Английская Википедия:Face (geometry)

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Шаблон:Short description 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 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 polygonШаблон:Efn on the boundary of a polyhedron. Other names for a polygonal face include polyhedron side and Euclidean plane tile.

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

Regular examples by Schläfli symbol

Polyhedron	Star polyhedron	Euclidean tiling	Hyperbolic tiling	4-polytope
{4,3}	{5/2,5}	{4,4}	{4,5}	{4,3,3}
Файл:Hexahedron.png The cube has 3 square faces per vertex.	Файл:Small stellated dodecahedron.png The small stellated dodecahedron has 5 pentagrammic faces per vertex.	Файл:Tile 4,4.svg The square tiling in the Euclidean plane has 4 square faces per vertex.	Файл:H2-5-4-primal.svg The order-5 square tiling has 5 square faces per vertex.	Файл:Hypercube.svg The tesseract has 3 square faces per edge.

Number of polygonal faces of a polyhedron

Any convex polyhedron's surface has Euler characteristic

<math>V - E + F = 2,</math>

where Шаблон:Mvar is the number of vertices, Шаблон:Mvar is the number of edges, and Шаблон:Mvar 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 cube 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 Шаблон:Mvar is called a Шаблон:Mvar-face. For example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, 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 Шаблон:Mvar-polytope (Шаблон:Mvar-dimensional polytope), Шаблон:Math.

For example, with this meaning, the faces of a cube comprise the cube itself (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. Formally, a face of a polytope Шаблон:Mvar is the intersection of Шаблон:Mvar with any closed halfspace whose boundary is disjoint from the interior of Шаблон:Mvar. 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.

An Шаблон:Mvar-dimensional simplex (line segment (Шаблон:Math), triangle (Шаблон:Math), tetrahedron (Шаблон:Math), etc.), defined by Шаблон:Math 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 Шаблон:Math faces in total. The number of them that are Шаблон:Mvar-faces, for Шаблон:Math, is the binomial coefficient <math>\binom{n+1}{k+1}</math>.

There are specific names for Шаблон:Mvar-faces depending on the value of Шаблон:Mvar and, in some cases, how close Шаблон:Mvar is to the dimensionality Шаблон:Mvar of the polytope.

Vertex or 0-face Шаблон:Anchor

Vertex is the common name for a 0-face.

Edge or 1-face Шаблон:Anchor

Edge is the common name for a 1-face.

Face or 2-face Шаблон:Anchor

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

Cell or 3-face Шаблон:Anchor

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 examples by Schläfli symbol

4-polytopes		3-honeycombs
{4,3,3}	{5,3,3}	{4,3,4}	{5,3,4}
Файл:Hypercube.svg The tesseract has 3 cubic cells (3-faces) per edge.	Файл:Schlegel wireframe 120-cell.png The 120-cell has 3 dodecahedral cells (3-faces) per edge.	Файл:Partial cubic honeycomb.png The cubic honeycomb fills Euclidean 3-space with cubes, with 4 cells (3-faces) per edge.	Файл:Hyperbolic orthogonal dodecahedral honeycomb.png The order-4 dodecahedral honeycomb fills 3-dimensional hyperbolic space with dodecahedra, 4 cells (3-faces) per edge.

Facet or (n − 1)-face Шаблон:Anchor

In higher-dimensional geometry, the facets (also called hyperfaces) of a Шаблон:Mvar-polytope are the (Шаблон:Math)-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.
• 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 Шаблон:Anchor

In related terminology, the (Шаблон:Math)-faces of an Шаблон:Mvar-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 or simply faces.
• The ridges of a 5D polytope or 4-honeycomb are its 3-faces or cells.

Peak or (n − 3)-face Шаблон:Anchor

The (Шаблон:Math)-faces of an Шаблон:Mvar-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 or simply faces.

See also

• Face lattice

External links

• Шаблон:Mathworld
• Шаблон:Mathworld
• Шаблон:Mathworld