A geodesicpolyhedron is a convex polyhedron made from triangles. They usually have icosahedral symmetry, such that they have 6 triangles at a vertex, except 12 vertices which have 5 triangles. They are the dual of corresponding Goldberg polyhedra with mostly hexagonal faces. Geodesic polyhedra are a good approximation to a sphere for many purposes, and appear in many different contexts. The most well-known may be the geodesic domes designed by Buckminster Fuller, which geodesic polyhedra are named after. Geodesic grids used in geodesy also have the geometry of geodesic polyhedra. The capsids of some viruses have the shape of geodesic polyhedra, and fullerene molecules have the shape of Goldberg polyhedra. Geodesic polyhedra are available as geometric primitives in the Blender 3D modeling software package, which calls them icospheres: they are an alternative to the UV sphere, having a more regular distribution of vertices than the UV sphere. The Goldberg–Coxeter construction is an expansion of the concepts underlying geodesic polyhedra.
Geodesic notation
In Magnus Wenninger's Spherical models, polyhedra are given geodesic notation in the form b,c, where ' is the Schläfli symbol for the regular polyhedron with triangular faces, and q-valence 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: 1,0 for a tetrahedron, 1,0 for an octahedron, and 1,0 for an icosahedron. The dual notation for Goldberg polyhedra is b,c, with valence-3 vertices, with q-gonal and hexagonal faces. There are 3 symmetry classes of forms: 1,0 for a tetrahedron, 1,0 for a cube, and 1,0 for a dodecahedron. Values for b,c are divided into three 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 bsteps 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 1,0, and pentakis dodecahedron, 1,1 is seen as a regular dodecahedron with pentagonal faces divided into 5 triangles. The primary face of the subdivision is called a principal polyhedral triangle or the breakdown structure. Calculating a single PPT allows the entire figure to be created. The frequency of a geodesic polyhedron is defined by the sum of ν = b + c. A harmonic is a subfrequency and can be any whole divisor of ν. Class II always have a harmonic of 2, since ν = 2b. The triangulation number' is T = b2 + bc + c2. This number times the number of original faces expresses how many triangles the new polyhedron will have.
Elements
The number of elements are specified by the triangulation number. Two different geodesic polyhedra may have the same number of elements, for instance, 5,3 and 7,0 both have T=49.
Symmetry
Icosahedral
Octahedral
Tetrahedral
Base
Icosahedron = 1,0
Octahedron = 1,0
Tetrahedron = 1,0
Image
Symbol
b,c
b,c
b,c
Vertices
Faces
Edges
Construction
Geodesic polyhedra are constructed by subdividing faces of simpler polyhedra, and then projecting the new vertices onto the surface of a sphere. A geodesic polyhedron has straight edges and flat faces that approximate a sphere, but it can also be made as a spherical polyhedron with true geodesic curved edges on the surface of a sphere. and spherical triangle faces.
Geodesic polyhedra are the dual of Goldberg polyhedra. Goldberg polyhedra are also related in that applying a kis operator creates new geodesic polyhedra, and truncating vertices of a geodesic polyhedron creates a new Goldberg polyhedron. For example, Goldberg G kised, becomes 4,1, and truncating that becomes G. And similarly 2,1 truncated becomes G, and that kised becomes 6,3.
Examples
Class I
Class II
Class III
Spherical models
's book Spherical Models explores these subdivisions in building polyhedron models. After explaining the construction of these models, he explained his usage of triangular grids to mark out patterns, with triangles colored or excluded in the models.
An artistic model created by Father Magnus Wenninger called Order in Chaos, representing a chiral subset of triangles of a 16-frequency icosahedral geodesic sphere, 16,0
A virtual copy showing icosahedral symmetry great circles. The 6-fold rotational symmetry is illusionary, not existing on the icosahedron itself.
A single icosahedral triangle with a 16-frequency subdivision
