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A triangulation of a polyhedron, or rectilinear triangulation, is a representation of the polyhedron (cf. [[Polyhedron, abstract|Polyhedron, abstract]]) as the space of a geometric [[Simplicial complex|simplicial complex]]  $  K $, 
that is, a decomposition of it into closed simplices such that any two simplices either do not intersect or intersect along a common face. Rectilinear triangulations of polyhedra serve as the main tool for studying them. Any polyhedron has a triangulation and any two triangulations of it have a common subdivision.

The closed star  $   \mathop{\rm St} ( \sigma , T) $
of a simplex  $  \sigma $
in a triangulation  $  T $
is the union of the simplices of  $  T $
containing  $  \sigma $. 
There is a representation of the closed star of a simplex  $  \sigma \in T $
as the union (or join, cf. [[Union of sets|Union of sets]]) of  $  \sigma $
and its [[Link|link]]:  $   \mathop{\rm St} ( \sigma , T) = \sigma \star  \mathop{\rm lk} ( \delta , T) $. 
In particular, the star of a vertex is a [[Cone|cone]] over its link. If a simplex  $  \sigma \in T $
is represented as the join of two of its faces  $  \delta $
and  $  \gamma $, 
then  $   \mathop{\rm lk} ( \sigma , T) =  \mathop{\rm lk} ( \delta ,  \mathop{\rm lk} ( \gamma , T)) $. 
The link of a simplex does not depend on  $  T $: 
If  $  \sigma $
is a simplex in rectilinear triangulations  $  T _ {1} $, 
$  T _ {2} $
of the same polyhedron, then the polyhedra  $  |  \mathop{\rm lk} ( \sigma , T _ {1} ) | $
and  $  |  \mathop{\rm lk} ( \sigma , T _ {2} ) | $
are PL-homeomorphic. The open star of a simplex  $  \sigma \in T $
is defined as the union of the interiors of those simplices of  $  T $
containing  $  \sigma $
as a face. The open stars of the vertices of a triangulation of a polyhedron  $  P $
form an open covering of  $  P $. 
The nerve of this covering (cf. [[Nerve of a family of sets|Nerve of a family of sets]]) is simplicially isomorphic to the triangulation. Two triangulations  $  T _ {1} $
and  $  T _ {2} $
of polyhedra  $  P _ {1} $
and  $  P _ {2} $
are combinatorially equivalent if certain subdivisions of them are simplicially isomorphic. In order that two triangulations  $  T _ {1} $
and  $  T _ {2} $
be combinatorially equivalent it is necessary and sufficient that  $  P _ {1} $
and  $  P _ {2} $
be PL-homeomorphic. A triangulation of a manifold is said to be combinatorial if the star of any of its vertices is combinatorially equivalent to a simplex. In this case the star of any simplex of the triangulation is also combinatorially equivalent to a simplex.

If  $  P $
is a closed subpolyhedron of a polyhedron  $  Q $, 
then any triangulation  $  K $
of  $  P $
can be extended to some triangulation  $  L $
of  $  Q $. 
In this case one says that the pair of geometric simplicial complexes  $  ( L, K) $
triangulates the pair  $  ( Q, P) $. 
A triangulation of the direct product  $  \sigma \times \delta \in \mathbf R  ^ {m} \times \mathbf R  ^ {n} $
of two simplices  $  \sigma \in \mathbf R  ^ {m} $, 
$  \delta \in \mathbf R  ^ {n} $
can be constructed as follows. The vertices of the triangulation are the points  $  c _ {ij} = ( a _ {i} b _ {j} ) $, 
$  0 \leq  i \leq   \mathop{\rm dim}  \delta $, 
where  $  a _ {i} $
are the vertices of  $  \sigma $
and  $  g _ {j} $
are the vertices of  $  \delta $. 
The vertices  $  c _ {i _ {0}  j _ {0} } \dots c _ {i _ {p}  j _ {p} } $, 
where  $  i _ {0} \leq  \dots \leq  i _ {k} $, 
span a  $  k $-
dimensional simplex if and only if none of these coincide and  $  j _ {0} \leq  \dots \leq  j _ {k} $. 
A triangulation of the direct product of two simplicial complexes with ordered vertices can be carried out in the same way.

A triangulation of a topological space, or curvilinear triangulation, is a pair  $  ( K, f  ) $, 
where  $  K $
is a geometric simplicial complex and  $  f:  | K | \rightarrow X $
is a homeomorphism. Two triangulations  $  ( K, f  ) $
and  $  ( L, g) $
of a space  $  X $
coincide if  $  g  ^ {-} 1 f:  | K | \rightarrow | L | $
is a simplicial isomorphism. If  $  \sigma $
is a simplex of a complex  $  K $
and  $  ( K, f  ) $
is a triangulation of  $  X $, 
then the space  $  f ( \sigma ) $
endowed with the homeomorphism  $  f \ \mid  _  \sigma  :  \sigma \rightarrow f ( \sigma ) $
is called a topological simplex. The star and the link of a topological simplex of a triangulated topological space are defined in the same way as in the case of rectilinear triangulations. If a point  $  a \in X $
is a vertex of triangulations  $  ( K, f  ) $
and  $  ( L, g) $
of  $  X $, 
then its links in these triangulations are homotopy equivalent.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,   "Combinatorial topology" , Graylock , Rochester  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.B. Fuks,   V.A. Rokhlin,   "Beginner's course in topology. Geometric chapters" , Springer  (1981)  (Translated from Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.C. Zeeman,   "Seminar on combinatorial topology" , IHES  (1963)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Seifert,   W. Threlfall,   "Lehrbuch der Topologie" , Chelsea, reprint  (1970)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I.M. Singer,   J.A. Thorpe,   "Lecture notes on elementary topology and geometry" , Springer  (1967)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L.C. Glaser,   "Geometrical combinatorial topology" , '''1–2''' , v. Nostrand  (1970)</TD></TR></table>