|
In geometry, a simplex (plural: ''simplexes'' or ''simplices'') is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. Specifically, a ''k''-simplex is a ''k''-dimensional polytope which is the convex hull of its ''k'' + 1 vertices. More formally, suppose the ''k'' + 1 points are affinely independent, which means are linearly independent. Then, the simplex determined by them is the set of points . For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, and a 4-simplex is a 5-cell. A single point may be considered a 0-simplex, and a line segment may be considered a 1-simplex. A simplex may be defined as the smallest convex set containing the given vertices. A regular simplex〔 Chapter IV, five dimensional semiregular polytope〕 is a simplex that is also a regular polytope. A regular ''n''-simplex may be constructed from a regular (''n'' − 1)-simplex by connecting a new vertex to all original vertices by the common edge length. In topology and combinatorics, it is common to “glue together” simplices to form a simplicial complex. The associated combinatorial structure is called an abstract simplicial complex, in which context the word “simplex” simply means any finite set of vertices. == Examples == * A 0-simplex is a point. * A 1-simplex is a line segment. * A 2-simplex is a triangle. * A 3-simplex is a tetrahedron. == Elements == The convex hull of any nonempty subset of the ''n''+1 points that define an n-simplex is called a ''face'' of the simplex. Faces are simplices themselves. In particular, the convex hull of a subset of size ''m''+1 (of the ''n''+1 defining points) is an m-simplex, called an ''m''-face of the n-simplex. The 0-faces (i.e., the defining points themselves as sets of size 1) are called the vertices (singular: vertex), the 1-faces are called the edges, the (''n'' − 1)-faces are called the facets, and the sole ''n''-face is the whole ''n''-simplex itself. In general, the number of ''m''-faces is equal to the binomial coefficient . Consequently, the number of ''m''-faces of an ''n''-simplex may be found in column (''m'' + 1) of row (''n'' + 1) of Pascal's triangle. A simplex ''A'' is a coface of a simplex ''B'' if ''B'' is a face of ''A''. ''Face'' and ''facet'' can have different meanings when describing types of simplices in a simplicial complex; see simplical complex for more detail. The regular simplex family is the first of three regular polytope families, labeled by Coxeter as ''αn'', the other two being the cross-polytope family, labeled as ''βn'', and the hypercubes, labeled as ''γn''. A fourth family, the infinite tessellation of hypercubes, he labeled as ''δn''. The number of ''1''-faces (edges) of the ''n''-simplex is the (''n''-1)th triangle number, the number of ''2''-faces of the ''n''-simplex is the (''n''-2)th tetrahedron number, the number of ''3''-faces of the ''n''-simplex is the (''n''-3)th 5-cell number, and so on. = 3.( ) | 3 | 3 | 1 | | | | | | | | | 7 |- ! Δ3 | ''3-simplex'' (tetrahedron) | = 4.( ) | 4 | 6 | 4 | 1 | | | | | | | | 15 |- ! Δ4 | ''4-simplex'' (5-cell) | = 5.( ) | 5 | 10 | 10 | 5 | 1 | | | | | | | 31 |- ! Δ5 | ''5-simplex'' | = 6.( ) | 6 | 15 | 20 | 15 | 6 | 1 | | | | | | 63 |- ! Δ6 | ''6-simplex'' | = 7.( ) | 7 | 21 | 35 | 35 | 21 | 7 | 1 | | | | | 127 |- ! Δ7 | ''7-simplex'' | = 8.( ) | 8 | 28 | 56 | 70 | 56 | 28 | 8 | 1 | | | | 255 |- ! Δ8 | ''8-simplex'' | = 9.( ) | 9 | 36 | 84 | 126 | 126 | 84 | 36 | 9 | 1 | | | 511 |- ! Δ9 | ''9-simplex'' | = 10.( ) | 10 | 45 | 120 | 210 | 252 | 210 | 120 | 45 | 10 | 1 | | 1023 |- ! Δ10 | ''10-simplex'' | = 11.( ) | 11 | 55 | 165 | 330 | 462 | 462 | 330 | 165 | 55 | 11 | 1 | 2047 |} An (''n''+1)-simplex can be constructed as a join (∨ operator) of an ''n''-simplex and a point, ( ). An (''m''+''n''+1)-simplex can be constructed as a join of an ''m''-simplex and an ''n''-simplex. The two simplices are oriented to be completely normal from each other, with translation in a direction orthogonal to both of them. A 1-simplex is a joint of two points: ( )∨( ) = 2.( ). A general 2-simplex (scalene triangle) is the join of 3 points: ( )∨( )∨( ). An isosceles triangle is the join of a 1-simplex and a point: ∨( ). An equilateral triangle is 3.( ) or . A general 3-simplex is the join of 4 points: ( )∨( )∨( )∨( ). A 3-simplex with mirror symmetry can be expressed as the join of an edge and 2 points: ∨( )∨( ). A 3-simplex with triangular symmetry can be expressed as the join of an equilateral triangle and 1 point: 3.( )∨( ) or ∨( ). A regular tetrahedron is 4.( ) or and so on. In some conventions,〔Kozlov, Dimitry, ''Combinatorial Algebraic Topology'', 2008, Springer-Verlag (Series: Algorithms and Computation in Mathematics)〕 the empty set is defined to be a (−1)-simplex. The definition of the simplex above still makes sense if ''n'' = −1. This convention is more common in applications to algebraic topology (such as simplicial homology) than to the study of polytopes. 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「Simplex」の詳細全文を読む スポンサード リンク
|