Geodesic polyhedron

A geodesic polyhedron is a convex polyhedron made from triangles that approximates a sphere. They usually have icosahedral symmetry which have 6 triangles at a vertex, except 12 vertices which have 5 triangles. It is the dual of a corresponding Goldberg polyhedron, with mostly hexagonal faces.

In Magnus Wenninger's Spherical models, polyhedra are given geodesic notation in the form {3,q+}_b,c, where {3,q} is the Schläfli symbol for the regular polyhedron with triangular faces, with gonal faces, and q-valance vertices. The + symbol indicates the valence of the vertices being increased. b,c represent a subdivision description, with 1,0 representing the base form. There are 3 symmetry classes of forms: {3,3+}_1,0 for a tetrahedron, {3,4+}_1,0 for a octahedron, and {3,5+}_1,0 for a icosahedron.

The dual notation for Goldberg polyhedra is {q+,3}_b,c, with valence-3 vertices, with q-gonal and hexagonal faces. There are 3 symmetry classes of forms: {3+,3}_1,0 for a tetrahedron, {4+,3}_1,0 for a cube, and {5+,3}_1,0 for a dodecahedron.

Values for b,c are divided into 3 classes:

Subdivisions in class III here do not line up simply with the original edges. The subgrids can be extracted by looking at a triangular tiling, positioning a large triangle on top of grid vertices and walking paths from one vertex b steps in one direction, and a turn, either clockwise or counterclockwise, and then another c steps to the next primary vertex.

For example the icosahedron is {3,5+}_1,0, and pentakis dodecahedron, {3,5+}_1,1 is seen as a regular dodecahedron with pentagonal faces divided into 5 triangles.

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