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Anicosahedron and related symmetry polyhedra can be used to define a high geodesic polyhedron by dividing triangularfaces into smaller triangles, and projecting all the new vertices onto a sphere. Higher order polygonal faces can be divided into triangles by adding new vertices centered on each face. The new faces on the sphere are notequilateral triangles, but they are approximately equal edge length. All vertices are valence-6 except 12 vertices which are valence 5.

Ageodesic polyhedron is a convexpolyhedron made fromtriangles. They usually haveicosahedral symmetry, such that they have 6 triangles at avertex, except 12 vertices which have 5 triangles. They are thedual of correspondingGoldberg polyhedra, of which all but the smallest one (which is aregular dodecahedron) have 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 thegeodesic domes, hemispherical architectural structures designed byBuckminster Fuller, which geodesic polyhedra are named after.Geodesic grids used ingeodesy also have the geometry of geodesic polyhedra. Thecapsids of someviruses have the shape of geodesic polyhedra,^[1][2] and somepollen grains are based on geodesic polyhedra.^[3]Fullerene molecules have the shape ofGoldberg polyhedra. Geodesic polyhedra are available asgeometric primitives in theBlender 3D modeling software package, which calls themicospheres: they are an alternative to theUV sphere, having a more regular distribution.^[4][5] TheGoldberg–Coxeter construction is an expansion of the concepts underlying geodesic polyhedra.

InMagnus Wenninger'sSpherical models, polyhedra are givengeodesic notation in the form{3,q+}_b,c, where{3,q} is theSchläfli 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: {3,3+}_1,0 for atetrahedron, {3,4+}_1,0 for anoctahedron, and {3,5+}_1,0 for anicosahedron.

The dual notation forGoldberg polyhedra is{q+,3}_b,c, with valence-3 vertices, withq-gonal and hexagonal faces. There are 3 symmetry classes of forms: {3+,3}_1,0 for atetrahedron, {4+,3}_1,0 for acube, and {5+,3}_1,0 for adodecahedron.

Values forb,c are divided into three classes:

Class I (b=0 or c=0):{3,q+}_b,0 or{3,q+}_0,b represent a simple division with original edges being divided intob sub-edges.

Class II (b=c):{3,q+}_b,b are easier to see from thedual polyhedron {q,3} withq-gonal faces first divided into triangles with a central point, and then all edges are divided intob sub-edges.

Class III:{3,q+}_b,c have nonzero unequal values forb,c, and exist in chiral pairs. Forb > c we can define it as a right-handed form, andc > 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 atriangular tiling, positioning a large triangle on top of grid vertices and walking paths from one vertexb steps in one direction, and a turn, either clockwise or counterclockwise, and then anotherc steps to the next primary vertex.

For example, theicosahedron is {3,5+}_1,0, andpentakis dodecahedron, {3,5+}_1,1 is seen as aregular dodecahedron with pentagonal faces divided into 5 triangles.

The primary face of the subdivision is called aprincipal polyhedral triangle (PPT) or thebreakdown structure. Calculating a single PPT allows the entire figure to be created.

Thefrequency of a geodesic polyhedron is defined by the sum ofν =b +c. Aharmonic is a subfrequency and can be any whole divisor ofν. Class II always have a harmonic of 2, sinceν = 2b.

Thetriangulation number isT =b² +bc +c². This number times the number of original faces expresses how many triangles the new polyhedron will have.

The number of elements are specified by the triangulation number${\displaystyle T=b^2+bc+c^2}$. Two different geodesic polyhedra may have the same number of elements, for instance, {3,5+}_5,3 and {3,5+}_7,0 both have T=49.

Symmetry	Icosahedral	Octahedral	Tetrahedral
Base	Icosahedron {3,5} = {3,5+}1,0	Octahedron {3,4} = {3,4+}1,0	Tetrahedron {3,3} = {3,3+}1,0
Image
Symbol	{3,5+}b,c	{3,4+}b,c	{3,3+}b,c
Vertices	${\displaystyle 10T+2}$	${\displaystyle 4T+2}$	${\displaystyle 2T+2}$
Faces	${\displaystyle 20T}$	${\displaystyle 8T}$	${\displaystyle 4T}$
Edges	${\displaystyle 30T}$	${\displaystyle 12T}$	${\displaystyle 6T}$

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 aspherical polyhedron (atessellation on asphere) with truegeodesic curved edges on the surface of a sphere andspherical triangle faces.

Conway	u3I = (kt)I	(k)tI	ktI
Image
Form	3-frequency subdividedicosahedron	Kistruncated icosahedron	Geodesic polyhedron (3,0)	Spherical polyhedron

In this case, {3,5+}_3,0, with frequency${\displaystyle \nu =3}$ and triangulation number${\displaystyle T=9}$, each of the four versions of the polygon has 92 vertices (80 where six edges join, and 12 where five join), 270 edges and 180 faces.

Geodesic polyhedra are the duals ofGoldberg polyhedra. Goldberg polyhedra are also related in that applying akis operator (dividing faces into triangles with a center point) creates new geodesic polyhedra, andtruncating vertices of a geodesic polyhedron creates a new Goldberg polyhedron. For example, Goldberg G(2,1)kised, becomes {3,5+}_4,1, and truncating that becomes G(6,3). And similarly {3,5+}_2,1 truncated becomes G(4,1), and thatkised becomes {3,5+}_6,3.

Class I geodesic polyhedra

Frequency	(1,0)	(2,0)	(3,0)	(4,0)	(5,0)	(6,0)	(7,0)	(8,0)	(m,0)
T	1	4	9	16	25	36	49	64	m2
Face triangle									...
Icosahedral									more
Octahedral									more
Tetrahedral									more

Class II geodesic polyhedra

Frequency	(1,1)	(2,2)	(3,3)	(4,4)	(5,5)	(6,6)	(7,7)	(8,8)	(m,m)
T	3	12	27	48	75	108	147	192	3m2
Face triangle									...
Icosahedral									more
Octahedral									more
Tetrahedral									more

Class III geodesic polyhedra

Frequency	(2,1)	(3,1)	(3,2)	(4,1)	(4,2)	(4,3)	(5,1)	(5,2)	(m,n)
T	7	13	19	21	28	37	31	39	m2+mn+n2
Face triangle									...
Icosahedral									more
Octahedral									more
Tetrahedral									more

Magnus Wenninger's bookSpherical Models explores these subdivisions in buildingpolyhedron 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.^[6]

Example model

An artistic model created by FatherMagnus Wenninger calledOrder in Chaos, representing a chiral subset of triangles of a 16-frequency icosahedralgeodesic sphere, {3,5+}16,0

A virtual copy showingicosahedral symmetrygreat circles. The 6-fold rotational symmetry is illusionary, not existing on the icosahedron itself.

A single icosahedral triangle with a 16-frequency subdivision

• Conway polyhedron notation – Method of describing higher-order polyhedra

• Williams, Robert (1979).The Geometrical Foundation of Natural Structure: A source book of Design. pp. 142–144, Figure 4-49, 50, 51 Custers of 12 spheres, 42 spheres, 92 spheres.
• Pugh, Antony (1976). "Chapter 6. The Geodesic Polyhedra of R. Buckminster Fuller and Related Polyhedra".Polyhedra: a visual approach.
• Wenninger, Magnus (1979).Spherical Models.Cambridge University Press.ISBN 978-0-521-29432-4.MR 0552023. Archived fromthe original on July 4, 2008. Reprinted by Dover (1999),ISBN 978-0-486-40921-4.
• Popko, Edward S. (2012). "Chapter 8. Subdivision schemas, 8.1 Geodesic Notation, 8.2 Triangulation number 8.3 Frequency and Harmonics 8.4 Grid Symmetry 8.5 Class I: Alternates and fords 8.5.1 Defining the Principal triangle 8.5.2 Edge Reference Points".Divided spheres: Geodesics & the Orderly Subdivision of the Sphere.

• Geodesic polyhedra

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