Wilson Dome, Pukekohe, New Zealand
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 faceNow 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
4 frequency (4v) with 16 triangles per icosa face.
And so on...
One frequency (1v) tetrahedron
One frequency (1v) Octahedron - half an octahedron is a pyramid.Two frequency (2v) octahedron
Three frequency (3v) octahedron
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.