12, 8), and if Δ is the first simplicial complex pictured above, its f-vector is (1, 18, 23, 8, 1). A complete characterization of the possible f-vectors of simplicial complexes is given by the Kruskal–Katona theorem. By using the f-vector of a simplicial d-complex Δ as coefficients of a polynomial (written in decreasing order of exponents), we obtain the f-polynomial of Δ. In our two examples above, the f-polynomials would be and , respectively. Combinatorists are often quite interested in the h-vector of a simplicial complex Δ, which is the sequence of coefficients of the polynomial that results from plugging x − 1 into the f-polynomial of Δ. Formally, if we write FΔ(x) to mean the f-polynomial of Δ, then the h-polynomial of Δ is and the h-vector of Δ is We calculate the h-vector of the octahedron boundary (our first example) as follows: So the h-vector of the boundary of the octahedron is (1, 3, 3, 1). It is not an accident this h-vector is symmetric. In fact, this happens whenever Δ is the boundary of a simplicial polytope (these are the Dehn–Sommerville equations). In general, however, the h-vector of a simplicial complex is not even necessarily positive. For instance, if we take Δ to be the 2-complex given by two triangles intersecting only at a common vertex, the resulting h-vector is (1, 3, −2). A complete characterization of all simplicial polytope h-vectors is given by the celebrated g-theorem of Stanley, Billera, and Lee. Simplicial complexes can be seen to have the same geometric structure as the contact graph of a sphere packing (a graph where vertices are the centers of spheres and edges exist if the corresponding packing elements touch each other) and as such can be used to determine the combinatorics of sphere packings, such as the number of touching pairs (1-simplices), touching triplets (2-simplices), and touching quadruples (3-simplices) in a sphere packing. Computational problems The simplicial complex recognition problem is: given a finite simplicial complex, decide whether it is homeomorphic to a given geometric object. This problem is undecidable for any d-dimensional manifolds for d ≥ 5. See also Abstract simplicial complex Barycentric subdivision Causal dynamical triangulation Delta set Polygonal chain 1 dimensional simplicial complex Tucker's lemma References External links Norman J. Wildberger. "Simplices and simplicial complexes". A Youtube talk.. Topological spaces Algebraic topology Simplicial sets Triangulation (geometry)