Face (geometry)

k-face

In higher-dimensional geometry, the faces of a polytope are features of all dimensions. A face of dimension k is sometimes called a k-face. For example, the polygonal faces of an ordinary polyhedron are 2-faces. The word "face" is defined differently in different areas of mathematics. For example, many but not all authors allow the polytope itself and the empty set as faces of a polytope, where the empty set is for consistency given a "dimension" of −1. For any n-dimensional polytope, faces have dimension ${\displaystyle k}$ with ${\displaystyle -1\le k\le n}$.

For example, with this meaning, the faces of a cube comprise the cube itself (a 3-face), its (square) facets (2-faces), its (line segment) edges (1-faces), its (point) vertices (0-faces), and the empty set.

In some areas of mathematics, such as polyhedral combinatorics, a polytope is by definition convex. In this setting, there is a precise definition: a face of a polytope P in Euclidean space ${\displaystyle {\mathbf{R}}^n}$ is the intersection of P with any closed halfspace whose boundary is disjoint from the relative interior of P. According to this definition, the set of faces of a polytope includes the polytope itself and the empty set. For convex polytopes, this definition is equivalent to the general definition of a face of a convex set, given below.

In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, the requirement of convexity is relaxed. One precise combinatorial concept that generalizes some earlier types of polyhedra is the notion of a simplicial complex. More generally, there is the notion of a polytopal complex.

An n-dimensional simplex (line segment (n = 1), triangle (n = 2), tetrahedron (n = 3), etc.), defined by n + 1 vertices, has a face for each subset of the vertices, from the empty set up through the set of all vertices. In particular, there are 2^{n + 1} faces in total. The number of k-faces, for k ∈ {−1, 0, ..., n}, is the binomial coefficient ${\displaystyle (\genfrac{}{}{0ex}{}{n+1}{k+1})}$.

There are specific names for k-faces depending on the value of k and, in some cases, how close k is to the dimension n of the polytope.

Vertex or 0-face

Vertex is the common name for a 0-face.

Edge or 1-face

Edge is the common name for a 1-face.

Face or 2-face

The use of face in a context where a specific k is meant for a k-face but is not explicitly specified is commonly a 2-face.

Cell or 3-face

A cell is a polyhedral element (3-face) of a 4-dimensional polytope or 3-dimensional tessellation, or higher. Cells are facets for 4-polytopes and 3-honeycombs.

Examples:

Facet or (n − 1)-face

In higher-dimensional geometry, the facets of a n-polytope are the (n − 1)-faces (faces of dimension one less than the polytope itself). A polytope is bounded by its facets.

For example:

• The facets of a line segment are its 0-faces or vertices.
• The facets of a polygon are its 1-faces or edges.
• The facets of a polyhedron or plane tiling are its 2-faces. However, in some contexts, a facet of a polyhedron means any polygon formed from a subset of three or more vertices of a 2-face.
• The facets of a 4D polytope or 3-honeycomb are its 3-faces or cells.
• The facets of a 5D polytope or 4-honeycomb are its 4-faces.

Ridge or (n − 2)-face

In related terminology, the (n − 2)-faces of an n-polytope are called ridges (also subfacets). A ridge is seen as the boundary between exactly two facets of a polytope or honeycomb.

For example:

• The ridges of a 2D polygon or 1D tiling are its 0-faces or vertices.
• The ridges of a 3D polyhedron or plane tiling are its 1-faces or edges.
• The ridges of a 4D polytope or 3-honeycomb are its 2-faces.
• The ridges of a 5D polytope or 4-honeycomb are its 3-faces or cells.

Peak or (n − 3)-face

The (n − 3)-faces of an n-polytope are called peaks. A peak contains a rotational axis of facets and ridges in a regular polytope or honeycomb.

For example:

• The peaks of a 3D polyhedron or plane tiling are its 0-faces or vertices.
• The peaks of a 4D polytope or 3-honeycomb are its 1-faces or edges.
• The peaks of a 5D polytope or 4-honeycomb are its 2-faces.

Face of a convex set

The notion of a face can be generalized from convex polytopes to all convex sets, as follows. Let ${\displaystyle C}$ be a convex set in a real vector space ${\displaystyle V}$. A face or extreme set of ${\displaystyle C}$ is a convex subset ${\displaystyle F\subseteq C}$ such that whenever a point ${\displaystyle p\in F}$ lies strictly between two points ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle C}$, both ${\displaystyle x}$ and ${\displaystyle y}$ must be in ${\displaystyle F}$. Equivalently, for any ${\displaystyle x,y\in C}$ and any real number ${\displaystyle 0<\theta <1}$ such that ${\displaystyle \theta x+(1-\theta )y}$ is in ${\displaystyle F}$, ${\displaystyle x}$ and ${\displaystyle y}$ must be in ${\displaystyle F}$.