Polyhedra - Department of Mathematics

2. Polyhedra
2.1. What is a Polyhedron?
A polyhedron is a surface in the three dimensional space consisting of polygons,
with each edge of the polyhedron shared by exactly two polygons [1].
2.2. Convex Polyhedra
In a convex polyhedron, the line segment joining any two vertices of the
polyhedron lies entirely in the interior of the polyhedron. A convex polyhedron has no
holes or indentations.
2.2.1. The Platonic Polyhedra
Interior
of cube
Regular faces (square)
Figure 2.1. Cube
Let P be a polyhedron whose faces are congruent regular polygons. The following
statements are equivalent [2]:
• All the vertices of P lie on a sphere.
• All the dihedral angles of P are equal.
• All the vertex figures are regular polygons.
• All the solid angles are congruent.
• All the vertices have the same valency.
The above statements are the characteristics of the Platonic polyhedra. There are
only five Platonic polyhedra: the tetrahedron, the cube, the octahedron, the
dodecahedron and the icosahedron.
Naming Polyhedra
Polyhedra are named according to the Greek names for their number of faces.
Number In Greek Name of polyhedra
4 Tetra- Tetrahedra
6 Hexa- Cube (Hexahedra)
8 Octa- Octahedra
10 Dodeca- Dodecahedra
12 Icosa- Icosahedra
Table 2.1. Naming of the five Platonic polyhedra
3