# Simplicial complex

**Simplicial complexes** are structures composed of simplices in a particular way. They provide a bridge between abstract and topological structures.

## Definition[edit | edit source]

### Abstract[edit | edit source]

An abstract simplicial complex is a set of finite sets π such that:

- .

That is it is a set which is closed under subsets.

### Geometric[edit | edit source]

A geometric simplicial complex is a set of simplices π, such that:

- If then every element of x is in π,
- The intersection of two simplices, x and y , in π is an element of both x and y .

## Terminology[edit | edit source]

There are a number of terms used commonly with simplicial complexes which are unique or conflict with similar terms used for polytopes.

- Cell
- A rank k simplex of a simplicial complex realized in k -dimensional space.
- Closed star
- The
**closed star**denoted is the closure of the star. - Closure
- If π is a simplicial complex, then the
**closure**of , denoted , is the smallest subcomplex of π containing S . - Face
- An element of a simplex.
- Facet
- A simplex in a complex that is not an element of another simplex in that complex.
- Homogeneous
- A k -complex where every simplex is an element of a k -simplex. Alternatively: A simplex where every facet has the same rank. Pure complexes may also be called pure.
- Polyhedron
- The space described by a simplicial complex.
- Pure
- Homogeneous
- Star
- If π is a simplicial complex, then the
**star**of , denoted , is the smallest set of all simplices in π that intersect with S .

## Applications in polytopes[edit | edit source]

While simplicial complexes are not themselves polytopes, simplicial complexes give a method of deriving topological properties for polytopes which don't necessarily have an induced topology, including abstract, skew, and self-intersecting polytopes.

For a polytope π we construct an analogous abstract simplicial complex π by taking the elements of π as vertices in π. The simplicies in π are then sets of mutually incident elements. π is then a homogeneous k -complex where every flag of π corresponds to a k -simplex in π attached to other simplices along a k-1 -element iff those two flags are adjacent.

The simplicial complex can then be converted to an analogous cell complex to create a topology of π.

## External links[edit | edit source]

- Wikipedia contributors. "Simplicial complex".
- Weisstein, Eric W. "Simplicial Complex" at MathWorld.
- nLab contributors. "Simplicial complex" on nLab.