The 18th-century Swiss mathematician Leonhard Euler developed two equations that have come to be known as Euler's formula. One of these equations relates the number of vertices, faces, and edges on a polyhedron. The other formula relates the five most common mathematical constants to each other. These two equations ranked second and first, respectively, as the most elegant mathematical results according to "The Mathematical Intelligencer."

Euler's formula for polyhedra is sometimes also called the Euler-Descartes theorem. It states that the number of faces, plus the number of vertices, minus the number of edges on a polyhedron always equals two. It is written as F + V - E = 2. For example, a cube has six faces, eight vertices, and 12 edges. Plugging into Euler's formula, 6 + 8 - 12 does, in fact, equal two.

There are exceptions to this formula, because it only holds true for a polyhedron that does not intersect itself. Well-known geometrical shapes including spheres, cubes, tetrahedra, and octagons are all non-intersecting polyhedra. An intersecting polyhedron would be created, however, if someone were to join two of the vertices of a non-intersecting polyhedron. This would result in the polyhedron having the same number of faces and edges, but one fewer vertice, so it is obvious that the formula is no longer true.

On the other hand, a more general version of Euler's formula can be applied to polyhedra that intersect themselves. This formula is often used in topology, which is the study of spatial properties. In this version of the formula, F + V - E is equal to a number called Euler's characteristic, which is often symbolized by the Greek letter chi. For example, both the donut-shaped torus and the Mobius strip have an Euler's characteristic of zero. Euler's characteristic can also be less than zero.

The second Euler's formula includes the mathematical constants e, i, Π, 1, and 0. E, which is often called Euler's number and is an irrational number that rounds to 2.72. The imaginary number i is defined as the square root of -1. Pi (Π), the relationship between the diameter and circumference of a circle, is approximately 3.14 but, like e, is an irrational number.

This formula is written as e^{(i*Π)} + 1 = 0. Euler discovered that if Π was substituted for x in the trigonometric identity e^{(i*Π)} = cos(x) + i*sin(x), the result was what we now know as Euler's formula. In addition to relating these five fundamental constants, the formula also demonstrates that raising an irrational number to the power of an imaginary irrational number can result in a real number.