Using Trigonometry to Express the Equations of Piecewise Spirals

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 )

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 )

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 )

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 )

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 )

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 )