Introduction:
A simplicial complex is a special type of graph wherein the notion of a vertex is replaced with a new higher dimensional analog, called a simplex. A simplex is, in simplest terms, an n-dimensional collection of vertices enclosed by faces.
Definitions:
Graph:
"A graph G = (V, E) consists of V, a nonempty set of vertices (or nodes) and E, a set of edges. Each edge has either one or two vertices associated with it, called its endpoints. An edge is said to connect its endpoints."1
Simplex:
"A simplex...is the generalization of a tetrahedral region of space to n dimensions."2
A simplex of n dimensions is referred to as an n-simplex.
| Dimension | Name |
| 0 | 0-Simplex or Point |
| 1 | 1-Simplex or Line Segment |
| 2 | 2-Simplex or Triangle |
| 3 | 3-Simplex or Tetrahedron |
| n | n-Simplex |
Simplicial Complex:
"...a simplicial complex K in Rnis a collection of simplices in Rn such that...every face of K is in K, and...the intersection of any two faces of K is a face of each of them."3
Simplicial Complexes:
In an connected planar graph, Euler's Formula E + 2 = V + F for which V is the number of vertices, F is the number of faces, and E is the number of edges holds.Whereas ordinary graphs are composed of 0-dimensional points (called vertices), a simplicial complex removes this restriction. Where a regular graph would have a vertex, a simplicial complex can have any simplex (including a 0-dimensional point through an n-simplex).
A graph is a thing made of point, some of which are linked by line segments. Generalize the idea to points that can also be grouped into triangles, or tetrahedra, etc.
For graphs we know Euler's formula E+2=V+F. Give this a geometric meaning.
Discuss (maybe in the 2-dimensional case) what might replace this formula compare a "triangulated" sphere to a "triangulated" doughnut.
1 Found in Discrete Mathematics and Its Applications, Kenneth H. Rosen.
2 Found at http://mathworld.wolfram.com/Simplex.html.
3 Found at http://mathworld.wolfram.com/SimplicialComplex.html, adapted from J. R. Munkres, "Simplicial Complexes and Simplicial Maps," 1993.
