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For example, a 0-dimensional simplex is a point, a 1-dimensional simplex is a line segment, ... The volume of an n-simplex in n-dimensional space with vertices ...
Simplicial homology arose as a way to study topological spaces whose building blocks are n-simplices, the n-dimensional analogs of triangles. This includes a point (0-simplex), a line segment (1-simplex), a triangle (2-simplex) and a tetrahedron (3-simplex).
An example of a non-homogeneous complex is a triangle with a line segment attached to one of its vertices. Pure simplicial complexes can be thought of as triangulations and provide a definition of polytopes. A facet is a maximal simplex, i.e., any simplex in a complex that is not a face of any larger simplex. [2]
An example of simplicial complex, and the corresponding simplex tree data structure. Notice the two lowest nodes have a path of 4 to the node, indicating the 2 3-dimensional simplexes composed of 4 vertices each. In topological data analysis, a simplex tree is a type of trie used to represent efficiently any general simplicial complex.
For any face X in K of dimension n, let F(X) = Δ n be the standard n-simplex. The order on the vertex set then specifies a unique bijection between the elements of X and vertices of Δ n, ordered in the usual way e 0 < e 1 < ... < e n. If Y ⊆ X is a face of dimension m < n, then this bijection specifies a unique m-dimensional face of Δ n.
The method uses the concept of a simplex, which is a special polytope of n + 1 vertices in n dimensions. Examples of simplices include a line segment in one-dimensional space, a triangle in two-dimensional space, a tetrahedron in three-dimensional space, and so forth.
An n-apeirotope is an infinite n-polytope: a 2-apeirotope or apeirogon is an infinite polygon, a 3-apeirotope or apeirohedron is an infinite polyhedron, etc. There are two main geometric classes of apeirotope: [15] Regular honeycombs in n dimensions, which completely fill an n-dimensional space.
A 2-dimensional geometric simplicial complex with vertex V, link(V), and star(V) are highlighted in red and pink. As in the previous construction, by the topology induced by gluing, the closed sets in this space are the subsets that are closed in the subspace topology of every simplex in the complex.