Polyhedron - Basis For Definition

Basis For Definition

Defining a polyhedron as a solid bounded by flat faces and straight edges is not very precise and, to a modern mathematician, quite unsatisfactory. Grünbaum (1994, p. 43) observed, "The Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others ... at each stage ... the writers failed to define what are the 'polyhedra' ...." Since then rigorous definitions of "polyhedron" have been given within particular contexts. However such definitions are not always compatible in other contexts.

Any polyhedron can be built up from different kinds of element or entity, each associated with a different number of dimensions:

• 3 dimensions: The body is bounded by the faces, and is usually the volume enclosed by them.
• 2 dimensions: A face is a polygon bounded by a circuit of edges, and usually including the flat (plane) region inside the boundary. These polygonal faces together make up the polyhedral surface.
• 1 dimension: An edge joins one vertex to another and one face to another, and is usually a line segment. The edges together make up the polyhedral skeleton.
• 0 dimensions: A vertex (plural vertices) is a corner point.
• -1 dimension: The null polytope is a kind of non-entity required by abstract theories.

More generally in mathematics and other disciplines, "polyhedron" is used to refer to a variety of related constructs, some geometric and others purely algebraic or abstract.