The golden rectangle and the dilative rotation
When we divide a golden rectangle in a square and another golden rectangle this new rectangle is similar to the initial one. A dilative rotation transforms one into the other.
We can see that dilative rotation in three ways: a rotation followed by an expansion,
an expansion followed 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.
REFERENCES
Coxeter - Introduction to Geometry (John Whiley and sons)
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The diagonal of a regular pentagon are in golden ratio to its sides and the point of intersection of two diagonals of a regular pentagon are said to divide each other in the golden ratio or 'in extreme and mean 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.
A Dilative Rotation is a combination of a rotation an a dilatation from the same point.
Two equiangular spirals contains all vertices of golden rectangles.
One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.
The twelve vertices of an icosahedron lie in three golden rectangles. Then we can calculate the volume of an icosahedron
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.
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.
In an equiangular spiral the angle between the position vector and the tangent is constant.
Two transformations of an equiangular spiral with the same general efect.