Polyhedron

Polyhedron, in geometry, a solid bounded by flat surfaces with each surface bounded by straight sides. In other words, a polyhedron is a solid bounded by polygons. Each of the flat surfaces is called a face. A straight side bounding a face is called an edge. A point at the end of an edge is called a vertex. Figure 1, a pyramid with a square base and four triangular sides, is an example of a polyhedron.

In a regular polyhedron all of the faces are regular polygons that are congruent (equal in size and shape). The only regular polyhedra are the five shown in figure 2. They are the tetrahedron, which has four triangular faces; the cube, which has six square faces; the octahedron, which has eight triangular faces; the dodecahedron, whose 12 faces are all regular pentagons; and the icosahedron, which has 20 triangular faces. These are sometimes referred to as the Platonic solids because they appear in the writing of the Greek philosopher Plato, representing fire, air, earth, water, and the universe as a whole.

A convex polyhedron is one in which a line segment connecting any two vertices of the polyhedron contains only points that are on a face or inside the polyhedron. For convex polyhedrons, the relationship between the number of vertices v, faces f and edges e is given by v + f - e = 2. For example, the cube has 8 vertices, 6 faces, and 12 edges, which gives 8 + 6 - 12 = 2. The value of v + f - e for a general polyhedron is called the Euler characteristic of the surface of the polyhedron, named after the Swiss mathematician Leonhard Euler. It can be calculated for general polyhedra using the methods of algebraic topology, a branch of mathematics.