+ Icosahedral Goldberg polyhedra, ,equilateral and spherical
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 (1902–1990). They are defined by three properties: each face is either a pentagon or hexagon, exactly three faces meet at each vertex, and they have rotational icosahedral symmetry. They are not necessarily mirror-symmetric; e.g. and are enantiomorphs of each other. A Goldberg polyhedron is a dual polyhedron of a geodesic polyhedron.
A later generalization also considers polyhedra made from hexagons with exceptional squares or equilateral triangles rather than pentagons, with the symmetry of the octahedron or tetrahedron, respectively, rather than the icosahedron.
A consequence of Euler's polyhedron formula is that a Goldberg polyhedron always has exactly 12 pentagonal faces. Icosahedral symmetry ensures that the pentagons are always regular. If the vertices are not constrained to a sphere, the polyhedron can be constructed with planar equilateral (but not in general equiangular) faces.
Simple examples of Goldberg polyhedra include the dodecahedron and truncated icosahedron. Other forms can be described by taking a chess knight move from one pentagon to the next: first take steps in one direction, then turn 60° to the left and take steps. Such a polyhedron is denoted . A dodecahedron is , and a truncated icosahedron is .
A similar technique can be applied to construct polyhedra with tetrahedral symmetry and octahedral symmetry. These polyhedra will have four triangles or six squares rather than pentagons. These variations are given Roman numeral subscripts denoting the number of sides on the non-hexagon faces: , , and .
Elements The number of vertices, edges, and faces of can be computed from and , with
depending on one of three symmetry systems:Clinton’s Equal Central Angle Conjecture, JOSEPH D. CLINTON The number of non-hexagonal faces can be determined using the Euler characteristic, as demonstrated here.
Symmetry Icosahedral Octahedral TetrahedralBase Dodecahedron Cube TetrahedronImage Symbol Vertices Edges Faces Faces by type and and and
Construction Most Goldberg polyhedra can be constructed using Conway polyhedron notation starting with (T)etrahedron, (C)ube, and (D)odecahedron seeds. The chamfer operator, , replaces all edges by hexagons, transforming to , with a multiplier of 4. The truncated kis operator, , generates , transforming to , with a multiplier of 9.
For class 2 forms, the dual kis operator, , transforms into , with a multiplier of 3. For class 3 forms, the whirl operator, , generates , with a multiplier of 7. A clockwise and counterclockwise whirl generator, generates in class 1. In general, a whirl can transform a into for and the same chiral direction. If chiral directions are reversed, becomes if , and if .
Class I
Class I Goldberg polyhedra are those for which .
+ Class I polyhedra Frequency Icosahedral (Goldberg)regular dodecahedronchamfered dodecahedronmoreOctahedralcubechamfered cubemoreTetrahedraltetrahedronchamfered tetrahedronmore
Class II Class II Goldberg polyhedra are those for which .
+ Class II polyhedra Frequency Icosahedral (Goldberg)truncated icosahedronmoreOctahedraltruncated octahedronmoreTetrahedraltruncated tetrahedronmore
Class III
Class III Goldberg polyhedra are those for which and neither is zero.
+ Class III polyhedra Frequency Icosahedral (Goldberg)moreOctahedralmoreTetrahedralmore
See also Capsid Geodesic sphere Fullerene#Other buckyballs Conway polyhedron notation Goldberg–Coxeter construction
Notes
References
Joseph D. Clinton, Clinton’s Equal Central Angle Conjecture
External links Dual Geodesic Icosahedra Goldberg variations: New shapes for molecular cages Flat hexagons and pentagons come together in new twist on old polyhedral, by Dana Mackenzie, February 14, 2014