Whichever polyhedron is the basis of our geodesic, we must decide upon a pattern to map on to those of its faces we are going to use in our final structure. We may map any pattern at all onto a face, and choose as many different patterns as there are faces, or indeed map more than one pattern on to a face, or create layers of patterns on to faces to create double or treble layer grid domes. Here we will use the icosahedron as the base polyhedron and have a look at just a few possible face patterns we could choose.
2 Frequency Triangulation
This is the simplest version, where we simply divide each edge of the triangle
in two and join each created mid-point to the next.
3 Frequency Triangulation
This is exactly as above except that we divide each edge into three.
6 Frequency Triangulation
Again a simple pattern, this time dividing each edge into six.
Projected 6 Frequency Pattern
Here the pattern from the image immediately above has been projected to create
the geodesic triangle from the now patterned icosahedral one.
Geodesic projection of another single layer face pattern
This time a greater variation in final triangle size results from a variation in
triangle size in the pattern mapped on to the original icosahedral triangle.
Geodesic projection of a star onto a dodecahedral pentagon
This time a pentagon has been used since a star does not fit on to a triangular face. The dodecahedron is a suitable base polyhedron for this.
Geodesic projection of another pattern mapped on to an icosahedral face.
Another more complex pattern is used here. The resulting dome is dimpled.
A double layer of hexagons
Here we have a double layer grid that sits within a triangle.
Geodesic projection of above hexagonal double layer grid
The double layer grid now curves around the surface of our sphere.
Geodesic projection of above double layer grid created from the top five triangles of an icosahedron