Euler’s formula
Euler’s formula, either of two important mathematical theorems discovered by the Swiss mathematician Leonhard Euler.
The more well-known of the two Euler formulas is eiθ = cos θ + i sin θ, where e is the exponential function; i is the imaginary unit, the square root of -1; cos and sin are the trigonometric functions cosine and sine, respectively, and θ is the value of an angle. This formula is sometimes called Euler’s identity.
When θ = π, the Euler formula becomeseiπ = cos π + i sin π = -1.This last result can be rewritten as eiπ + 1 = 0,which relates five important mathematical constants—0, 1, e, i, and π—in a single equation.
The second formula is a numerical relationship between the faces F, edges E, and vertices V of a polyhedron (a solid shape with polygons as its faces):V - E + F = 2.This result is also called Euler’s polyhedral formula.
For example, a cube has 6 faces, 12 edges, and 8 vertices; that is,8 - 12 +6 = 2.The left-hand side of the formula is more generally known as the Euler characteristic in topology.