The golden rectangle and the dilative rotation

We can see that dilative rotation in three ways: a rotation followed by an expansion, an expansion followd by a rotation or a continuous dilative rotation. In this case, the vertices follow two equiangular spirals.

Four straight lines contain all the vertices of the rectangles. These four straight lines concur in a point. This one is the center of the transformation. The angle of the turn is a right angle and the homotetic reason is the inverse of the golden number.

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.
	The golden spiral The golden spiral is a good approximation of an equiangular spiral.
	Dilative rotation A Dilative Rotation is a combination of a rotation an a dilatation from the same point.
	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.
	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.
	Multiplying two complex numbers We can see it as a dilatative rotation.
	Standar Paper Size DIN A There is a standarization of the size of the paper that is called DIN A. Successive paper sizes in the series A1, A2, A3, A4, and so forth, are defined by halving the preceding paper size along the larger dimension.
	Equiangular spiral In an equiangular spiral the angle between the position vector and the tangent is constant.
	Dilation and rotation in an equiangular spiral Two transformations of an equiangular spiral with the same general efect.