The golden rectangle and two equiangular spirals

In one animation we can see the dilatation that takes an spiral into the other.

In the second animation we can see the rotation that transforms an spiral into the other.

LINKS

	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.
	Dilation and rotation in an equiangular spiral Two transformations of an equiangular spiral with the same general efect.
	Equiangular spiral In an equiangular spiral the angle between the position vector and the tangent is constant.
	The golden spiral The golden spiral is a good approximation of an equiangular spiral.
	The golden rectangle and two equiangular spirals Two equiangular spirals contains all vertices of golden rectangles.
	The golden rectangle A golden rectangle is made of an square and another golden rectangle.
	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.
	Regular dodecahedron One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.
	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
	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.
	Dilative rotation A Dilative Rotation is a combination of a rotation an a dilatation from the same point.