Outline of the Technology
3.3 Icosahedron
Icosahedron(plural icosahedrons or icosahedra) noun
20sided figure: a solid geometric figure having 20 sides or faces
[Late 16th century. Via late Latin icosahedrum from Greek eikosaedron, from eikosi 'twenty' + hedra 'base'.]
20sided figure: a solid geometric figure having 20 sides or faces
[Late 16th century. Via late Latin icosahedrum from Greek eikosaedron, from eikosi 'twenty' + hedra 'base'.]
icosahedral, adjective
icosahedron (noun)
twenty and over: icosahedron
angular figure: tetragon, polygon, pentagon, hexagon, heptagon, octagon, nonagon, decagon, dodecahedron, icosahedron
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 center 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
The Icosahedron is the most nearly spherical of all the Platonic Solids, but it is not really all that close an approximation of a sphere. To make a rounder sphere you need to relax the requirement that it be completely regular, but it can still be very nearly regular and have all the symmetries of the Icosahedron. To do this you subdivide each of the faces of an Icosahedron and move each of the new vertices away from the center of the sphere until it is the same distance from the center as the original vertices. For example you can subdivide each edge of an Icosahedron into three edges which would subdivide each triangular face into nine smaller triangles. This would be called a ThreeFrequency Icosahedral geodesic. (Some people call this a "Class 1" geodesic, but I think Icosahedral geodesic is more descriptive.)
The exact placement of the new vertices is one of the subtler issues in geodesic design. There are two traditional ways to do this, known as Method 1 and Method 2. In Method 1, you subdivide the edges of the polyhedron into equal length line segments, and then project the vertices out to the sphere radius. In Method 2, you subdivide the angle between polyhedron vertices into equal angles, then find the point on the sphere at that angle. Method 1 yields triangles that are more nearly equilateral. Method 2 gives smoother arcs, and less variation is triangle size.
The number of faces on a subdivided triangle of frequency f is f2, and thus the total number of faces in an Icosahedral geodesic is:
20 f2
To compute the number of edges, note that each face has three edges, and each edge is shared by the two adjacent faces. Thus the number of edges is:
30 f2
To compute the number of vertices, note that each edge has two ends, and that six edges will meet at each vertex except for those that were vertices of the original Icosahedron, where five edges will meet. Thus the number of vertices is:
(2 (30 f2)  12 (5)) ÷ 6 + 12 = 10 f2 + 2
So, for example, a threefrequency geosphere would have 180 triangles, 270 edges, and 92 vertices.
Here are examples of 1, 2, 3, and 4frequency Icosahedral Geodesics that have been colorcoded to indicate which struts share the same length. These are all 3/4 domes, which means that they are 3/4 of a full sphere. For example, if you look at the 1frequency dome below, you can see that it is made up of fifteen triangles, whereas the full Icosahedron has twenty. Also notice that a 1frequency geodesic is just the unsubdivided generative shape.
Geodesics can also be made by subdividing the other Platonic solids. For the Tetrahedron and the Octahedron, the same procedure is used as described above. To use the Cube or the Dodecahedron as the generative shape, you must first triangulate it by adding a new vertex at the center of each face and connecting it to each of the surrounding vertices. The triangulated Cube is also known as the Tetrakis Hexahedron, and the triangulated Dodecahedron is also known as the Pentakis Dodecahedron. Some people call geospheres made by subdividing the Dodecahedron "Class 2", but again I think that dodecahedral geodesic is more descriptive. Using the same logic as above, the number of faces, edges, and vertices is:
Efficiency. A sphere is already efficient: it encloses the most volume with the least surface. Thus, any dome that is a portion of a sphere has the least surface through which to lose heat or intercept potentially damaging winds. A geodesic dome uses a pattern of selfbracing triangles in a pattern that gives maximum structural advantage, thus theoretically using the least material possible. (A "geodesic" line on a sphere is the shortest distance between any two points.)
icosahedron (noun)
twenty and over: icosahedron
angular figure: tetragon, polygon, pentagon, hexagon, heptagon, octagon, nonagon, decagon, dodecahedron, icosahedron
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 center 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
The Icosahedron is the most nearly spherical of all the Platonic Solids, but it is not really all that close an approximation of a sphere. To make a rounder sphere you need to relax the requirement that it be completely regular, but it can still be very nearly regular and have all the symmetries of the Icosahedron. To do this you subdivide each of the faces of an Icosahedron and move each of the new vertices away from the center of the sphere until it is the same distance from the center as the original vertices. For example you can subdivide each edge of an Icosahedron into three edges which would subdivide each triangular face into nine smaller triangles. This would be called a ThreeFrequency Icosahedral geodesic. (Some people call this a "Class 1" geodesic, but I think Icosahedral geodesic is more descriptive.)
The exact placement of the new vertices is one of the subtler issues in geodesic design. There are two traditional ways to do this, known as Method 1 and Method 2. In Method 1, you subdivide the edges of the polyhedron into equal length line segments, and then project the vertices out to the sphere radius. In Method 2, you subdivide the angle between polyhedron vertices into equal angles, then find the point on the sphere at that angle. Method 1 yields triangles that are more nearly equilateral. Method 2 gives smoother arcs, and less variation is triangle size.
The number of faces on a subdivided triangle of frequency f is f2, and thus the total number of faces in an Icosahedral geodesic is:
20 f2
To compute the number of edges, note that each face has three edges, and each edge is shared by the two adjacent faces. Thus the number of edges is:
30 f2
To compute the number of vertices, note that each edge has two ends, and that six edges will meet at each vertex except for those that were vertices of the original Icosahedron, where five edges will meet. Thus the number of vertices is:
(2 (30 f2)  12 (5)) ÷ 6 + 12 = 10 f2 + 2
So, for example, a threefrequency geosphere would have 180 triangles, 270 edges, and 92 vertices.
Here are examples of 1, 2, 3, and 4frequency Icosahedral Geodesics that have been colorcoded to indicate which struts share the same length. These are all 3/4 domes, which means that they are 3/4 of a full sphere. For example, if you look at the 1frequency dome below, you can see that it is made up of fifteen triangles, whereas the full Icosahedron has twenty. Also notice that a 1frequency geodesic is just the unsubdivided generative shape.
Geodesics can also be made by subdividing the other Platonic solids. For the Tetrahedron and the Octahedron, the same procedure is used as described above. To use the Cube or the Dodecahedron as the generative shape, you must first triangulate it by adding a new vertex at the center of each face and connecting it to each of the surrounding vertices. The triangulated Cube is also known as the Tetrakis Hexahedron, and the triangulated Dodecahedron is also known as the Pentakis Dodecahedron. Some people call geospheres made by subdividing the Dodecahedron "Class 2", but again I think that dodecahedral geodesic is more descriptive. Using the same logic as above, the number of faces, edges, and vertices is:
Generative shape

Faces

Edges

Vertices

Tetrahedron  4 f2  6 f2  2 f2 + 2 
Tetrakis Hexahedron  24 f2  36 f2  12 f2 + 2 
Octahedron  8 f2  12 f2  4 f2 + 2 
Icosahedron  20 f2  30 f2  10 f2 + 2 
Pentakis Dodecahedron  60 f2  90 f2  30 f2 + 2 
Efficiency. A sphere is already efficient: it encloses the most volume with the least surface. Thus, any dome that is a portion of a sphere has the least surface through which to lose heat or intercept potentially damaging winds. A geodesic dome uses a pattern of selfbracing triangles in a pattern that gives maximum structural advantage, thus theoretically using the least material possible. (A "geodesic" line on a sphere is the shortest distance between any two points.)