Mathematical structure, an assemblage of a number of simplexes. No, "simplicial complex" is not an oxymoron.

Simplicial complexes, often called just "complexes", are used in Homotopy theory but are also very useful in modeling spatial objects in various computer graphics-related disciplines.

Each complex has its own dimension, (let's call it k), such that k >= 0, and contains one or more k-simplexes, none of which have points in common, except that two simplexes may share a face. Each complex is constructed with reference to a Euclidean Space whose dimension >= k.

As is pointed out in the node on simplexes, each k-simplex has (k+1) "faces" forming its outer boundary. Each face is a (k-1)-simplex.

So we can recursively take the k-simplexes, all the (k-1)-simplexes that are faces of the k-simplexes, all the (k-2)-simplexes which are faces of the faces, and so on, until we have 0-simplexes, which are all of the simplexes' vertices. The assemblage of all these simplexes is a simplicial complex.

For computer graphics cognoscenti, a wire frame is a 2-complex constructed in 3-dimensional space.

A simplicial complex is not a topological space (although it can serve as a base for the discrete topology of the set of its constituent simplexes). However, you can take the space the complex is constructed in, and restrict it to the set of all the points in any of a complex's simplexes. The resulting subspace is called the polyhedron of that complex.
(a) a family of finite sets, closed under subsets.

(b) a collection of simplices meeting face-to-face.

The first type of complex can be made into the second, by embedding the members of the sets in general position in some space of suffiently high dimension, and forming simplices by taking the convex hull of each set. The sets of a simplicial complex are also known as faces.

--back to combinatorics--

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