Sunday, August 17, 2014

Spiral Inequality

(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

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 )