Mathematical structure, an assemblage of a number of simplex
es. No, "simplicial complex" is not an oxymoron.
Simplicial complexes, often called just "complexes", are used in Homotopy
theory but are also very useful in modeling spatial objects in various computer graphics-related disciplines.
Each complex has its own dimension, (let's call it k
), such that k
>= 0, and contains one or more k
-simplexes, none of which have points in common, except that two simplexes may share a face. Each complex is constructed with reference to a Euclidean Space whose dimension >= k
As is pointed out in the node on simplexes
, each k
-simplex has (k
+1) "faces" forming its outer boundary. Each face is a (k
So we can recursive
ly take the k
-simplexes, all the (k
-1)-simplexes that are faces of the k
-simplexes, all the
-2)-simplexes which are faces of the faces, and
so on, until we have 0-simplexes, which are all of the simplexes' vertices
. The assemblage of all these simplexes is a simplicial complex.
For computer graphics cognoscenti
, a wire frame
is a 2-complex constructed in 3-dimensional space.
A simplicial complex is not
a topological space
(although it can serve as a base
for the discrete topology
of the set of its constituent simplexes). However, you can take the space the complex is constructed in, and restrict it to the set of all the points in any of a complex's simplexes. The resulting subspace is called the polyhedron
of that complex.