Face (geometry)

In the geometry of polyhedra and higher dimensional polytopes, the word face is used with two related but inconsistent meanings: a face may either refer to a two-dimensional element of a polyhedron, or an element of any dimension of a more general polytope (in any number of dimensions).

Polygonal face

In three-dimensional geometry a face usually means a two-dimensional polygon (2-face) on the boundary of a polyhedron. Other names for a polygonal face include side for a polyhedron, and tile in a Euclidean plane tessellation.

For example, any of the six squares that bound a cube is a face of the cube. Sometimes the same word is also used to refer to the 2-dimensional features of 4-polytopes. With this meaning, the 4-dimensional tesseract has 24 square faces, each sharing two of 8 cubic cells.

{4,3}	{4,4}	{4,5}	{4,3,3}
The cube has 3 square faces per vertex	The square tiling in the Euclidean plane has 4 square faces per vertex	The order-5 square tiling has 5 square faces per vertex in the hyperbolic plane as seen in this Poincaré disk model projection.	The tesseract has 3 square faces per edge

Other types of nonface polygons associated with polyhedra and tessellations include Petrie polygons and facets (flat polygons formed by the polyhedron edges and vertices that are not faces of the polyhedron).

General face

In higher-dimensional geometry the faces of a polytope are features of all dimensions. A face of dimension k is called a k-face. The set of faces of a polytope includes the polytope itself and the empty set.

For example, with this meaning, the faces of a cube include the empty set, its vertices (0-faces), edges (1-faces) and squares (2-faces), and the cube itself (3-face).

All of the following are the faces of a 4-dimensional polytope:

• 4-face – the 4-dimensional 4-polytope itself
• 3-faces – 3-dimensional cells (polyhedral faces)
• 2-faces – 2-dimensional faces (polygonal faces)
• 1-faces – 1-dimensional edges
• 0-faces – 0-dimensional vertices
• the empty set, which for consistency may be defined as having dimension −1

In some areas of mathematics, such as polyhedral combinatorics, a polytope is by definition convex. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P. From this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set.

In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, the requirement for convexity is relaxed. Abstract theory still requires that the set of faces include the polytope itself and the empty set.

Related terms

Facets, ridges, and peaks

In higher-dimensional geometry, the facets of a polytope are the faces of dimension one less than the polytope itself.

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.
• The facets of a 4-polytope or 3-honeycomb are its 3-faces.
• The facets of a 5-polytope or 4-honeycomb are its 4-faces.

In related terminology, a (n − 2)-face of an n-dimensional polytope is called a ridge. A (n − 3)-face of an n-dimensional polytope is called a peak.

Cells

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:

Regular 4-polytopes		Regular 3-honeycombs
{4,3,3}	{5,3,3}	{4,3,4}	{5,3,4}
The tesseract has 8 cubic cells in 4-dimensions, with 3 cells per edge, seen in this Schlegel diagram projection, with the 8th cell projected as the exterior.	The 120-cell has 120 dodecahedron cells in 4-dimensions, with 3 cells per edge.	The cubic honeycomb fills Euclidean 3-space with cubes, with 4 cells per edge.	The order-4 dodecahedral honeycomb fills 3-dimensional hyperbolic space with dodecahedra, 4 cells per edge, seen here in projection with the Beltrami-Klein model

4-faces

A 4-face is a 4-polytope element of a higher dimensional polytope or tessellation. The simplest 4-face is the 5-cell, made of 5 tetrahedral facets.

It is sometimes called a hypercell for being one dimension higher than a cell (3-face). Another proposed name used with uniform polytopes teron is shortened from tetron, constructed from the prefix tetra- meaning four, similar to the shortened metric prefix tera- defined as 1000⁴.

A 5-dimensional polytope or 4-dimensional tessellation can be considered constructed of 4-dimensional 4-face, 3-dimensional cells (3-faces), 2-dimenaional faces (2-faces), 1-dimensional edges (1-faces), and 0-dimensional vertices (0-faces).

For example, the 4-dimensional tesseractic honeycomb and the 5-dimensional 5-cube (5-hypercube) are both constructed from tesseractic 4-faces.

5-faces

A 5-face is a 5-polytope element of a higher dimensional polytope or tessellation. The simplest 5-face is the 5-simplex made of 6 5-cell facets.

A 6-dimensional polytope or 5-dimensional tessellation can be considered constructed of 5-dimensional 5-face, 4-dimensional 4-face, 3-dimensional cells (3-faces), 2-dimensional faces (2-faces), 1-dimensional edges (1-faces), and 0-dimensional vertices (0-faces).

6-faces

A 6-face is a 6-polytope element of a higher dimensional polytope or tessellation. The simplest 6-face is the 6-simplex made of 7 5-simplex facets.

A 7-dimensional polytope or 6-dimensional tessellation can be considered constructed of 6-dimensional 6-face, 5-dimensional 5-face, 4-dimensional 4-face, 3-dimensional cells (3-faces), 2-dimensional faces (2-faces), 1-dimensional edges (1-faces), and 0-dimensional vertices (0-faces).

See also

• Euler characteristic
• Face lattice

External links

• Olshevsky, George, Face at Glossary for Hyperspace.
• Olshevsky, George, Cell at Glossary for Hyperspace.
• Olshevsky, George, Ridge at Glossary for Hyperspace.
• Olshevsky, George, Peak at Glossary for Hyperspace.
• Weisstein, Eric W., "Face", MathWorld.
• Weisstein, Eric W., "Facet", MathWorld.
• Weisstein, Eric W., "Side", MathWorld.