.000-07:00

Inequalities (Relations) for the Olympic Rings:

(Blue Ring) min(max(R₁(x,y), -y), max(R₁(x,y), y, -R₄(x,y))) < 0
(Black Ring) min(max(R₂(x,y), -y, -R₄(x,y)), max(R₂(x,y), y, -R₅(x,y))) < 0
(Red Ring) min(max(R₃(x,y), -y, -R₅(x,y)), max(R₃(x,y), y)) < 0
(Yellow Ring) min(max(R₄(x,y), -y, -R₁(x,y)), max(R₄(x,y), y, -R₂(x,y))) < 0
(Green Ring) min(max(R₅(x,y), -y, -R₂(x,y)), max(R₅(x,y), y, -R₃(x,y))) < 0

Functions:

R₁(x,y) = (sqrt((x+2d)² + (y-h)²) - a)² - b²
R₂(x,y) = (sqrt(x² + (y-h)²) - a)² - b²
R₃(x,y) = (sqrt((x-2d)² + (y-h)²) - a)² - b²
R₄(x,y) = (sqrt((x+d)² + (y+h)²) - a)² - b²
R₅(x,y) = (sqrt((x-d)² + (y+h)²) - a)² - b²

Constants:

a = 49.25
b = 4.75
d = 58.5
h = 25

Download Examples:

olympic_rings.zip

( Mathematical softwares used: Graph, gnuplot )

Using Trigonometry to Express the Equations of Piecewise Spirals

2016-07-05T23:09:00.000-07:00

Circle Spirals (spirals made of semicircles):

Equations: sin(π sqrt((x + k sgn(y)/2)² + y²)) = 0

k = 1, 2, 3, ..., 8

Equations: cos(π sqrt((x + k sgn(y)/2)² + y²)) = 0

k = 1, 2, 3, ..., 8

Square Spirals (spirals made of half-squares):

Equations: sin(π(abs(x + k sgn(y)/2) + abs(y))) = 0

k = 1, 2, 3, 4

Equations: cos(π(abs(x + k sgn(y)/2) + abs(y))) = 0

k = 1, 2, 3, 4

Regular Polygon Spirals (spirals made of regular semi-polygons):

Equations: sin(π p(x + k sgn(y)/2, y)) = 0

k = 1, 2, 3, ..., 8

p(x,y) = sqrt(x² + y²) sec(π/n) cos(mod(arctan(y/x),2π/n) - π/n)

n = 6

Equations:cos(π p(x + k sgn(y)/2, y)) = 0

k = 1, 2, 3, ..., 8

p(x,y) = sqrt(x² + y²) sec(π/n) cos(mod(arctan(y/x),2π/n) - π/n)

n = 6

( Mathematical softwares used: Graph, gnuplot )

Folding Star Polygons using Equations (用方程式摺星形多邊形)

2016-06-22T21:42:00.000-07:00

Regular star polygon {8/3}

Regular star polygon {12/4}

Regular star polygon {12/5}

Regular star polygon {10/4}

Regular star polygon {15/6}

( Mathematical softwares used: Graph, gnuplot )

Related posts:

Folding Stars using Equations (用方程式摺星星)

2016-06-17T14:23:00.000-07:00

Example 1.

(n = 5, ratio = 1)

gnuplot script 1:

n = 5
ratio = 1.0
A = 2*pi/n
c = 2*cos(A)
d = ratio*c*(1 - cos(A))/(1 + cos(A))
max(x,y) = x < y ? y : x
f1(x,y) = max(max(abs(y - c) - d, y*cos(A) - x*sin(A) - c/cos(A)), \
max(d - abs(y*cos(A) - x*sin(A) - c), y*cos(3*A) - x*sin(3*A) - c + d))
f2(x,y) = f1(y*sin(A) + x*cos(A), y*cos(A) - x*sin(A))
f3(x,y) = f1(y*sin(2*A) + x*cos(2*A), y*cos(2*A) - x*sin(2*A))
f4(x,y) = f1(y*sin(3*A) + x*cos(3*A), y*cos(3*A) - x*sin(3*A))
f5(x,y) = f1(y*sin(4*A) + x*cos(4*A), y*cos(4*A) - x*sin(4*A))
set xrange [-2.5:2.5]
set yrange [-2.5:2.5]
set size ratio -1
set samples 256
set isosamples 256
set contour base
set cntrparam levels discrete 0.0
unset key
unset surface
set table 'f1.dat'
splot f1(x,y)
unset table
set table 'f2.dat'
splot f2(x,y)
unset table
set table 'f3.dat'
splot f3(x,y)
unset table
set table 'f4.dat'
splot f4(x,y)
unset table
set table 'f5.dat'
splot f5(x,y)
unset table
plot 'f1.dat' w l lw 2 lc rgb "#0099ff", \
'f2.dat' w l lw 2 lc rgb "#0099ff", \
'f3.dat' w l lw 2 lc rgb "#0099ff", \
'f4.dat' w l lw 2 lc rgb "#0099ff", \
'f5.dat' w l lw 2 lc rgb "#0099ff"

Example 2.

(n = 6, ratio = 1)

gnuplot script 2:

n = 6
ratio = 1.0

A = 2*pi/n

c = 1

d = ratio*c/3

max(x,y) = x < y ? y : x

f1(x,y) = max(max(abs(y - c) - d, y - sqrt(3)*x - 4*c), \

max(d - abs(y*cos(A) - x*sin(A) - c), y*cos(4*A) - x*sin(4*A) - c + d))

f2(x,y) = f1(y*sin(A) + x*cos(A), y*cos(A) - x*sin(A))

f3(x,y) = f1(y*sin(2*A) + x*cos(2*A), y*cos(2*A) - x*sin(2*A))

f4(x,y) = f1(y*sin(3*A) + x*cos(3*A), y*cos(3*A) - x*sin(3*A))

f5(x,y) = f1(y*sin(4*A) + x*cos(4*A), y*cos(4*A) - x*sin(4*A))

f6(x,y) = f1(y*sin(5*A) + x*cos(5*A), y*cos(5*A) - x*sin(5*A))

set xrange [-2.5:2.5]

set yrange [-2.5:2.5]

set size ratio -1

set samples 256

set isosamples 256

set contour base

set cntrparam levels discrete 0.0

unset key

unset surface

set table 'f1.dat'

splot f1(x,y)

unset table

set table 'f2.dat'

splot f2(x,y)

unset table

set table 'f3.dat'

splot f3(x,y)

unset table

set table 'f4.dat'

splot f4(x,y)

unset table

set table 'f5.dat'

splot f5(x,y)

unset table

set table 'f6.dat'

splot f6(x,y)

unset table

plot 'f1.dat' w l lw 2 lc rgb "#0099ff", \

'f2.dat' w l lw 2 lc rgb "#0099ff", \

'f3.dat' w l lw 2 lc rgb "#0099ff", \

'f4.dat' w l lw 2 lc rgb "#0099ff", \

'f5.dat' w l lw 2 lc rgb "#0099ff", \

'f6.dat' w l lw 2 lc rgb "#0099ff"

Example 3.

(n = 8, ratio = 1)

Download Graph files:

folding_stars.zip

( Mathematical softwares used: Graph, gnuplot )

Related posts:

Folding Star Polygons using Equations

Momotaro Peach Boy (桃太郎)

2016-06-12T07:20:00.000-07:00

Inequalities:

(x² + 4(y+1)²) [(x² + y² - 3)sqrt(x² + y²) - a(1 - by/sqrt(x² + y²))^ln(c + abs(x)/sqrt(x² + y²))] < d

a = 5, -1.25, -2.1, -2.2, -2.36, -2.6

b = 0.84, c = 0.5, d = 0.1

( Softwares used: Graph, PhotoScape )

Lion Man

2016-06-10T04:30:00.000-07:00

Equations:

(x² + y² - 3) sqrt(x² + y²)

= a(1 + sign(a)by/sqrt(x² + y²))^ln(c + abs(x)/sqrt(x² + y²))

Examples:

Download page

Weird Face - a family of implicit polar equations

2016-06-05T23:28:00.000-07:00

( Softwares used: gnuplot, PhotoScape )

Face Slimming

2016-06-05T23:15:00.000-07:00

The main equations are:

r³ - 3r + 1.95 = 4 abs(sinθ)^(c + sinθ)

(c = 1, 2, 3, ..., 11)

( Softwares used: gnuplot, PhotoScape )