A simplicial complex is a union of spaces known as simplicies, that are convex hulls of points in general position. In Euclidean space they can be thought of as a generalisation of the triangle. For the first few dimensions they are: the point, the line segment, the triangle and the tetrahedron.
Definition of a simplex
Given n+1 points , in a space with dimension of at least n, such that no three points are colinear, the set
is a simplex, and more specifically an n-simplex since it has n+1 verticies.
The standard n-simplex
For ease we will sometimes need a standardised co-ordinate system for a simplex. The standard n-simplex, a subset of Euclidean (n+1)-space, , is
Note that the set is expressed in a very similar manner, but the co-ordinates are fixed in .
Face of a Simplex
A face of an n-simplex is any (n-1)-simplex formed by an n-1 element subset of the verticies of .
Defintion of a simplicial complex
A simplicial complex is a union of simplicies such that the intersection of any two simplicies is a simplex. Alternatively if is a simplicial complex then
1. Every face of a simplex in is a simplex in .
2. The intersection of two simplicies is a face of both of A and B.
The triangulation of a polygon in the plane is a simplcial complex. In fact it hints at the existence of simplicial complexes existing for all polytopes, e.g. every polyhedron can be expressed as tetrahedrons meeting full face to full face.