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 a Greek word meaning "face". Table of Geometry, from the 1728 Cyclopaedia. ... A polyhedron is a geometric shape which in mathematics is defined by three related meanings. ... A polygon (from the Greek poly, for many, and gonos, for angle) is a closed planar path composed of a finite number of sequential straight line segments. ... In plane geometry, a square is a polygon with four equal sides and four right angles. ... A cube [1] (or regular hexahedron) is a three-dimensional Platonic solid composed of six square faces, facets or sides, with three meeting at each vertex. ...

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. In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ...

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 R³ is entirely on one hyperplane of R⁴. If R⁴ 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. Convex Geometry is the branch of geometry studying convex bodies: compact, convex sets in Euclidean space. ...

All of the following are faces of a 4-dimensional polychoron: In geometry, a four-dimensional polytope is sometimes called a polychoron (plural: polychora) (from Greek poly meaning many and choros meaning room or space), 4-polytope, or polyhedroid. ...

• the 4-dimensional polychoron itself
• any 3-dimensional cell
• any 2-dimensional polygonal "face" (using the common definition of face)
• any 1-dimensional edge
• any 0-dimensional vertex
• the empty set

A cell is a three-dimensional object that is part of the boundary of a higher-dimensional object, such as a polychoron. ... In geometry, a vertex (Latin: whirl, whirlpool; plural vertices) is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet). ...

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. A facet of an n-dimensional simplex is its (n-1)-dimensional face. ...