Either of two important mathematical theorems of Leonhard Euler .
The first is a topological invariance (see topology ) relating the number of faces, vertices, and edges of any polyhedron . It is written F + V = E + 2, where F is the number of faces, V the number of vertices, and E the number of edges. A cube, for example, has 6 faces, 8 vertices, and 12 edges, and satisfies this formula. The second formula, used in trigonometry , says e i x = cos x + i sin x where e is the base of the natural logarithm and i is the square root of -1 (see irrational number ). When x is equal to 03C0; or 2 03C0; , the formula yields two elegant expressions relating 03C0; , e , and i : e i 03C0; = -1 and e 2 i 03C0; = 1.