It’s most intuitive to start with the geometric viewpoint, in which case an -simplex is defined to be the convex hull of affinely independent points in . These points are called the vertices of the simplex. Here are examples for
A simplicial complex is then a collection of simplices glued together along lower-dimensional simplices. More formally, if is a (geometric) simplex, then a face of is a subset formed by taking the convex hull of a subset of the vertices of .
Finally, a (geometric) simplicial complex is a collection of simplices such that
- If and is a face of , then , and
- If and , then is a face of both and .
Property (1) gives us downward closure, and property (2) specifies how simplices can be glued together (only along faces). For instance, the first picture depicts a simplicial complex. The second does not.