The golden spiral
Drawing up an arc of circumference in each square we get a golden spiral.
These vertices also belong to an equiangular spiral
When studying the golden rectangle and dilative rotation also we can found that spiral.
"This true spiral is closely approximated by the artificial spiral formed by circular quadrants inscribed in the successive squares. (But the true spiral cuts the sides of the squares at very small angles, instead of touching them)" Coxeter.
Using the zoom we can see how the equiangular spiral cuts the sides of the squares. Dragging with the mouse right button we can move all the drawing. We can see (two different animations) how a dilative rotation transform one golden rectangle into another.
REFERENCES
Coxeter - Introduction to Geometry (John Whiley and sons)
LINKS
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The golden rectangle and two equiangular spirals
Two equiangular spirals contains all vertices of golden rectangles.
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Equiangular spiral
In an equiangular spiral the angle between the position vector and the tangent is constant.
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The golden rectangle and the dilative rotation
A golden rectangle is made of an square an another golden rectangle. These rectangles are related through an dilative rotation.
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Regular dodecahedron
One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.
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The icosahedron and its volume
The twelve vertices of an icosahedron lie in three golden rectangles. Then we can calculate the volume of an icosahedron
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Dilative rotation
A Dilative Rotation is a combination of a rotation an a dilatation from the same point.
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