- Face (geometry)
In

geometry , a**face**of apolyhedron is any of thepolygons that make up its boundaries. For example, any of the squares that bound acube is a face of the cube. The suffix "-hedron" is derived from the Greek word "hedra" which means "face".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, more specifically called an**n-face**.**Formal Definition**In

convex geometry , a**face**of a polytope "P" is the intersection of anysupporting 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**R**^{3}is entirely on one hyperplane of**R**^{4. 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.}All of the following are the

**n-faces**of a 4-dimensional polychoron:

***4-face**- the 4-dimensionalpolychoron itself

***3-face**- any 3-dimensional cell

***2-face**- any 2-dimensional polygonal**face**(using the common definition of face)

***1-face**- any 1-dimensional edge

***0-face**- any 0-dimensional vertex

* the empty set.**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.**External links***

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**Face (disambiguation)**— The face forms a part of the human body: the front of the head.Face may also refer to:In literature:* Face , a novel by Benjamin Zephaniah. * The Face (magazine), a British hip culture and lifestyle magazine (1980–2004) * The Face (novel) (2003) … Wikipedia**geometry**— /jee om i tree/, n. 1. the branch of mathematics that deals with the deduction of the properties, measurement, and relationships of points, lines, angles, and figures in space from their defining conditions by means of certain assumed properties… … Universalium**face**— /feɪs / (say fays) noun 1. the front part of the head, from the forehead to the chin inclusive. 2. a person, especially with regard to familiarity or some other quality: it s time for some fresh faces on council. 3. a look or expression on the… … Australian English dictionary**Face (géométrie)**— Pour les articles homonymes, voir Face. Un cube : les surface en rouge sont les faces du cube. Chaque sommet est entouré par trois faces. En géométrie, les f … Wikipédia en Français**face**— noun 1》 the front part of a person s head from the forehead to the chin, or the corresponding part in an animal. ↘an aspect: the unacceptable face of social drinking. 2》 an expression on someone s face. 3》 the surface of a thing, especially… … English new terms dictionary**Face configuration**— In geometry, a face configuration is notational description of a face transitive polyhedron. It represents a sequential count of the number of faces that exist at each vertex around a face.There is no single standard accepted representation, but… … Wikipedia**Face diagonal**— In geometry, a face diagonal of a polyhedron is a diagonal on one of the faces, in contrast to a space diagonal passing through the interior of the polyhedron. A cuboid has twelve face diagonals (and four space diagonals); the cuboid s face… … Wikipedia**Edge (geometry)**— For edge in graph theory, see Edge (graph theory) In geometry, an edge is a one dimensional line segment joining two zero dimensional vertices in a polytope. Thus applied, an edge is a connector for a one dimensional line segment and two zero… … Wikipedia**Cell (geometry)**— In geometry, a cell is a three dimensional element that is part of a higher dimensional object. In polytopes A cell is a three dimensional polyhedron element that is part of the boundary of a higher dimensional polytope, such as a polychoron (4… … Wikipedia**Coordination geometry**— The term coordination geometry is used in a number of related fields of chemistry and solid state chemistry/physics. Contents 1 Molecules 2 Inorganic coordination complexes 3 Crystallography usage 4 … Wikipedia