Goldberg polyhedron

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Icosahedral Goldberg polyhedra with pentagons in red
Conway polyhedron Dk5k6st.png
GP(1,4) = {5+,3}1,4
Conway polyhedron dadkt5daD.png
GP(4,4) = {5+,3}4,4
Goldberg polyhedron 7 0.png
GP(7,0) = {5+,3}7,0
Goldberg polyhedron 5 3.png
GP(3,5) = {5+,3}3,5
Goldberg 10 0 equilateral-spherical.png
GP(10,0) = {5+,3}10,0
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 by Michael Goldberg (1902–1990) in 1937. 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. GP(5,3) and GP(3,5) are enantiomorphs of each other. A Goldberg polyhedron is a dual polyhedron of a geodesic sphere.

A consequence of Euler's polyhedron formula 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 and truncated icosahedron. Other forms can be described by taking a chess knight move from one pentagon to the next: first take m steps in one direction, then turn 60° to the left and take n steps. Such a polyhedron is denoted GP(m,n). A dodecahedron is GP(1,0) and a truncated icosahedron is GP(1,1).

A similar technique can be applied to construct polyhedra with tetrahedral symmetry and octahedral symmetry. 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: GPIII(n,m), GPIV(n,m), and GPV(n,m).

Elements[edit]

The number of vertices, edges, and faces of GP(m,n) can be computed from m and n, with T = m2 + mn + n2 = (m + n)2 − mn, depending on one of three symmetry systems:[1] The number of non-hexagonal faces can be determined using the Euler characteristic, as demonstrated here.

Symmetry Icosahedral Octahedral Tetrahedral
Base Dodecahedron
GPV(1,0) = {5+,3}1,0
Cube
GPIV(1,0) = {4+,3}1,0
Tetrahedron
GPIII(1,0) = {3+,3}1,0
Image Dodecahedron Cube Tetrahedron
Symbol GPV(m,n) = {5+,3}m,n GPIV(m,n) = {4+,3}m,n GPIII(m,n) = {3+,3}m,n
Vertices
Edges
Faces
Faces by type 12 {5} and 10(T − 1) {6} 6 {4} and 4(T − 1) {6} 4 {3} and 2(T − 1) {6}

Construction[edit]

Most Goldberg polyhedra can be constructed using Conway polyhedron notation starting with (T)etrahedron, (C)ube, and (D)odecahedron seeds. The chamfer operator, c, replaces all edges by hexagons, transforming GP(m,n) to GP(2m,2n), with a T multiplier of 4. The truncated kis operator, y = tk, generates GP(3,0), transforming GP(m,n) to GP(3m,3n), 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, ww = wrw generates GP(7,0) in class 1. In general, a whirl can transform a GP(a,b) into GP(a + 3b,2ab) for a > b and the same chiral direction. If chiral directions are reversed, GP(a,b) becomes GP(2a + 3b,a − 2b) if a ≥ 2b, and GP(3a + b,2b − a) if a < 2b.

Examples[edit]

Class I 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
Icosahedral (Goldberg) Dodecahedron.svg
regular dodecahedron
Truncated rhombic triacontahedron.png
chamfered dodecahedron
Conway polyhedron Dk6k5tI.png Conway polyhedron dk6k5at5daD.png Goldberg polyhedron 5 0.png Conway polyhedron tkt5daD.png Goldberg polyhedron 7 0.png Conway polyhedron dk6k5adk6k5at5daD.png more
Octahedral Hexahedron.svg
cube
Truncated rhombic dodecahedron2.png
chamfered cube
Octahedral goldberg polyhedron 03 00.svg Octahedral goldberg polyhedron 04 00.svg Octahedral goldberg polyhedron 05 00.svg Octahedral goldberg polyhedron 06 00.svg Octahedral goldberg polyhedron 07 00.svg Octahedral goldberg polyhedron 08 00.svg more
Tetrahedral Tetrahedron.svg
tetrahedron
Alternate truncated cube.png
chamfered tetrahedron
Tetrahedral Goldberg polyhedron 03 00.svg Tetrahedral Goldberg polyhedron 04 00.svg Tetrahedral Goldberg polyhedron 05 00.svg Tetrahedral Goldberg polyhedron 06 00.svg Tetrahedral Goldberg polyhedron 07 00.svg Tetrahedral Goldberg polyhedron 08 00.svg more
Class II 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
Icosahedral (Goldberg) Truncated icosahedron.png
truncated icosahedron
Conway polyhedron dkt5daD.png Conway polyhedron dkdktI.png Conway polyhedron dadkt5daD.png Conway du5zI.png Conway cyzD.png Conway wrwdkD.png Conway cccdkD.png more
Octahedral Truncated octahedron.png
truncated octahedron
Conway polyhedron dkt4daC.png Conway polyhedron tktO.png Conway polyhedron dk6k4adk6k4adkC.png Octahedral goldberg polyhedron 05 05.svg more
Tetrahedral Uniform polyhedron-33-t12.png
truncated tetrahedron
Conway polyhedron tktT.png more
Class III polyhedra
Frequency (1,2) (1,3) (2,3) (1,4) (2,4) (3,4) (1,5) (m,n)
T 7 13 19 21 28 37 31 m2+mn+n2
Icosahedral (Goldberg) Conway polyhedron Dk5sI.png Goldberg polyhedron 3 1.png Goldberg polyhedron 3 2.png Conway polyhedron Dk5k6st.png Conway polyhedron dk6k5adk5sD.png Goldberg polyhedron 4 3.png Goldberg polyhedron 5 1.png more
Octahedral Conway polyhedron wC.png more
Tetrahedral Conway polyhedron wT.png more

See also[edit]

Notes[edit]

  1. ^ Clinton’s Equal Central Angle Conjecture, JOSEPH D. CLINTON

References[edit]

  • Goldberg, Michael (1937). "A class of multi-symmetric polyhedra". Tohoku Mathematical Journal.
  • Joseph D. Clinton, Clinton’s Equal Central Angle Conjecture
  • Hart, George (2012). "Goldberg Polyhedra". In Senechal, Marjorie (ed.). Shaping Space (2nd ed.). Springer. pp. 125–138. doi:10.1007/978-0-387-92714-5_9. [1]
  • Hart, George (June 18, 2013). "Mathematical Impressions: Goldberg Polyhedra". Simons Science News.
  • Schein, S.; Gayed, J. M. (2014-02-25). "Fourth class of convex equilateral polyhedron with polyhedral symmetry related to fullerenes and viruses". Proceedings of the National Academy of Sciences. 111 (8): 2920–2925. doi:10.1073/pnas.1310939111. ISSN 0027-8424. PMC 3939887. PMID 24516137.

External links[edit]