Friday, July 25, 2014

Tiling using Spiral Hexaskelions


The spirals are logarithmic spirals:
r(θ) = a exp(bθ), b = 0.25.


The spirals are logarithmic spirals:
r(θ) = a exp(bθ), b = golden ratio ~ 0.618.


The spirals are Archimedean spirals.
(1 < number of spiral turns < 2)


The spirals are Archimedean spirals.
(2 < number of spiral turns < 3)


The spirals are logarithmic spirals:
r(θ) = a exp(bθ), b = 0.05, 0.1, 0.15, 0.2, ..., 0.8.


( Mathematical software used: GeoGebra )

Thursday, July 24, 2014

Spiral Starfish - a logarithmic spiral pentaskelion

A pentaskelion with five logarithmic spirals r(θ) = a*exp(b*θ),
b ~ 0.30635 (same as that of the golden spiral).

b = 0.25

b = 0.5

b = golden ratio ~ 0.618


( Mathematical software used: gnuplot )

Saturday, July 5, 2014

Flower of Golden Triangle Spirals





The Graph file used for this post can be found here.


( Mathematical software used: Graph )

Tuesday, July 1, 2014

Double Spirals - images of nested regular polygons under conformal mappings

(n = 6, Draw type = Lines)

(n = 4, Draw type = Dots)


These are images of nested regular polygons (n=6 & n=4)
under the Mobius transformation z -> 3(z-1)/(z+1).


( Mathematical software used: Graph )

Sunday, June 29, 2014

Sinusoidal Strings - made from a ring of circles

Sinusoidal Strings of Pebbles (r=0.1, n=32)

(r=0.15, n=42)

(r=0.4, n=60)

A ring of circles can be expressed as follows:
x = cos(t-mod(t,s)) + r*cos(n*mod(t,s))
y = sin(t-mod(t,s)) + r*sin(n*mod(t,s))
r=0.1, n=32, s=2*pi/n, t=0...2*pi.

Note that mod(t,s) = t - s*floor(t/s).

gnuplot Examples:


Method 1:
r = 0.1
n = 32
s = 2*pi/n
f(t) = cos(s*floor(t/s)) + r*cos(n*(t-s*floor(t/s)))
g(t) = sin(s*floor(t/s)) + r*sin(n*(t-s*floor(t/s)))
theta(t) = atan2(g(t), f(t))
set terminal wxt enhanced font "Arial, 10"
set xtics ('-2π' -2*pi, '-π' -pi, '0' 0, 'π' pi, '2π' 2*pi)
set xrange [-2*pi-0.4:2*pi+0.4]
set yrange [-pi-0.2:pi+0.2]
set size ratio -1
set samples 10000
set multiplot
set parametric
unset key
do for [k=-2:2:2]{
  plot [-pi:pi] theta(t) + k*pi, f(t) w p pt 0.5 lc rgb "#0099cc"
}
do for [k=-2:2:2]{
  plot [-pi:pi] theta(t) + k*pi, g(t) w p pt 0.5 lc rgb "#ff00ff"
}
unset multiplot


Method 2:
r = 0.1
n = 32
s = 2*pi/n
f(t) = cos(s*floor(t/s)) + r*cos(n*(t-s*floor(t/s)))
g(t) = sin(s*floor(t/s)) + r*sin(n*(t-s*floor(t/s)))
angle(t) = atan2(g(t), f(t))
theta(t) = angle(t - 2*pi*floor((t+pi)/(2*pi))) + 2*pi*floor((t+pi)/(2*pi))
set terminal wxt enhanced font "Arial, 10"
set xtics ('-2π' -2*pi, '-π' -pi, '0' 0, 'π' pi, '2π' 2*pi)
set xrange [-2*pi-0.4:2*pi+0.4]
set yrange [-pi-0.2:pi+0.2]
set size ratio -1
set samples 10000
set parametric
unset key
plot [-4*pi:4*pi] theta(t), f(t) w p pt 0.5 lc rgb "#0099cc", \
                      theta(t), g(t) w p pt 0.5 lc rgb "#ff00ff"


The result plot:

Graph Examples:

  • Method 1 ( sinusoidal_strings_01.grf )
  • Method 2 ( sinusoidal_strings_02.grf )


( Mathematical softwares used: Graph, gnuplot )

Friday, December 6, 2013

Infinite Families of Tangent Circles (3)

Type 1.



Equations of the Circles:
(x + a(1 - p^ceil(k/2)))² + (y + b(1 - p^floor(k/2)))² = d^k
a = (1 - r)/(1 + r²)
b = r(1 - r)/(1 + r²)
d = r²
p = -r²
0 < r < 1
k = 0, 1, 2, 3, ....




Type 2.



Equations of the Circles:
(x + a(1 - p^floor(k/2)))² + (y + b(1 - p^ceil(k/2)))² = d(1 - r^(k+1))²
a = r²/(1 + r²)
b = r/(1 + r²)
d = (1/(1-r))²
p = -r²
0 < r < 1
k = 0, 1, 2, 3, ....




Type 3.



Equations of the Circles:
(See the equations for Type 1 and Type 2.)




( Mathematical software used: Graph )

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