Abstract
Polyhedra are ideal frameworks for subdividing spheres, they jump-start the process because their forms symmetrically distribute and initial set of points on the sphere. This chapter defines the characteristics of Polyhedra and the classic Platonic and Archimedean solids that many spherical subdivisions use as references. Their symmetry groups, edge-face-vertex and duals relationships are described as well as their various axes of symmetry. Several techniques show how to define precise Polyhedra to deriving an initial point set for subsequent subdivision. The vertices of many Polyhedra lay on a circumscribed sphere while inscribed spheres often define tangents to center face points. These relationships are useful when defining points for various gridding systems. Great circles can subdivide Platonic and Archimedean solids into highly symmetrical work areas such as Schwarz triangles. Subdividing one of these areas, in effect, subdivides the entire sphere since its geometry can be reflected, mirrored, and rotated over to cover the entire sphere without gaps or overlaps.
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