A geodesic polyhedron is a convex

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...

made from triangles. They usually have

icosahedral symmetry In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the ...

, such that they have 6 triangles at a vertex, except 12 vertices which have 5 triangles. They are the

dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...

of corresponding

Goldberg polyhedra In mathematics, and more specifically in polyhedral combinatorics, a Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described in 1937 by Michael Goldberg (mathematician), Michael Goldberg (1902–1990 ...

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 dome A geodesic dome is a hemispherical thin-shell structure (lattice-shell) based on a geodesic polyhedron. The triangular elements of the dome are structurally rigid and distribute the structural stress throughout the structure, making geodesic dom ...

s designed by

Buckminster Fuller Richard Buckminster Fuller (; July 12, 1895 – July 1, 1983) was an American architect, systems theorist, writer, designer, inventor, philosopher, and futurist. He styled his name as R. Buckminster Fuller in his writings, publishing more t ...

, which geodesic polyhedra are named after.

Geodesic grid A geodesic grid is a spatial grid based on a geodesic polyhedron or Goldberg polyhedron. Construction A geodesic grid is a global Earth reference that uses triangular tiles based on the subdivision of a polyhedron (usually the icosahedron, and ...

s used in

geodesy Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equivale ...

also have the geometry of geodesic polyhedra. The

capsids A capsid is the protein shell of a virus, enclosing its genetic material. It consists of several oligomeric (repeating) structural subunits made of protein called protomers. The observable 3-dimensional morphological subunits, which may or may ...

of some

virus A virus is a submicroscopic infectious agent that replicates only inside the living cells of an organism. Viruses infect all life forms, from animals and plants to microorganisms, including bacteria and archaea. Since Dmitri Ivanovsky's 1 ...

es have the shape of geodesic polyhedra, and

fullerene A fullerene is an allotrope of carbon whose molecule consists of carbon atoms connected by single and double bonds so as to form a closed or partially closed mesh, with fused rings of five to seven atoms. The molecule may be a hollow sphere, ...

molecules have the shape of

. Geodesic polyhedra are available as

geometric primitive In vector computer graphics, CAD systems, and geographic information systems, geometric primitive (or prim) is the simplest (i.e. 'atomic' or irreducible) geometric shape that the system can handle (draw, store). Sometimes the subroutines that ...

s in the Blender 3D modeling software package, which calls them icospheres: they are an alternative to the

UV sphere UV mapping is the 3D modeling process of projecting a 3D model's surface to a 2D image for texture mapping. The letters "U" and "V" denote the axes of the 2D texture because "X", "Y", and "Z" are already used to denote the axes of the 3D object i ...

, having a more regular distribution of vertices than the UV sphere. The

Goldberg–Coxeter construction The Goldberg–Coxeter construction or Goldberg–Coxeter operation (GC construction or GC operation) is a graph operation defined on regular polyhedral graphs with degree 3 or 4. It also applies to the dual graph of these graphs, i.e. graphs wit ...

is an expansion of the concepts underlying geodesic polyhedra.

Geodesic notation

Magnus Wenninger Father Magnus J. Wenninger Order of Saint Benedict, OSB (October 31, 1919Banchoff (2002)– February 17, 2017) was an American mathematician who worked on constructing polyhedron models, and wrote the first book on their construction. Early life ...

's ''Spherical models'', polyhedra are given geodesic notation in the form _''b'',''c'', where is the

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

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 In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...

, _1,0 for an

octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...

, and _1,0 for an

icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...

. The dual notation for

is _''b'',''c'', with valence-3 vertices, with ''q''-gonal and hexagonal faces. There are 3 symmetry classes of forms: _1,0 for a

, _1,0 for a

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...

, and _1,0 for a

dodecahedron In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...

. Values for ''b'',''c'' are divided into three classes: : Class I (b=0 or c=0): _''b'',0 or _0,''b'' represent a simple division with original edges being divided into ''b'' sub-edges. : Class II (b=c): _''b'',''b'' are easier to see from the

dual polyhedron In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. ...

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 In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilater ...

, 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

is _1,0, and

pentakis dodecahedron In geometry, a pentakis dodecahedron or kisdodecahedron is the polyhedron created by attaching a pentagonal pyramid to each face of a regular dodecahedron; that is, it is the Kleetope of the dodecahedron. It is a Catalan solid, meaning that i ...

, _1,1 is seen as a

regular dodecahedron A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges ...

with pentagonal faces divided into 5 triangles. The primary face of the subdivision is called a principal polyhedral triangle (PPT) 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'' = 2''b''. The triangulation number is ''T'' = ''b''² + ''bc'' + ''c''². 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

T=b^2+bc+c^2

. Two different geodesic polyhedra may have the same number of elements, for instance, _5,3 and _7,0 both have T=49.

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 In geometry, a spherical polyhedron or spherical tiling is a tessellation, tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhe ...

tessellation A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...

on a

sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

) with true

geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...

curved edges on the surface of a sphere and

spherical triangle Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are grea ...

faces. In this case, _3,0, with frequency

\nu =3

and triangulation number

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.

Relation to Goldberg polyhedra

Geodesic polyhedra are the dual of Goldberg polyhedra. Goldberg polyhedra are also related in that applying a kis operator (dividing faces triangles with a center point) creates new geodesic polyhedra, and truncating vertices of a geodesic polyhedron creates a new Goldberg polyhedron. For example, Goldberg G(2,1) kised, becomes _4,1, and truncating that becomes G(6,3). And similarly _2,1 truncated becomes G(4,1), 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 model A polyhedron model is a physical construction of a polyhedron, constructed from cardboard, plastic board, wood board or other panel material, or, less commonly, solid material. Since there are 75 uniform polyhedra, including the five regular con ...

s. 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.Spherical Models, pp. 150–159

References

* Robert Williams ''The Geometrical Foundation of Natural Structure: A source book of Design'', 1979, pp. 142–144, Figure 4-49,50,51 Custers of 12 spheres, 42 spheres, 92 spheres * Antony Pugh, ''Polyhedra: a visual approach'', 1976, Chapter 6. The Geodesic Polyhedra of R. Buckminster Fuller and Related Polyhedra'' * Reprinted by Dover 1999 {{ISBN, 978-0-486-40921-4 * Edward S. Popko, ''Divided spheres: Geodesics & the Orderly Subdivision of the Sphere'' (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 Geodesic polyhedra