mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, and more specifically in

polyhedral combinatorics Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral co ...

, a Goldberg polyhedron is a

convex polyhedron A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

made from

hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...

s and

pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be sim ...

s. They were first described in 1937 by Michael Goldberg (1902–1990). They are defined by three properties: each

face The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may aff ...

is either a pentagon or hexagon, exactly three faces meet at each

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

, and they have

rotational 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 ...

. They are not necessarily mirror-symmetric; e.g. and are enantiomorphs of each other. A Goldberg polyhedron is a

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 oth ...

of a

geodesic sphere A geodesic polyhedron 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 polyhedr ...

. A consequence of

Euler's polyhedron formula In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...

is that a Goldberg polyhedron always has exactly twelve pentagonal faces. Icosahedral symmetry ensures that the pentagons are always regular and that there are always 12 of them. 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 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 ...

and

truncated icosahedron In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares ...

. Other forms can be described by taking a

chess Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to dist ...

knight A knight is a person granted an honorary title of knighthood by a head of state (including the Pope) or representative for service to the monarch, the Christian denomination, church or the country, especially in a military capacity. Knighthood ...

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 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection ...

and

octahedral symmetry A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...

. These polyhedra will have triangles or 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 ''GP''(''m'',''n'') can be computed from ''m'' and ''n'', with ''T'' = ''m''² + ''mn'' + ''n''² = (''m'' + ''n'')² − ''mn'', 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 Here is an adverb that means "in, on, or at this place". It may also refer to: Software * Here Technologies, a mapping company * Here WeGo (formerly Here Maps), a mobile app and map website by Here Television * Here TV (formerly "here!"), a ...

Construction

Most Goldberg polyhedra can be constructed using

Conway polyhedron notation In geometry, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations. Conway and Hart extended the idea of using o ...

starting with (T)etrahedron, (C)ube, and (D)odecahedron seeds. The

chamfer A chamfer or is a transitional edge between two faces of an object. Sometimes defined as a form of bevel, it is often created at a 45° angle between two adjoining right-angled faces. Chamfers are frequently used in machining, carpentry, ...

operator, ''c'', replaces all edges by hexagons, transforming ''GP''(''m'',''n'') to ''GP''(2''m'',2''n''), with a ''T'' multiplier of 4. The ''truncated kis'' operator, ''y'' = ''tk'', generates ''GP''(3,0), transforming ''GP''(''m'',''n'') to ''GP''(3''m'',3''n''), with a ''T'' multiplier of 9. For class 2 forms, the ''dual kis'' operator, ''z'' = ''dk'', transforms ''GP''(''a'',0) into ''GP''(''a'',''a''), with a ''T'' multiplier of 3. For class 3 forms, the ''whirl'' operator, ''w'', generates ''GP''(2,1), with a ''T'' multiplier of 7. A clockwise and counterclockwise whirl generator, ''w'' = ''wrw'' generates ''GP''(7,0) in class 1. In general, a whirl can transform a GP(''a'',''b'') into GP(''a'' + 3''b'',2''ab'') for ''a'' > ''b'' and the same chiral direction. If chiral directions are reversed, GP(''a'',''b'') becomes GP(2''a'' + 3''b'',''a'' − 2''b'') if ''a'' ≥ 2''b'', and GP(3''a'' + ''b'',2''b'' − ''a'') if ''a'' < 2''b''.

Examples

Notes

References

* * Joseph D. Clinton
''Clinton’s Equal Central Angle Conjecture''

* * {{Cite journal, last1=Schein, first1=S., last2=Gayed, first2=J. M., date=2014-02-25, title=Fourth class of convex equilateral polyhedron with polyhedral symmetry related to fullerenes and viruses, journal=Proceedings of the National Academy of Sciences, language=en, volume=111, issue=8, pages=2920–2925, doi=10.1073/pnas.1310939111, issn=0027-8424, pmc=3939887, pmid=24516137, bibcode=2014PNAS..111.2920S, doi-access=free

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 Goldberg polyhedra,