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Wilson Dome, Pukekohe, New Zealand

The Icosahedron
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I was once a speaker at the N.Z.Institute of Building Inspectors conference. I had the good fortune to follow two Forest Research Institute wood scientists, whom had spoken for two hours about the moisture content of timber framing. 
I started by saying, " I'm going to tell you about a four frequency oblate superellipsoidal icosahedron." 
Someone yelled out, "Spell it!
I retorted, "Clockwise or anti-clockwise?"  My audience was laughing and awake.

Consider a one frequency icosahedron (1v). It consists of 20 equilateral triangles. It seems to be the most useful polyhedron for dome building.Each vertex is the same distance from the centre of this polyhedron and thus each vertex is on the surface of an imaginary sphere. Note that one frequency is often written as 1v, two frequency as 2v and so on.........

1v icosahedron (20 equilateral triangles) 

    2v icosahedron with 4 triangles per icosa face
Now take one of the original equilateral triangles. It can be divided up into smaller triangles. Above is a 2 frequency (2v) icosahedron. The icosa face (or basic triangle of the icosahedron) has been broken up into four triangles. The side of the icosa face has been divided into two, thus two frequency. Each vertex is on the surface of an imaginary sphere. The higher the frequency, the more spherical the polyhedron looks.

3 frequency (3v) with 9 triangles  per icosa face 

                                4 frequency (4v) with 16 triangles per icosa face. 
                                           And so on...

bishop's dome
4v spheroidal icosahedron built as a chapel for the Catholic Bishop of 
the Waikato, NZ, the late Ed Gaines, a true gentleman and friend.

Other Polyhedra

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It is not just the icosahedron which can be used to make a dome. Other polyhedra (such as the tetrahedron, the octahedron, the dodecahedron and others) can be used. The same principles apply. 

One frequency (1v) tetrahedron

                                        One frequency (1v) Octahedron - half an octahedron is a pyramid.
Two frequency (2v) octahedron 

                                Three frequency (3v) octahedron

 
 
 

Other shapes and aspects

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It is not only the sphere, which can be used as the shape of a dome. In all the examples above, each vertex occurs on the surface of an imaginary sphere. Each vertex could occur on the surface of an ellipsoid (squashed or stretched), a super-spheroid, a super-ellipsoid or even a free form amorphous flow. And the whole triangulated network (i.e. the dome) can be rotated to have any point at the top, whether this is a vertex, the mid-point of an edge or the centroid of a face. 
This rotation can be done first, i.e. the spheroid is rotated and then squashed or stretched, or the stretching or squashing can be done first and then the ellipsoid ( or super-ellipsoid) can be rotated. For those really interested in the mathematics, the best reference is "Geodesic Math and How to Use It" by Prof. Hugh Kenner.
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