Outline of the Technology
3.2 Polyhedra
There exist only five polyhedra whose faces are all congruent regular polygons, and where the same number of faces meet at every vertex. This was discovered by the Greek philosopher Plato about 2,400 years ago. So these solids are collectively known as the Platonic Solids.
There are Only Five Regular Solids
A regular solid is a polyhedron in which all of the edges, vertices, and faces are identical. I.e. the edges are all the same length, the same number of edges meet at each vertex, and the faces are all identical regular polygons.
The ancient Geeks used the following simple reasoning to find all the regular solids. The minimum number of edges in a face (i.e. a polygon) is three. After all, a polygon is defined as an area of a plane enclosed by straight line segments, and there is no way to enclose an area with only one or two edges. Similarly the minimum number of faces that can meet at a vertex is three  since there is no way to enclose a volume with just one or two faces at a vertex.
There are Only Five Regular Solids
A regular solid is a polyhedron in which all of the edges, vertices, and faces are identical. I.e. the edges are all the same length, the same number of edges meet at each vertex, and the faces are all identical regular polygons.
The ancient Geeks used the following simple reasoning to find all the regular solids. The minimum number of edges in a face (i.e. a polygon) is three. After all, a polygon is defined as an area of a plane enclosed by straight line segments, and there is no way to enclose an area with only one or two edges. Similarly the minimum number of faces that can meet at a vertex is three  since there is no way to enclose a volume with just one or two faces at a vertex.
If you place three equilateral triangles around a vertex, the remaining opening can be filled with a fourth triangle. This gives you the Tetrahedron with four triangular faces and four vertices.
What if you try arranging four equilateral triangles around a vertex? You get a pyramid that is open at the bottom. If you add the additional triangles so that there are four faces around the vertices that bound the opening, you get the Octahedron with eight triangular faces and six vertices.
If you group five triangles around each vertex you get the lovely Icosahedron with twenty triangles and twelve vertices. If you try to put six triangles around a vertex you find that all the triangles lie in the same plane. With no curvature at a vertex there is no way the figure can ever be closed, so six or more triangles wont work.
Increasing the number of edges in the faces to four, of course yields a square. And three squares around each vertex of course gives you the alltoofamiliar cube. Naturally if you put four squares around a vertex you once again have a plane that can't enclose space, so four or more squares will never work.
Once again increasing the number of edges in each of the faces to five, you will see that with three regular pentagons around each vertex you get a solid with twelve pentagonal faces and twenty vertices. This is the splendid Dodecahedron. Finally note that: 1) You can't fit four pentagons around a vertex. 2) Three hexagons around a point will all lie in the same plane, and 3) You can't even fit three regular polygons around a vertex if the faces have more than six sides
What if you try arranging four equilateral triangles around a vertex? You get a pyramid that is open at the bottom. If you add the additional triangles so that there are four faces around the vertices that bound the opening, you get the Octahedron with eight triangular faces and six vertices.
If you group five triangles around each vertex you get the lovely Icosahedron with twenty triangles and twelve vertices. If you try to put six triangles around a vertex you find that all the triangles lie in the same plane. With no curvature at a vertex there is no way the figure can ever be closed, so six or more triangles wont work.
Increasing the number of edges in the faces to four, of course yields a square. And three squares around each vertex of course gives you the alltoofamiliar cube. Naturally if you put four squares around a vertex you once again have a plane that can't enclose space, so four or more squares will never work.
Once again increasing the number of edges in each of the faces to five, you will see that with three regular pentagons around each vertex you get a solid with twelve pentagonal faces and twenty vertices. This is the splendid Dodecahedron. Finally note that: 1) You can't fit four pentagons around a vertex. 2) Three hexagons around a point will all lie in the same plane, and 3) You can't even fit three regular polygons around a vertex if the faces have more than six sides
# of Faces

# of Sides in Each Face

# of Faces at Each Vertex

# of Vertices


Tetrahedron 
4

3

3

4

Octahedron 
8

3

4

6

Cube 
6

4

3

8

Dodecahedron 
12

5

3

20

Icosahedron 
20

3

5

12

These are called the Platonic Solids, not because they only think of each other as friends, but because they were given their names by the Greek Philosopher Plato (427?  347 B.C.).
Each Platonic Solid Has a Close Relative
If you place a new vertex in the center of each face of a polyhedron, connect each of those vertices to the corresponding vertices of the adjacent faces, and delete the original edges, you get a new polyhedron that is called the dual of the original. If you try this with the Platonic solids, you will see something interesting. The dual of Dodecahedron is the Icosahedron and the dual of the Icosahedron is the Dodecahedron. In other words the Icosahedron and the Dodecahedron are duals. Similarly the dual of the Octahedron is the Cube and viceversa, so they are duals as well. Are you beginning to see a pattern? But that leaves only one Platonic solid. The dual of the Tetrahedron is... the Tetrahedron, but in a different orientation! Thus the Tetrahedron is the most special polyhedron of all. Not only is it the simplest, but it is also the only one that is its own dual.
You can see these relationships in the table of Platonic solids above. Since the dual operation creates one face for each vertex and one vertex for each face, it's like interchanging the
# of Facescolumn with the # of Vertices column, as well as the # of Sidesin Each Face column with the # of Facesat Each Vertex column.
Geodesics are Built by Subdividing Regular Solids
Geodesic Spheres Can Only Be Neatly Cut in Certain Places
Normally geodesic domes are made by subdividing the triangles of an Icosahedron. But if for example you want to build a half, quarter, or an eight of a full sphere, you need to start with a shape that can be evenly cut that way, such as an Octahedron or a triangulated Cube. This cutoff procedure is referred to as "truncation".
Here is an example of a 5frequency Octahedral Geodesic Sphere.
Note at the center you can see a 4way vertex. There are six of these vertices in the entire sphere corresponding to the six vertices of the original Octahedron. All the other vertices were created by the subdivision process and are thus 6way.
Geodesics Don't Have to be Spheres
Here is an example of a fourfrequency Icosahedral geodesic paraboloid.
Geodesics Can be Made
Out of Great and Small Circles