Platonic solids why only five

Next we'll consider all possibilities for the number of faces meeting at a vertex of a regular polyhedron. For each possibility we actually construct such a polyhedron, a picture of which you can see close by on this page. Here are the possibilities:. But now things get a little more subtle. We have looked at all possibilities of congruent regular polygons meeting at a vertex of a polyhedron, but how do we know that there isn't another regular polyhedron for some of these cases?

For example, why is the cube the only polyhedron for which three squares meet at each vertex? The rest of this page gives the answer to this question, but the going will be much harder! The values of these numbers for each of the polyhedra are listed in this table:.

Our aim now is to show that for any pair of number n and m the values of the other parameters, f , e , and v are determined uniquely. We'll see below that this equations actually holds for all convex polyhedra. Given m and n the above three equations determine f , e , and v uniquely, and so there are only five possible regular polyhedra.

The result E is known as. The Greeks studied Platonic solids extensively, and they even associated them with the four classic elements: cube for earth, octahedron for air, icosahedron for water, and tetrahedron for fire. Interestingly, even though we can create infinitely many regular polygons, there are only five regular polyhedra. And the proof is fairly easy. Before we discuss the proof, let us familiarize ourselves with the different terms which we will use in the proof.

In the following discussion, vertex will refer to the corner of a Platonic solid, face will refer to the regular polygons that make up the solid, and side edges in 3D will refer the side of the polygon. Let be the number of sides of a regular polygon on a Platonic solid, and be the number of polygons meeting at each vertex. Let us represent each regular polygon with.

Notice that the interior angles of the regular polygon can be expressed as recall sum of interior angles of a polygon which is equal to. But this is also the same as counting all the edges of the little shapes.

There are s number of sides per face times F number of faces. Likewise, when we cut it up, what was one corner will now be several corners. And just to keep you well educated Hide Ads About Ads. Platonic Solids - Why Five? A Platonic Solid is a 3D shape where: each face is the same regular polygon the same number of polygons meet at each vertex corner There are only five of them Simplest Reason: Angles at a Vertex The simplest reason there are only 5 Platonic Solids is this: At each vertex at least 3 faces meet maybe more.

The faces can be triangles 3 sides , squares 4 sides , etc. Let us call this " s ", the number of s ides each face has.