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

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
• Regular Polygons and Regular Star Polygons

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

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

Tiling using Nested Stars (2)

(A) Tiling using Nested Six-Pointed Stars (and Nested Triangles)

(Clockwise Stars)

(Counterclockwise Stars)

(Type 1: Clockwise Stars & Clockwise Triangles)

(Type 2: Clockwise Stars & Counterclockwise Triangles)

(Type 3: Counterclockwise Stars & Clockwise Triangles)

(Type 4: Counterclockwise Stars & Counterclockwise Triangles)

(Type 1: Clockwise Stars & Clockwise Triangles)

(Type 2: Clockwise Stars & Counterclockwise Triangles)

(Type 3: Counterclockwise Stars & Clockwise Triangles)

(Type 4: Counterclockwise Stars & Counterclockwise Triangles)

(B) Tiling using Nested Four-Pointed Stars (and Nested Squares)

(Clockwise Stars)

(Counterclockwise Stars)

(Clockwise Stars)

(Counterclockwise Stars)

(Type 1: Clockwise Stars & Clockwise Squares)

(Type 2: Clockwise Stars & Counterclockwise Squares)

(Type 3: Counterclockwise Stars & Clockwise Squares)

(Type 4: Counterclockwise Stars & Counterclockwise Squares)

( Mathematical software used: GeoGebra )


Related posts:

• Nested Stars
• Tiling using Nested Stars (1)
• Tiling by Nested Polygons (1)
• Tiling by Nested Polygons (2)
• Tiling using Nested Crosses

Tiling using Nested Stars (1)

The stars and hexagons all rotate in the same direction (clockwise).

The stars rotate clockwise, but the hexagons rotate counterclockwise.

( Mathematical software used: GeoGebra )


Related posts:

• Nested Stars
• Tiling using Nested Stars (2)
• Tiling by Nested Polygons (1)
• Tiling by Nested Polygons (2)
• Tiling using Nested Crosses

Nested Stars

           (1)

---

           (2)

---

           (3)

---

           (4)

---

           (5)

---

           (6)

( Mathematical software used: GeoGebra )


Related posts:

• Nested Regular Polygons
• Nested Ellipses
• Tiling using Nested Stars (1)
• Tiling using Nested Stars (2)