Simplicial complex: Difference between revisions
No edit summary |
|||
| Line 6: | Line 6: | ||
* For any two simplices in the complex, either they do not intersect, or their intersection is a common face of both. | * For any two simplices in the complex, either they do not intersect, or their intersection is a common face of both. | ||
Here a '''simplex''' is a finite affine independent subset of the Euclidean space, and a '''face''' of a simplex is simply a subset of it. The elements of the subset are often called the '''vertices''' of the simplex, and a simplex of size <math>k+1</math> is termed a <math>k</math>-simplex. | Here a '''simplex''' is the set of all convex combinations of a finite affine independent subset of the Euclidean space, and a '''face''' of a simplex is simply a subset of it. The elements of the subset are often called the '''vertices''' of the simplex, and a simplex of size <math>k+1</math> is termed a <math>k</math>-simplex. | ||
The '''underlying space''' of a simplicial complex is defined as the union of all its simplices. A space that can be expressed as the underlying space of a simplicial complex is termed a [[polyhedron]], and such an expression i s termed a [[triangulation]]. | The '''underlying space''' of a simplicial complex is defined as the union of all its simplices. A space that can be expressed as the underlying space of a simplicial complex is termed a [[polyhedron]], and such an expression i s termed a [[triangulation]]. | ||
Revision as of 01:46, 22 May 2007
Definition
A (finite) simplicial complex is a finite collection of simplies in some Euclidean space such that:
- If any simplex belongs to the complex, so do all its faces
- For any two simplices in the complex, either they do not intersect, or their intersection is a common face of both.
Here a simplex is the set of all convex combinations of a finite affine independent subset of the Euclidean space, and a face of a simplex is simply a subset of it. The elements of the subset are often called the vertices of the simplex, and a simplex of size is termed a -simplex.
The underlying space of a simplicial complex is defined as the union of all its simplices. A space that can be expressed as the underlying space of a simplicial complex is termed a polyhedron, and such an expression i s termed a triangulation.