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Vertex, edge and face of a cube. The Euler characteristic χ was classically defined for the surfaces of polyhedra, according to the formula = + where V, E, and F are respectively the numbers of vertices (corners), edges and faces in the given polyhedron.
Euler's formula states that if a finite, connected, planar graph is drawn in the plane without any edge intersections, and v is the number of vertices, e is the number of edges and f is the number of faces (regions bounded by edges, including the outer, infinitely large region), then
where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Euler's polyhedron formula. Thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, and hence 6 faces.
Euler also made contributions to the understanding of planar graphs. He introduced a formula governing the relationship between the number of edges, vertices, and faces of a convex polyhedron. Given such a polyhedron, the alternating sum of vertices, edges and faces equals a constant: V − E + F = 2.
Let the number of vertices be V, edges be E, faces be F, components H, shells S, and let the genus be G (S and G correspond to the b 0 and b 2 Betti numbers respectively). Then, to denote a meaningful geometric object, the mesh must satisfy the generalized Euler–Poincaré formula. V – E + F = H + 2 * (S – G) The Euler operators preserve ...
Euler's formula relating the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by Cauchy [21] and L'Huilier, [22] and represents the beginning of the branch of mathematics known as topology.
(Since the graph is a triangulation, the charge on each face is 0.) Recall that the sum of all the degrees in the graph is equal to twice the number of edges; similarly, the sum of all the face lengths equals twice the number of edges. Using Euler's Formula, it's easy to see that the sum of all the charges is 12:
Suppose v, e, and f are the number of vertices, edges, and regions (faces). Since each region is triangular and each edge is shared by two regions, we have that 2e = 3f. This together with Euler's formula, v − e + f = 2, can be used to show that 6v − 2e = 12. Now, the degree of a vertex is the number of edges abutting it.