Euler's Formula
Simple though it might look, this little formula encapsulates a fundamental property of those 3D solids we call polyhedra, which have fascinated mathematicians for over 4000 years. Actually I could go further and tell that Euler's formula tells us something very deep about shape and space. The formula bears the name of the famous Swiss mathematician Leonhard Euler (1707 - 1783), who would have celebrated his 300th birthday this year.
What is a polyhedron?
Before we examine what Euler's formula tells us, let's look at polyhedra in a bit more detail. A polyhedron is a solid object whose surface is made up of a number of flat faces which themselves are bordered by straight lines. Each face is in fact a polygon, a closed shape in the flat 2D plane made up of points joined by straight lines.
Polygons are not allowed to have holes in them, as the figure below illustrates: the left-hand shape here is a polygon, while the right-hand shape isn't.
A polygon is called regular if all of its sides are the same length, and all the angles between them are the same; the triangle and square in figure one and the pentagon in figure two are regular.
A polyhedron is what you get when you move one dimension up. It's a closed, solid object whose surface is made up of a number of polygonal faces. We call the sides of these faces edges — two faces meet along each one of these edges. We call the corners of the faces vertices, so that any vertex lies on at least three different faces. To illustrate this, here are two examples of well-known polyhedra.
A polyhedron consists of just one piece. It cannot, for example, be made up of two (or more) basically separate parts joined by only an edge or a vertex. This means that neither of the following objects is a true polyhedron.
The proof
Playing around with various simple polyhedra will show you that Euler's formula always holds true. But if you are a mathematician, this isn it enough. You will want a proof, a water-tight logical argument that shows you that it really works for all polyhedra, including the ones you will never have the time to check.
Despite the formula's name, it wasn't in fact Euler who came up with the first complete proof. Its history is complex, spanning 200 years and involving some of the greatest names in maths, including René Descartes (1596 - 1650), Euler himself, Adrien-Marie Legendre (1752 - 1833) and Augustin-Louis Cauchy (1789 - 1857).
It is interesting to note that all these mathematicians used very different approaches to prove the formula, each striking in its ingenuity and insight. It is Cauchy's proof, though, that I do like to give you a flavour of here. His method consists of several stages and steps. The first stage involves constructing what is called a network.
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