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.
Golden Spiral in a furniture designed and made by Roberto Cardil (pine wood).
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The golden ratio
From Euclid's definition of the division of a segment into its extreme and mean ratio we introduce a property of golden rectangles and we deduce the equation and the value of the golden ratio.
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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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Dilation and rotation in an equiangular spiral
Two transformations of an equiangular spiral with the same general efect.
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The golden spiral
The golden spiral is a good approximation of an equiangular spiral.
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The golden rectangle
A golden rectangle is made of an square and another golden rectangle.
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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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Dilative rotation
A Dilative Rotation is a combination of a rotation an a dilatation from the same point.
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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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Leonardo da Vinci: Drawing of a dodecahedron made to Luca Pacioli's De divina proportione.
Leonardo da Vinci made several drawings of polyhedra for Luca Pacioli's book 'De divina proportione'. Here we can see an adaptation of the dodecahedron.
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