The golden rectangle is a beautiful contruction related with some spirals and the dodecahedron.

In this page we are going to study the relation between the golden ratio and the icosahedron.

We can put three golden rectangles and make a well known construction. These rectangles have 12 vertices. The distance between any pair of neighbouring points is equal to the short side of one golden rectangle. Then, these 12 points coincide with the 12 vertices of an icosahedron.

An icosahedron has twenty equilateral triangles. In our case, the length of the sides is 2.

The area of one of the twenty equilateral triangles that the icosahedron of side 2 have is:

The volume of an icosahedron of side 1 is one eigth of the volume of an icosahedron of side 2.

The volume of the icosahedron of side 1 is the same as the volume of two and a half pyramids. We need to calculate the height of one of this pyramids: the distance between the center and one triangular face.

There are two similar triangles and we can write:

Then the volume of one pyramid is:

And the volume of an icosahedron of side 1 is:

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If the twelve edges of an octahedron are dividen in the golden ratio (in some order) these vertices are the vertices of an icosahedron. In this image we can see an icosahedron, an octahedron and an tetrahedron, one inside the other.

We can build the structure of an icosahedron using six sticks and six elastic bands. It is one of the simplest examples of "tensegrity".

LINKS

	The golden rectangle A golden rectangle is made of an square and another golden rectangle.
	The golden spiral The golden spiral is a good approximation of an equiangular spiral.
	Regular dodecahedron One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.
	Volume of a regular dodecahedron (Flash version) One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.
	The volume of the tetrahedron The volume of a tetrahedron is one third of the prism that contains it.
	Volume of an octahedron The volume of an octahedron is four times the volume of a tetrahedron. It is easy to calculate and then we can get the volume of a tetrahedron.