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 )

Fractal Spacetree - a mothership-like fractal tree (碎形太空樹 - 像太空母艦般的碎形樹)

( Mathematical software used: GeoGebra )

Fractal Hair (2) - Super Saiyan Style Hair (碎形頭髮 2 - 超級賽亞頭)

hair = a ternary fractal tree

face = an implicit curve [f(x,y)=0]

( Mathematical software used: GeoGebra )

Tree Star - Star in a Quinary Fractal Tree (樹星 - 五元碎形樹裡的星星)

(GIF animation)

(GIF animation)

This looks like the Sierpinski pentagon without central pentagon.
(1322x1322 pixel²)

(GIF animation)

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(826x826 pixel²)

( Mathematical software used: GeoGebra )

Fractal Hair (1) - Fractal Wig (碎形頭髮 1 - 碎形假髮)

wig = a quaternary fractal tree

face = an implicit curve [f(x,y)=0]

Example Files:

• Download the GeoGebra(.ggb) file here.

( Mathematical software used: GeoGebra )

Binary Fractal Tree (二元碎形樹)

θ₁=30°, θ₂=35°, k₁=0.65, k₂=0.75

θ₁=15°, θ₂=45°, k₁= k₂=0.5√2

θ₁=30°, θ₂=60°, k₁=0.65, k₂=0.75

θ₁=20°, θ₂=80°, k₁=k₂=0.65

θ₁=10°, θ₂=90°, k₁= k₂=0.5√2

θ₁=20°, θ₂=90°, k₁= k₂=0.5√2

θ₁=θ₂=45°, k₁= k₂=0.5√2

θ₁=45°, θ₂=90°, k₁=0.7, k₂=0.65

θ₁=45°, θ₂=120°, k₁=0.75, k₂=0.6

θ₁=90°, θ₂=345°, k₁= k₂=0.5√2

θ₁=θ₂=90°, k₁= k₂=0.7

Example Files:

• Download the GeoGebra(.ggb) file here.

( Mathematical software used: GeoGebra )

Pythagoras Trees (畢達哥拉斯樹)

Two Pythagorean Tree-Fighters (top view)

Four Pythagoras Trees

Pythagoras Shoes (green)

Pythagoras Shoes (golden)

( Mathematical software used: GeoGebra )

Fractal Star - Star fractal using GeoGebra

(GIF animation)

(1275x1275 pixels)

(GIF animation)

(1240x1240 pixels)

(GIF animation)

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(GIF animation)

(1240x1240 pixels)

(GIF animation)

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(GIF animation)

(1179x1179 pixels)

(GIF animation)

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( Mathematical software used: GeoGebra )