Simplices are the building blocks of simplicial complexes, generalizing triangles and tetrahedra to higher dimensions. They're defined as convex hulls of affinely independent points, with dimensions ranging from 0 (points) to n (n-simplices).
Simplicial complexes are topological spaces constructed by gluing simplices together, following specific rules. They're crucial in algebraic topology, allowing us to represent complex structures using simple geometric shapes and study their properties through combinatorial methods.
Simplices and their dimensions
Definition and basic properties
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Simplex generalizes triangle or tetrahedron to arbitrary dimensions defined as convex hull of (n+1) affinely independent points in Euclidean space
Dimension of simplex equals one less than number of vertices (0-simplex point, 1-simplex line segment, 2-simplex triangle, 3-simplex tetrahedron)
Represent k-simplex as [v0, v1, ..., vk] where vi are vertices and ordering determines orientation
Standard n-simplex subset of R^(n+1) where sum of all coordinates equals 1 and all coordinates non-negative
Simplices serve as building blocks for more complex topological structures in simplicial complexes
Coordinates and boundaries
Barycentric coordinates uniquely specify any point within simplex as weighted sum of vertices
Boundary of k-simplex consists of all (k-1)-dimensional faces forming simplicial complex
Calculate barycentric coordinates using formula: p = ∑ i = 0 n λ i v i p = \sum_{i=0}^n \lambda_i v_i p = ∑ i = 0 n λ i v i where ∑ i = 0 n λ i = 1 \sum_{i=0}^n \lambda_i = 1 ∑ i = 0 n λ i = 1 and λ i ≥ 0 \lambda_i \geq 0 λ i ≥ 0
Boundary operator ∂ \partial ∂ maps k-simplex to sum of its (k-1)-faces with alternating signs based on vertex ordering
Examples and applications
0-simplex [v0] represents single point
1-simplex [v0, v1] represents line segment between two points
2-simplex [v0, v1, v2] represents triangle with vertices v0, v1, v2
3-simplex [v0, v1, v2, v3] represents tetrahedron with vertices v0, v1, v2, v3
Higher-dimensional simplices used in data analysis (simplicial complex models for high-dimensional data)
Simplices form basis for simplicial homology theory in algebraic topology
Constructing simplicial complexes
Fundamental properties and construction
Simplicial complex topological space constructed by gluing simplices along faces subject to specific rules
Two fundamental properties:
Every face of simplex in complex also in complex
Intersection of any two simplices either empty or face of both
Construct simplicial complex:
Start with collection of simplices
Ensure all faces included
Verify intersections of simplices are faces of both
Dimension of simplicial complex equals maximum dimension of constituent simplices
Represent simplicial complex by vertex set and list of maximal simplices (not contained in larger simplex)
Representation and description
F-vector records number of simplices of each dimension providing compact description of structure
Calculate f-vector components: f i = number of i-dimensional simplices f_i = \text{number of i-dimensional simplices} f i = number of i-dimensional simplices
Abstract simplicial complexes generalize geometric simplicial complexes defined by vertex sets and collections of subsets representing simplices
Nerve of a cover important construction method for simplicial complexes from topological data
Examples of simplicial complexes
Triangle with its edges and vertices forms 2-dimensional simplicial complex
Tetrahedron with all faces, edges, and vertices forms 3-dimensional simplicial complex
Simplicial complex modeling social network:
Vertices represent individuals
Edges represent connections
Triangles represent groups of three mutually connected individuals
Vietoris-Rips complex constructed from point cloud data used in topological data analysis
Faces and cofaces of simplices
Faces and their properties
Face of k-simplex any l-simplex (l ≤ k) formed by subset of vertices including simplex itself
Proper faces all faces except simplex itself and empty set
Calculate number of l-faces of k-simplex using binomial coefficient: ( k + 1 l + 1 ) \binom{k+1}{l+1} ( l + 1 k + 1 )
Boundary operator ∂ \partial ∂ maps simplex to alternating sum of its faces
Coface of simplex σ in simplicial complex any simplex τ containing σ as face
Link of simplex set of all simplices disjoint from it but together form larger simplex in complex
Star of simplex union of all simplices having it as face forming subcomplex of original complex
Facets maximal faces of simplicial complex not contained in larger simplex of complex
Applications and examples
Faces and cofaces crucial for understanding combinatorial structure of simplicial complexes
Use faces and cofaces to define various topological operations (homology groups)
Example: 2-simplex [v0, v1, v2] has faces:
0-faces: [v0], [v1], [v2]
1-faces: [v0, v1], [v1, v2], [v0, v2]
2-face: [v0, v1, v2] itself
Link of vertex in triangulation of surface consists of edges and vertices forming cycle around vertex
Boundary and interior of simplicial complexes
Boundary definition and properties
Boundary of simplicial complex consists of simplices that are faces of odd number of maximal simplices
Closed simplicial complex has no boundary
Calculate boundary using chain complex and boundary operator
Boundary of n-simplex consists of all its (n-1)-dimensional faces
Interior and closure
Interior of simplicial complex complement of boundary within complex itself
Open simplicial complex equals its interior
Closure of subset A of simplicial complex K smallest subcomplex of K containing A
Interior point of simplex point not contained in any proper face
Geometric realization and topological concepts
Geometric realization views simplicial complex as topological space
Boundary and interior have usual topological meanings in geometric realization
Euler characteristic key topological invariant related to boundary structure
Calculate Euler characteristic using f-vector: χ = ∑ i = 0 n ( − 1 ) i f i \chi = \sum_{i=0}^n (-1)^i f_i χ = ∑ i = 0 n ( − 1 ) i f i
Relative homology groups of simplicial complex with respect to boundary provide important topological information
Compute relative homology groups using chain complexes of simplicial complex and its boundary