Euclides, in his book The Elements, defines a proportion derived from a division of a segment into its "extreme and mean ratio".

Euclid's definition reads:

"A straight line is said to have been cut in extreme and mean ratio when ,as the whole line is to the greater segment, so is the greater to the less." (Book IV, Definition 3)

Now we call this ratio the golden section, the golden ratio or the divine proportion. It is usually denoted by the greek letter phi, the initial letter of Phidias's name.

Euclid's construction for the regular pentagon depends on this ratio. Two crossing diagonals of a regular pentagon divide one another in extreme and mean ratio.

Using a strip of paper we can made a knot and get a pentagon and a pentagram.

We also find the golden ratio in the dodecahedron and the icosahedron.

A rectangle is called a golden rectangle if it has its sides in the golden ratio. If we cut a golden rectangle into a square and a small rectangle this small rectangle is a golden rectangle.

We can deduce the value of the golden ratio.

The big rectangle and the small one are similar. We can write the ratio:

Clearing denominators we get a second degree equation:

The positive solution of this ecuation is the golden ratio:

LINKS

	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.
	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.
	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.
	Dilative rotation A Dilative Rotation is a combination of a rotation an a dilatation from the same point.
	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.
	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.