Click on your choice for a detailed explanation - Then look below.
The Theory Platonic And Archimedian Solids Face Variations
Projection Variations Truncations Variations Why Choose Geodesic?
The Platonic and Archimedian solids (see the pictures below) are polyhedra, 3D shapes, that are either completely regular in all respects, as are the platonic, or semi-regular, as are the archimedian.

The platonic solids have regular length edges, are regular in size and shape of faces, and have regular angles throughout. There are five of these.

The archimedian solids are variants of the platonic solids and have regular length edges, but have two or three different faces. The angles are regular within a face, but vary from face to face.

The choice between polyhedra as a starting point for a design depends upon the function and desired aesthetics of the final structure. For example, a dome that wants to sit as a conservatory in a corner between two wings of an existing building would be best as a quarter of a polyhedron with fourfold symmetry, such as the octahedron, but a free standing greenhouse could be based purely on desired aesthetics and any of the polyhedra could be used as a starting point.

The drawing below shows the five platonic polyhedra and how the archimedian polyhedra are derived from them.