FACE (GEOMETRY)

In geometry, a 'face' of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube. The suffix ''-hedron'' is derived from the Greek word ''hedra'' which means ''face''.
The (two-dimensional) polygons that bound higher-dimensional polytopes are also commonly called ''faces''. Formally, however, a face is ''any'' of the lower dimensional boundaries of the polytope, more specifically called an 'n-face'.

Contents

Formal Definition

Facets

External links

Formal Definition

In convex geometry, a 'face' of a polytope ''P'' is the intersection of any supporting hyperplane of ''P'' and ''P''. From this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. For example, a polyhedron 'R'³ is entirely on one hyperplane of 'R'4. If 'R'⁴ were spacetime, the hyperplane at t=0 supports and contains the entire polyhedron. Thus, by the formal definition, the polyhedron is a face of itself.
All of the following are the 'n-faces' of a 4-dimensional polychoron:

★ '4-face' - the 4-dimensional polychoron itself

★ '3-face' - any 3-dimensional cell

★ '2-face' - any 2-dimensional polygonal 'face' (using the common definition of face)

★ '1-face' - any 1-dimensional edge

★ '0-face' - any 0-dimensional vertex

★ the empty set

Facets

If the polytope lies in ''n''-dimensions, a face in the ''(n-1)''-dimension is called a 'facet.' For example, a cell of a polychoron is a facet, a "face" of a polyhedron is a facet, an edge of a polygon is a facet, etc. A face in the ''(n-2)''-dimension is called a ridge.

External links

★