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
LINKS
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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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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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