Goldberg polyhedron

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. GP(5,3) and GP(3,5) are enantiomorphs of each other. A Goldberg polyhe…

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Jump to content Main menu Main menu move to sidebar hide Navigation Main page Contents Current events Random article About Wikipedia Contact us Contribute Help Learn to edit Community portal Recent changes Upload file Special pages Search Search Appearance Donate Create account Log in Personal tools Donate Create account Log in Contents move to sidebar hide (Top) 1 Elements 2 Construction 3 Examples 4 See also 5 Notes 6 References 7 External links Toggle the table of contents Goldberg polyhedron 4 languages Español فارسی Română 中文 Edit links Article Talk English Read Edit View history Tools Tools move to sidebar hide Actions Read Edit View history General What links here Related changes Upload file Permanent link Page information Cite this page Get shortened URL Print/export Download as PDF Printable version In other projects Wikimedia Commons Wikidata item Appearance move to sidebar hide From Wikipedia, the free encyclopedia This is an old revision of this page, as edited by Antonissimo ( talk | contribs ) at 19:09, 24 July 2023 (Undid revision 1162207019 by Kwékwlos ( talk )— where do you see a yellow pentagon?) . The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision . Revision as of 19:09, 24 July 2023 by Antonissimo ( talk | contribs ) (Undid revision 1162207019 by Kwékwlos ( talk )— where do you see a yellow pentagon?) ( diff ) ← Previous revision | Latest revision ( diff ) | Newer revision → ( diff ) Convex polyhedron made from hexagons and pentagons Icosahedral Goldberg polyhedra with pentagons in red GP(1,4) = {5+,3} 1,4 GP(4,4) = {5+,3} 4,4 GP(7,0) = {5+,3} 7,0 GP(3,5) = {5+,3} 3,5 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 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. 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: GP III ( n , m ), GP IV ( n , m ), and GP V ( n , m ). 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: 1 The number of non-hexagonal faces can be determined using the Euler characteristic, as demonstrated here . Symmetry Icosahedral Octahedral Tetrahedral Base Dodecahedron GP V (1,0) = {5+,3} 1,0 Cube GP IV (1,0) = {4+,3} 1,0 Tetrahedron GP III (1,0) = {3+,3} 1,0 Image Symbol GP V ( m , n ) = {5+,3} m , n GP IV ( m , n ) = {4+,3} m , n GP III ( m , n ) = {3+,3} m , n Vertices 20 T {\displaystyle 20T} 8 T {\displaystyle 8T} 4 T {\displaystyle 4T} Edges 30 T {\displaystyle 30T} 12 T {\displaystyle 12T} 6 T {\displaystyle 6T} Faces 10 T + 2 {\displaystyle 10T+2} 4 T + 2 {\displaystyle 4T+2} 2 T + 2 {\displaystyle 2T+2} 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 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 (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 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 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 m 2 Icosahedral (Goldberg) regular dodecahedron chamfered dodecahedron more Octahedral cube chamfered cube more Tetrahedral tetrahedron chamfered tetrahedron 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 3 m 2 Icosahedral (Goldberg) truncated icosahedron more Octahedral truncated octahedron more Tetrahedral truncated tetrahedron more Class III polyhedra Frequency (1,2) (1,3) (2,3) (1,4) (2,4) (3,4) (5,1) ( m , n ) T 7 13 19 21 28 37 31 m 2 + mn + n 2 Icosahedral (Goldberg) more Octahedral more Tetrahedral more See also Capsid Geodesic sphere Fullerene#Other buckyballs Conway polyhedron notation Goldberg–Coxeter construction Notes ^ Clinton’s Equal Central Angle Conjecture, JOSEPH D. CLINTON References Goldberg, Michael (1937). "A class of multi-symmetric polyhedra" . Tohoku Mathematical Journal . 43 : 104– 108. 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 . ISBN 978-0-387-92713-8 . 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. Bibcode : 2014PNAS..111.2920S . doi : 10.1073/pnas.1310939111 . ISSN 0027-8424 . PMC 3939887 . PMID 24516137 . 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 Retrieved from " https://en.wikipedia.org/w/index.php?title=Goldberg_polyhedron&oldid=1166944792 " Category : Goldberg polyhedra Hidden categories: Articles with short description Short description matches Wikidata This page was last edited on 24 July 2023, at 19:09 (UTC) . This version of the page has been revised . Besides normal editing, the reason for revision may have been that this version contains factual inaccuracies, vandalism, outdated information, or material not compatible with the Creative Commons Attribution-ShareAlike 4.0 License . Privacy policy About Wikipedia Disclaimers Contact Wikipedia Legal & safety contacts Code of Conduct Developers Statistics Cookie statement Mobile view Search Search Toggle the table of contents Goldberg polyhedron 4 languages Add topic