Simplicial complex

From Polytope Wiki
Jump to navigation Jump to search
A simplicial 3-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“š 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“š, such that:

  • If then every element of x  is in π“š,
  • The intersection of two simplices, x  and y , in π“š 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“š is a simplicial complex, then the closure of , denoted , is the smallest subcomplex of π“š 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“š is a simplicial complex, then the star of , denoted , is the smallest set of all simplices in π“š that intersect with S .

Applications in polytopes[edit | edit source]

The topology of the icosahedron as a simplicial complex.

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“Ÿ we construct an analogous abstract simplicial complex π“š by taking the elements of π“Ÿ as vertices in π“š. The simplicies in π“š are then sets of mutually incident elements. π“š is then a homogeneous k -complex where every flag of π“Ÿ corresponds to a k -simplex in π“š 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“Ÿ.

External links[edit | edit source]