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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 Wolfram Language package PolyhedronOperations` . Geodesic polyhedra computed using a slightly different method are implemented in the Wolfram Language as GeodesicPolyhedron[poly].
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. R. Buckminster Fuller subsequently popularized so-called geodesic domes, and explored them far more thoroughly. Fuller's original dome was constructed from an icosahedron by adding isosceles triangles about each polyhedron vertex and slightly repositioning the polyhedron vertices. In such domes, neither the polyhedron vertices nor the centers of faces necessarily lie at exactly the same distances from the center. However, these conditions are approximately satisfied.
In the geodesic domes discussed by Kniffen (1994), the sum of polyhedron vertex angles is chosen to be a constant. Given a Platonic
solid, let 
be the number of edges, 
the number of vertices,
 |
(1)
|
be the number of edges meeting at a polyhedron vertex and 
be the number of edges of the constituent
polygon. Call the angle of the old polyhedron
vertex point 
and the angle of the new polyhedron vertex point
. Then
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
Solving for 
gives
 |
(5)
|
 |
(6)
|
and
 |
(7)
|
The polyhedron vertex sum is
 |
(8)
|
| solid |  |  |  |  |  |  |  |
| tetrahedron | 6 | 4 | 3 | 3 |  |  |  |
| octahedron | 12 | 6 | 4 | 3 |  |  |  |
| cube | 12 | 8 | 3 | 4 |  |  |  |
| dodecahedron | 30 | 20 | 3 | 5 |  |  |  |
| icosahedron | 30 | 12 | 5 | 3 |  |  |  |
Wenninger and Messer (1996) give general formulas for solving any geodesic chord factor and dihedral angle in a geodesic dome.
