Subdivision (simplicial complex)
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A subdivision (also called refinement) of a simplicial complex is another simplicial complex in which, intuitively, one or more simplices of the original complex have been partitioned into smaller simplices. The most commonly used subdivision is the barycentric subdivision, but the term is more general. The subdivision is defined in slightly different ways in different contexts.
Subdivision by starring
Let K be a geometric simplicial complex (GSC). A subdivision of K is a GSC L such that:[1]: 15 [2]: 3
- |K| = |L|, that is, the union of simplices in K equals the union of simplices in L (they cover the same region in space).
- each simplex of L is contained in some simplex of K.
As an example, let K be a GSC containing a single triangle {A,B,C} (with all its faces and vertices). Let D be a point on the face AB. Let L be the complex containing the two triangles {A,D,C} and {B,D,C} (with all their faces and vertices). Then L is a subdivision of K, since the two triangles {A,D,C} and {B,D,C} are both contained in {A,B,C}, and similarly the faces {A,D}, {D,B} are contained in the face {A,B}, and the face {D,C} is contained in {A,B,C}.
One way to obtain a subdivision of K is to pick an arbitrary point x in |K|, remove each simplex s in K that contains x, and replace it with the closure of the following set of simplices:
where is the join of the point x and the face t. This process is called starring at x.[1]: 15
A stellar subdivision is a subdivision obtained by sequentially starring at different points.[1]: 15
A derived subdivision is a subdivision obtained by the following inductive process.[2]: 3
- Star each 1-dimensional simplex (a segment) at some internal point;
- Star each 2-dimensional simplex at some internal point, over the subdivision of the 1-dimensional simplices;
- ... Star each k-dimensional simplex at some internal point, over the subdivision of the (k-1)-dimensional simplices.
The barycentric subdivision is a derived subdivision where the points used for starring are always barycenters of simplices. For example, if D, E, F, G are the barycenters of {A,B}, {A,C}, {B,C}, {A,B,C} respectively, then the first barycentric subdivision of {A,B,C} is the closure of {A,D,G}, {B,D,G}, {A,E,G}, {C,E,G}, {B,F,G}, {C,F,G}.
Iterated subdivisions can be used to attain arbitrarily fine triangulations of a given polyhedron.