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In geometry, a vertex configuration〔(Uniform Solution for Uniform Polyhedra ) (1993)〕〔(The Uniform Polyhedra ) Roman E. Maeder (1995)〕〔(Crystallography of Quasicrystals: Concepts, Methods and Structures ) by Walter Steurer, Sofia Deloudi, (2009) p.18-20, p.51-53〕〔(Physical Metallurgy: 3-Volume Set, Volume 1 ) edited by David E. Laughlin, (2014) p. 16-20〕 is a shorthand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For ''uniform polyhedra'' there is only one vertex type and therefore the ''vertex configuration'' fully defines the polyhedron. (Chiral polyhedra exist in mirror image pairs with the same ''vertex configuration''.) A ''vertex configuration'' is given as a sequence of numbers representing the number of sides of the faces going around the vertex. A ''a.b.c'' means a vertex has 3 faces around it, with ''a'', ''b'', and ''c'' sides. For example, ''3.5.3.5'' means a vertex has 4 faces, alternating triangles and pentagons. This vertex configuration defines the vertex-transitive icosidodecahedron. The notation is cyclic and therefore is equivalent with different starting points, so 3.5.3.5 is the same as 5.3.5.3. The order is important and so 3.3.5.5 is different from 3.5.3.5. (The first has two triangles followed by two pentagons.) Repeated elements can be collected as exponents so this example is also represented as (3.5)2. It has variously been called a vertex description,〔(Archimedean Polyhedra ) Steven Dutch〕〔(Uniform Polyhedra ) Jim McNeill〕〔(Uniform Polyhedra and their Duals ) Robert Webb〕 vertex type,〔(Symmetry-type graphs of Platonic and Archimedean solids ), Jurij Kovič, (2011)〕〔(3. General Theorems: Regular and Semi-Regular Tilings ) Kevin Mitchell, 1995〕 vertex symbol,〔Resources for Teaching Discrete Mathematics: Classroom Projects, History, modules, and articles, edited by Brian Hopkins〕〔(Vertex Symbol ) Robert Whittaker〕 vertex arrangement,〔Structure and Form in Design: Critical Ideas for Creative Practice By Michael Hann〕 vertex pattern,〔(Symmetry-type graphs of Platonic and Archimedean solids ) Jurij Kovič〕 face-vector.〔(Uniform Partitions of 3-space, their Relatives and Embedding ) MICHEL DEZA AND MIKHAIL SHTOGRIN〕 It is also called a Cundy and Rollett symbol for its usage for the Archimedean solids in their 1952 book ''Mathematical Models''.〔(Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere ) 6.4.1 Cundy-Rollett symbol, p.164〕〔( Physical Metallurgy: 3-Volume Set, Volume 1 ) edited by David E. Laughlin, (2014) p. 16〕〔(CRC Concise Encyclopedia of Mathematics, Second Edition ) By Eric W. Weisstein〕 == Vertex figures == A ''vertex configuration'' can also be represented as a polygonal vertex figure showing the faces around the vertex. This ''vertex figure'' has a 3-dimensional structure since the faces are not in the same plane for polyhedra, but for vertex-uniform polyhedra all the neighboring vertices are in the same plane and so this plane projection can be used to visually represent the vertex configuration. 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「vertex configuration」の詳細全文を読む スポンサード リンク
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