Simplicial Complex

For this definition, it is assumed that $X = {\mathbb{R}}^n$. Let $p_1$, $p_2$, $\ldots$, $p_{k+1}$, be $k+1$ linearly independent^6.5 points in ${\mathbb{R}}^n$. A $k$-simplex, $[p_1,\ldots,p_{k+1}]$, is formed from these points as

$$[p_1,\ldots,p_{k+1}] = \left\{ \sum_{i=1}^{k+1} \alpha_i p_i \in ... ...\geq 0 \mbox{ for all } i \mbox{ and } \sum_{i=1}^{k+1} \alpha_i = 1 \right\} ,$$ (6.3)

in which $\alpha_i p_i$ is the scalar multiplication of $\alpha_i$ by each of the point coordinates. Another way to view (6.3) is as the convex hull of the $k+1$ points (i.e., all ways to linearly interpolate between them). If $k=2$, a triangular region is obtained. For $k=3$, a tetrahedron is produced.

For any $k$-simplex and any $i$ such that $1 \leq i \leq k+1$, let $\alpha_i=0$. This yields a $(k-1)$-dimensional simplex that is called a face of the original simplex. A 2-simplex has three faces, each of which is a 1-simplex that may be called an edge. Each 1-simplex (or edge) has two faces, which are 0-simplexes called vertices.

To form a complex, the simplexes must fit together in a nice way. This yields a high-dimensional notion of a triangulation, which in ${\mathbb{R}}^2$ is a tiling composed of triangular regions. A simplicial complex, ${\cal K}$, is a finite set of simplexes that satisfies the following:

1. Any face of a simplex in ${\cal K}$ is also in ${\cal K}$.
2. The intersection of any two simplexes in ${\cal K}$ is either a common face of both of them or the intersection is empty.

Figure 6.15 illustrates these requirements. For $k > 0$, a $k$-cell of ${\cal K}$ is defined to be interior, $\operatorname{int}([p_1,\ldots,p_{k+1}])$, of any $k$-simplex. For $k = 0$, every 0-simplex is a 0-cell. The union of all of the cells forms a partition of the point set covered by ${\cal K}$. This therefore provides a cell decomposition in a sense that is consistent with Section 6.2.2.

Figure 6.15: To become a simplicial complex, the simplex faces must fit together nicely.