# Difference between simplex and simplicial complex

First I know the definition of simplex intuitively as follows,

Simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension.

And the defintion of simplicial complex as follows from wiki, Simplicial complex is a toplogical space of certain kind, constructed by gluing together points, line segements, triangles and their $n-$ dimensional counterparts.

Here what i want to know the difference between two object (simplex and simplicial complex).

For my short knowledge, I found some statement in the textbook.

Finite simplicial complexes is consisted of a finite number of simplicies.

It seems to me simplex is key object and simplicial complex is a set of this key object.

But i am not sure.

• The way you cut out six squares in a cross-formation (like this) and fold and glue them together to make a cube, you can cut out simplices and glue them together along their edges to make a simplicial complex. This gets more difficult to imagine the higher the dimensions are, but the idea is the same. Dec 11, 2014 at 12:40
• A simplex is to a simplicial complex as a Lego brick is to a dinosaur assembled from such bricks. Dec 11, 2014 at 13:14
• To Arthur, and Georges Elencwajg, Thanks ! Dec 11, 2014 at 17:39

You can think of a simplicial complex simply (haha) as a set of sets. with certain conditions. Let $X$ be a set with simplices $\sigma_1, ..., \sigma_n$. The following must apply for $X$ to be a simplicial complex:
1. If $\sigma_i,\sigma_j \in X$ then $\sigma_i \cap \sigma_j \in X$. Meaning that simplices only intersect at their faces and
2. If $\sigma \in X$ then if $\tau \subseteq \sigma$ then $\tau \in X$. Meaning that all faces of a simplex are also simplices.
This is the definition for an abstract simplicial complex, there are more rigorous definitions for mapping them into $\mathbb{R}^n$ but this I think it the most useful definition for gaining intuitive understanding of these objects.