Wilson Dome, Pukekohe,
New Zealand
The Icosahedron
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.
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...
## Other PolyhedraIt 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 aspectsIt 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. Please submit all questions and comments to geodesicsnz@geocities.com |