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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, they have rotational icosahedral symmetry. They are not necessarily mirror-symmetric; e.g. ''G''(5,3) and ''G''(3,5) are enantiomorphs of each other. A consequence of Euler's polyhedron formula is that there will be exactly twelve pentagons. Icosahedral symmetry ensures that the pentagons are always regular, although many of the hexagons may not be. Typically all of the vertices lie on a sphere. It is a dual polyhedron of a geodesic sphere, with all triangle faces and 6 triangles per vertex, except for 12 vertices with 5 triangles. 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 ''G''(''m'',''n''). A dodecahedron is ''G''(1,0) and a truncated icosahedron is ''G''(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: GIII(n,m), GIV(n,m), and GV(n,m). == Polyhedral elements == The number of vertices, edges, and faces of ''G''(''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〕 |- align=center |Octahedral GIV(m,n) ||8''T''||12''T''||4''T'' + 2||6 and 4(''T'' − 1) |- align=center |Icosahedral GV(m,n) ||20''T''||30''T''||10''T'' + 2||12 and 10(''T'' − 1) |} 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Goldberg polyhedron」の詳細全文を読む スポンサード リンク
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