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A key concept in defining simplicial homology is the notion of an orientation of a simplex. By definition, an orientation of a k-simplex is given by an ordering of the vertices, written as (v 0,...,v k), with the rule that two orderings define the same orientation if and only if they differ by an even permutation.
The standard simplex or probability simplex [2] is the (k − 1)-dimensional simplex whose vertices are the k standard unit vectors in , or in other words {: + + =, =, …,}. In topology and combinatorics , it is common to "glue together" simplices to form a simplicial complex .
A simplicial k-complex is a simplicial complex where the largest dimension of any simplex in equals k. For instance, a simplicial 2-complex must contain at least one triangle, and must not contain any tetrahedra or higher-dimensional simplices.
A complete graph with n nodes represents the edges of an (n – 1)-simplex. Geometrically K 3 forms the edge set of a triangle, K 4 a tetrahedron, etc. The Császár polyhedron, a nonconvex polyhedron with the topology of a torus, has the complete graph K 7 as its skeleton. [15] Every neighborly polytope in four or more dimensions also has a ...
Let K and L be two geometric simplicial complexes (GSC). A simplicial map of K into L is a function : such that the images of the vertices of a simplex in K span a simplex in L. That is, for any simplex , ((())).
Let K be a geometric simplicial complex (GSC). A subdivision of K is a GSC L such that: [1]: 15 [2]: 3 |K| = |L|, that is, the union of simplices in K equals the union of simplices in L (they cover the same region in space). each simplex of L is contained in some simplex of K.
The technical term for the set of points in the support of a K-dimensional Dirichlet distribution is the open standard (K − 1)-simplex, [3] which is a generalization of a triangle, embedded in the next-higher dimension.
A 3-simplex, with barycentric subdivisions of 1-faces (edges) 2-faces (triangles) and 3-faces (body). In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex (a triangle for points in a plane, a tetrahedron for points in three-dimensional space, etc.).