Tuesday, November 4, 2014

Tic-Tac-Toe (井字遊戲)



Let f(a,b,x,y) = 1/exp(1/(a² - x²)²) + 1/exp(1/(a² - y²)²) - b

Equations:

Blue grid (#) ..... f(2, 0.999, x, y) = 0
Green nought (o) ..... f(0.125, 0.85, x, y) = 0
Red cross (x) ..... f(0.125, 0.999, 2(y+x), 2(y-x)) = 0


(Tessellations)


When the parameter 'a' changes from 0 to 2, the graph
changes shape from a plus sign(+) to a number sign(#).


In order to avoid discontinuities, we redefine f as follows:
f(a,b,x,y) = 1/exp(1/(s + (a² - x²)²)) + 1/exp(1/(s + (a² - y²)²)) - b,
where s(=0.001) is a small positive number.


Graph example:
  • Click here to download the example file (tic_tac_toe.zip).


( Mathematical software used: Graph )

Sunday, August 17, 2014

Spiral Inequality (螺旋不等式)

(1) Archimedean spiral





(2) Fermat's spiral (Parabolic spiral)





(3) Hyperbolic spiral (Reciprocal spiral)





(4) Lituus





(5) Logarithmic spiral





( Mathematical software used: gnuplot )

Sunday, August 10, 2014

Spiral Polyskelion

Spiral Types:


(1) Archimedean spiral: r(θ) = a*θ


(1 < number of spiral turns < 2)

(2 < number of spiral turns < 3)


(2) Fermat's spiral (Parabolic spiral): r(θ) = a*sqrt(θ)


(1 < number of spiral turns < 2)

(2 < number of spiral turns < 3)


(3) Hyperbolic spiral (Reciprocal spiral): r(θ) = a/θ




(4) Lituus: r(θ) = a/sqrt(θ)




(5) Logarithmic spiral: r(θ) = a*exp(b*θ)


(b = 0.25)

(b = 0.618)


( Mathematical software used: GeoGebra )

Friday, August 8, 2014

Spiral of Spirals

Spirals (green, magenta) around a Spiral Dodecaskelion (grey)

(GIF animation)

A spiral of spirals of spirals.
The spirals are logarithmic spirals: r(θ) = a exp(bθ), b = 0.2.

Spirals of spirals of spirals of ..... (b = 0.2)


A spiral of spirals of spirals.
The spirals are logarithmic spirals with b = 0.30635.

Spirals of spirals of spirals of ..... (b = 0.30635)


Logarithmic spirals
The pair of big spirals: b = 0.1
The small spirals: b = 0.2

The small logarithmic spirals: b = 0.5

The small logarithmic spirals: b = 0.2, 0.5

The small logarithmic spirals: b = 0.5, 0.8


The spirals are Archimedean spirals.
Small spirals: 1 < number of spiral turns < 2

The spirals are Archimedean spirals.
Small spirals: 2 < number of spiral turns < 3


( Mathematical softwares used: GeoGebra, gnuplot )

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 (螺旋海星)

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 )