Polyhedron

A polyhedron is a three-dimensional closed surface or solid, bounded by plane figures called polygons.

The word polyhedron comes from the Greek prefix poly- , which means "many," and the root word hedron which refers to "surface." A polyhedron is a solid whose boundaries consist of planes. Many common objects in the world around us are in the shape of polyhedrons. The cube is seen in everything from dice to clock-radios; CD cases, and sticks of butter, are in the shape of polyhedrons called parallelpipeds. The pyramids are a type of polyhedron, as are geodesic domes. Most shapes formed in nature are irregular. In an interesting exception, however, crystals grow in mathematically perfect, and frequently complex, polyhedrons.

The bounding polygons of a polyhedron are called the faces. The line segments along which the faces meet are called the edges. The points at which the ends of edges intersect (think of the corner of a cereal box) are the vertices. Vertices are connected through the body of the polyhedron by an imaginary line called a diagonal.

A polyhedron is classified as convex if a diagonal contains only points inside of the polyhedron. Convex polyhedrons are also known as Euler polyhedrons, and can be defined by the equation E = v + f- e = 2, where v is the number of vertices, f is the number of faces, and e is the number of edges. The intersection of a plane and a polyhedron is called the cross section of the polyhedron. The cross-sections of a convex polyhedron are all convex polygons.

Polyhedrons are classified and named according to the number and type of faces. A polyhedron with four sides is a tetrahedron, but is also called a pyramid. The six-sided cube is also called a hexahedron. A polyhedron with six rectangles as sides also has many names—a rectangular parallelepided, rectangular prism, or box.

A polyhedron whose faces are all regular polygons congruent to each other, whose polyhedral angels are all equal, and which has the same number of faces meet at each vertex is called a regular polyhedron. Only five regular polyhedrons exist: the tetrahedron (four triangular faces), the cube (six square faces), the octahedron (eight triangular faces—think of two pyramids placed bottom to bottom), the dodecahedron (12 pentagonal faces), and the icosahedron (20 triangular faces).

Other common polyhedrons are best described as the same as one of previously named that has part of it cut off, or truncated, by a plane. Imagine cutting off the corners of a cube to obtain a polyhedron formed of triangles and squares, for example.