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Face (geometry)

In solid geometry, a face is a flat (planar) surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In solid geometry, a face is a flat (planar) surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by flat 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). In elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include side of a polyhedron, and tile of a Euclidean plane tessellation. 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. Some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets (flat polygons formed by coplanar vertices which do not lie in the same face of the polyhedron). Any convex polyhedron's surface has Euler characteristic 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 cube has 12 edges and 8 vertices, and hence 6 faces. 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, 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 cube include the empty set, its vertices (0-faces), edges (1-faces) and squares (2-faces), and the cube itself (3-face).

[ "Polyhedron", "Vertex (geometry)", "Geometry", "Computer vision", "Artificial intelligence", "Conway polyhedron notation", "Flexible polyhedron", "Vertex arrangement", "Tetradecahedron", "Small rhombidodecahedron" ]
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