# Triangulation in Geometry

More specifically, it can be defined as the underlying space of a simplicial complex. Here, a polyhedron can be viewed as an intersection of halfspaces (Webster 1994).
Then, a convex polytope may be defined as the convex hull of a finite set of points (which are always bounded), or as a bounded intersection of a finite set of half-spaces (Webster 1994). More specifically, it can be defined as a finite region of -dimensional space enclosed by a finite number of hyperplanes. The part of the polytope that lies in one of the bounding hyperplanes is called a cell (Weisstein 2002).
Simplex is the generalization of a tetrahedral region of space to -D. The boundary of a -simplex has -faces (vertices), -faces (edges), and -faces, where is a binomial coefficient. The simplex named because it represents the simplest possible polytope in any given space (Weisstein 2002).
Simplicial complex is a space with a triangulation. Formally, a simplicial complex in is a collection of simplices in such that: (i) every face of a simplex of is in , and (ii) the intersection of any two simplices of is a face of each of them (Munkres 1993). Objects in the space made up of only the simplices in the triangulation of the space are called simplicial subcomplexes (Weisstein 2002).
Usually, surface is a -D submanifold of -D Euclidean space. More generally, surface is an -D submanifold of an -D manifold, or in general, any codimension-1 subobject in an object like a Banach space or an infinitedimensional manifold. A surface with a finite number of triangles in its triangulation is called compact surface (Weisstein 2002).
Genus is a topologically invariant property of a surface defined as the largest number of nonintersecting simple closed curves that can be drawn on the surface without separating it. In fact, it is the number of holes in a surface (Weisstein 2002). The geometric genus of a surface is related to the Euler characteristic by .
Our final key definition is for Betti numbers. Betti numbers are topological objects which were proved to be invariants by Poincar, and used by him to extend the polyhedral formula to higher dimensional spaces. Informally, the Betti number is the maximum number of cuts that can be made without dividing a surface into two separate pieces. Formally, the -th Betti number is the rank of the -th homology group of a topological space (Griffiths 1976. Weisstein 2002).
2. Euler’s theorem
Euler’s theorem states relation between the number of vertices , edges , and faces of a simply connected (i.e., genus ) polyhedron () or polygon (), i.e. it states the polyhedral formula .
Initial Euler’s proof of the polyhedral formula is not irreproachable (Cromwell 1999). This proof is based on the principle that polyhedrons can be truncated. Euler’s proof begins with a polyhedron consisting of a large number of vertice