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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
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.
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.
None of its faces intersect except at their edges. The same number of faces meet at each of its vertices . Each Platonic solid can therefore be assigned a pair { p , q } of integers, where p is the number of edges (or, equivalently, vertices) of each face, and q is the number of faces (or, equivalently, edges) that meet at each vertex.
From the fact that each facet of a three-dimensional polyhedron has at least three edges, it follows by double counting that 2e ≥ 3f, and using this inequality to eliminate e and f from Euler's formula leads to the further inequalities e ≤ 3v − 6 and f ≤ 2v − 4. By duality, e ≤ 3f − 6 and v ≤ 2f − 4.
Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics". [2] When x = π, Euler's formula may be rewritten as e iπ + 1 = 0 or e iπ = −1, which is known as Euler's identity.
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.