The golden rectangle
A rectangle can be divided in two pieces: a square with the smaller side and another rectangle.
For a certain proportion of the sides of the initial rectangle, by that procedure we obtain a similar rectangle to the previous one.
Then we have a golden rectangle.
If we start from a golden rectangle that procedure can be repeated indefinitely.
The animation shows that this division of a golden rectangle and we can imagine how the process can be considered infinite.
We can see 4 straight lines, orthogonal in pairs, that they contain all the vertices of those infinites rectangles. Each one of these pairs of straight lines bisects the other pair.
This golden or divine proportion can be expresed in this way: the ratio of the larger part to the smaller is equal to the ratio of the whole to the larger part.
Then we obtain the golden number
This infinite process suggests that the sides of a golden rectangle are incommensurable or, otherwise, that the golden number is irrational.
REFERENCES
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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 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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The golden spiral
The golden spiral is a good approximation of an equiangular spiral.
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The golden rectangle and two equiangular spirals
Two equiangular spirals contains all vertices of golden rectangles.
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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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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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Dilative rotation
A Dilative Rotation is a combination of a rotation an a dilatation from the same point.
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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.
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