Leonhard Euler developed two elegant formulas: one for the number of vertices, faces, and edges on a polyhedron, and the other for relating e, i, Π, 1, and 0. Euler’s formula for polyhedra is F + V – E = 2, but it only holds for non-intersecting polyhedra. A more general version can be applied to intersecting polyhedra. Euler’s second formula is e(i*Π) + 1 = 0, which relates the five fundamental constants and demonstrates that raising an irrational number to the power of an imaginary irrational number can result in a real number.
The 18th-century Swiss mathematician Leonhard Euler developed two equations known as Euler’s formula. One such equation concerns 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 were ranked second and first respectively as the most elegant mathematical results according to “The Mathematical Intelligencer”.
Euler’s formula for polyhedra is also sometimes 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. Inserting Euler’s formula, 6 + 8 – 12 is, in fact, equal to two.
There are exceptions to this formula, because it only holds for a non-intersecting polyhedron. Well-known geometric shapes including spheres, cubes, tetrahedrons 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 less vertex, so it stands to reason that the formula is no longer true.
On the other hand, a more general version of Euler’s formula can be applied to intersecting polyhedra. This formula is often used in topology, which is the study of spatial properties. In this version of the formula, F + V – E equals a number called the Euler characteristic, often symbolized by the Greek letter chi. For example, both the donut-shaped torus and the Mobius strip have a zero Euler characteristic. The Euler characteristic can also be less than zero.
Euler’s second 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 were replaced by x in the trigonometric identity e(i*Π) = cos(x) + i*sin(x), the result was what we know today 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.
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