Geodesic polyhedron

A geodesic polyhedron is a convex polyhedron made from triangles which approximates a sphere. 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, of which all but the smallest one have mostly hexagonal faces. The Goldberg–Coxeter construction is an expansion of the concepts underlying geodesic polyhedra.
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, hemispherical architectural structures 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 some pollen grains are based on geodesic polyhedra. 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.

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 spherical polyhedron constructed from geodesic triangles.

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:

Class I : b'',0 or _0,b represent a simple division with original edges being divided into b sub-edges.
Class II : _b,''b are easier to see from the dual polyhedron with q''-gonal faces first divided into triangles with a central point, and then all edges are divided into b sub-edges.
Class III: _b,''c have nonzero unequal values for b'',c, and exist in chiral pairs. For b > c we can define it as a right-handed form, and c > b is a left-handed form.

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 _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.

Conway	u₃I = I	tI	ktI
Image
Form	3-frequency subdivided icosahedron	Kis truncated icosahedron	Geodesic polyhedron

In this case, _3,0, with frequency and triangulation number, each of the versions of the polygon has 92 vertices, 270 edges and 180 faces.

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

Relation to Goldberg polyhedra

Geodesic polyhedra are the duals 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

Frequency
T	1	4	9	16	25	36	49	64	m²
Face triangle									...
Icosahedral									more
Octahedral									more
Tetrahedral									more

Class II

Frequency
T	3	12	27	48	75	108	147	192	3m²
Face triangle									...
Icosahedral									more
Octahedral									more
Tetrahedral									more

Class III

Frequency
T	7	13	19	21	28	37	31	39	m²+mn+''n''²
Face triangle									...
Icosahedral									more
Octahedral									more
Tetrahedral									more

Spherical models

Magnus Wenninger'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