AMS :: Feature Column from the AMS
Euler's Formula. For any polyhedron that doesn't intersect itself, the. Number of Faces; plus the Number of Vertices (corner points); minus the Number of Edges. These numbers - 6 faces, 12 edges, and 8 vertices - are actually related to each other. This relationship is written as a math formula like this. This is level 1; Count the number of faces, edges and vertices. Can you find a connection between the number of faces, vertices and edges of any polyhedron.
Using Euler's formula in a similar way we can discover that there is no simple polyhedron with ten faces and seventeen vertices.
The prism shown below, which has an octagon as its base, does have ten faces, but the number of vertices here is sixteen. The pyramid, which has a 9-sided base, also has ten faces, but has ten vertices. But Euler's formula tells us that no simple polyhedron has exactly ten faces and seventeen vertices. Both these polyhedra have ten faces, but neither has seventeen vertices.
It's considerations like these that lead us to what's probably the most beautiful discovery of all.
- Faces, Edges and Vertices
- Euler's polyhedron formula
- Euler's Polyhedral Formula: Part II
It involves the Platonic Solids, a well-known class of polyhedra named after the ancient Greek philosopher Platoin whose writings they first appeared. From left to right we have the tetrahedon with four faces, the cube with six faces, the octahedron with eight faces, the dodecahedron with twelve faces, and the icosahedron with twenty faces.
Although their symmetric elegance is immediately apparent when you look at the examples above, it's not actually that easy to pin it down in words. It turns out that it is described by two features. The first is that Platonic solids have no spikes or dips in them, so their shape is nice and rounded.
In other words, this means that whenever you choose two points in a Platonic solid and draw a straight line between them, this piece of straight line will be completely contained within the solid — a Platonic solid is what is called convex.
The second feature, called regularity, is that all the solid's faces are regular polygons with exactly the same number of sides, and that the same number of edges come out of each vertex of the solid. The cube is regular, since all its faces are squares and exactly three edges come out of each vertex. You can verify for yourself that the tetrahedron, the octahedron, the icosahedron and the dodecahedron are also regular. Now, you might wonder how many different Platonic Solids there are.
Ever since the discovery of the cube and tetrahedron, mathematicians were so attracted by the elegance and symmetry of the Platonic Solids that they searched for more, and attempted to list all of them.
This is where Euler's formula comes in.
You can use it to find all the possibilities for the numbers of faces, edges and vertices of a regular polyhedron. What you will discover is that there are in fact only five different regular convex polyhedra! This is very surprising; after all, there is no limit to the number of different regular polygons, so why should we expect a limit here? The five Platonic Solids are the tetrahedron, the cube, the octahedron, the icosahedron and the dodecahedron shown above.
But if you're a mathematician, this isn't enough. You'll want a proof, a water-tight logical argument that shows you that it really works for all polyhedra, including the ones you'll never have the time to check.
Adrien-Marie Legendre, - Despite the formula's name, it wasn't in fact Euler who came up with the first complete proof. Augustin-Louis Cauchy, - It's interesting to note that all these mathematicians used very different approaches to prove the formula, each striking in its ingenuity and insight.
It's Cauchy's proof, though, that I'd 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. Forming a network Imagine that you're holding your polyhedron with one face pointing upward. Now imagine "removing" just this face, leaving the edges and vertices around it behind, so that you have an open "box". Next imagine that you can hold onto the box and pull the edges of the missing face away from one another.
If you pull them far enough the box will flatten out, and become a network of points and lines in the flat plane. The series of diagrams below illustrates this process as applied to a cube. Turning the cube into a network. As you can see from the diagram above, each face of the polyhedron becomes an area of the network surrounded by edges, and this is what we'll call a face of the network.
These are the interior faces of the network. There is also an exterior face consisting of the area outside the network; this corresponds to the face we removed from the polyhedron. So the network has vertices, straight edges and polygonal faces.
The network has faces, edges and vertices. When forming the network you neither added nor removed any vertices, so the network has the same number of vertices as the polyhedron — V. The network also has the same number of edges — E — as the polyhedron. Now for the faces; all the faces of the polyhedron, except the "missing" one, appear "inside" the network. The missing face has become the exterior face which stretches away all round the network.
So, including the exterior face, the network has F faces. We'll now go on to transform our network to make this value easier to calculate. Transforming the Network There are three types of operation which we can perform upon our network. We'll introduce three steps involving these. Step 1 We start by looking at the polygonal faces of the network and ask: If there is, we draw a diagonal as shown in the diagram below, splitting the face into two smaller faces.
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3D Shapes - Maths GCSE Revision
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