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.

  • 1
    $\begingroup$ 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. $\endgroup$
    – Arthur
    Dec 11, 2014 at 12:40
  • 5
    $\begingroup$ A simplex is to a simplicial complex as a Lego brick is to a dinosaur assembled from such bricks. $\endgroup$ Dec 11, 2014 at 13:14
  • $\begingroup$ To Arthur, and Georges Elencwajg, Thanks ! $\endgroup$
    – phy_math
    Dec 11, 2014 at 17:39

2 Answers 2


A simplicial complex is made out of simplicies which I'm sure you already now. But the conditions that these simplices have create the desired topological and algebraic structure.

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.


Simply put, simplicies are building blocks. Simplicial complexes are "things" constructed from simplicies.


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