In solid geometry, a face is a flat surface (a planar

region In geography, regions, otherwise referred to as zones, lands or territories, are areas that are broadly divided by physical characteristics ( physical geography), human impact characteristics ( human geography), and the interaction of humanity an ...

) that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a ''

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all o ...

''. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, the term is also used to mean an element of any dimension of a more general polytope (in any number of dimensions)..

Polygonal face

In elementary geometry, a face is a

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...

on the boundary of a

. Other names for a polygonal face include polyhedron side and Euclidean plane ''

tile Tiles are usually thin, square or rectangular coverings manufactured from hard-wearing material such as ceramic, stone, metal, baked clay, or even glass. They are generally fixed in place in an array to cover roofs, floors, walls, edges, or ...

''. For example, any of the six squares that bound a

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...

is a face of the cube. Sometimes "face" is also used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, each sharing two of 8 cubic cells.

Number of polygonal faces of a polyhedron

Any convex polyhedron's surface has

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...

V - E + F = 2,

where ''V'' is the number of vertices, ''E'' is the number of edges, and ''F'' is the number of faces. This equation is known as Euler's polyhedron formula. Thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a

has 12 edges and 8 vertices, and hence 6 faces.

''k''-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. For example, the polygonal faces of an ordinary polyhedron are 2-faces. In

set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...

, the set of faces of a polytope includes the polytope itself and the empty set, where the empty set is for consistency given a "dimension" of −1. For any ''n''-polytope (''n''-dimensional polytope), −1 ≤ ''k'' ≤ ''n''. For example, with this meaning, the faces of a

comprise the cube itself (3-face), its (square) facets (2-faces), (linear) edges (1-faces), (point) vertices (0-faces), and the empty set. 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 ridges (

al faces) *1-faces – 1-dimensional edges *0-faces – 0-dimensional vertices *the empty set, which has dimension −1 In some areas of mathematics, such as

polyhedral combinatorics Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral co ...

, a polytope is by definition convex. Formally, a face of a polytope ''P'' is the intersection of ''P'' with any

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

halfspace Half-space may refer to: * Half-space (geometry), either of the two parts into which a plane divides Euclidean space * Half-space (punctuation), a spacing character half the width of a regular space * (Poincaré) Half-space model, a model of 3-dime ...

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.

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 (also called hyperfaces) 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 In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...

are its 0-faces or vertices. *The facets of a

are its 1-faces or edges. *The facets of a

or plane tiling are its 2-faces. *The facets of a 4D polytope or 3-honeycomb are its

3-face In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a ''polyhedron''. In more technical treatments of the geometry of polyhedra ...

s or cells. *The facets of a 5D polytope or 4-honeycomb are its

4-face In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a ''polyhedron''. In more technical treatments of the geometry of polyhedra ...

Ridge or (''n'' − 2)-face

In related terminology, the (''n'' − 2)-''face''s of an ''n''-polytope are called ridges (also subfacets)., p. 87; , p. 71. A ridge is seen as the boundary between exactly two facets of a polytope or honeycomb. For example: *The ridges of a 2D

or 1D tiling are its 0-faces or vertices. *The ridges of a 3D

or plane tiling are its 1-faces or edges. *The ridges of a 4D polytope or 3-honeycomb are its 2-faces or simply faces. *The ridges of a 5D polytope or 4-honeycomb are its 3-faces or

cells Cell most often refers to: * Cell (biology), the functional basic unit of life Cell may also refer to: Locations * Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...

Peak or (''n'' − 3)-face

The (''n'' − 3)-''face''s 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

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 or simply faces.

Notes

References

External links

* * * {{mathworld , urlname=Side , title=Side Elementary geometry Convex geometry Polyhedra Planar surfaces de:Fläche (Mathematik)