From Wolfram MathWorld
A geodesic dome is a triangulation of a Platonic solid or other polyhedron to produce a close approximation to a sphere (or hemisphere). The th order geodesation operation replaces each polygon of the polyhedron by the projection onto the circumsphere of the order- regular tessellation of that polygon. The above figure shows base solids (top row) and geodesations of orders 1 to 3 (from top to bottom) of the cube, dodecahedron, icosahedron, octahedron, and tetrahedron (from left to right), computed using Geodesate[poly, n] in the Mathematica package PolyhedronOperations` . The first geodesic dome was built in Jena, Germany in 1922 on top of the Zeiss optics company as a projection surface for their planetarium projector. In the geodesic domes discussed by Kniffen (1994), the sum of polyhedron vertex angles is chosen to be a constant. be the number of edges, the number of vertices, be the number of edges meeting at a polyhedron vertex and be the number of edges of the constituent polygon. .
Related: Dome construction