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Face (geometry)
In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object. For example, a cube has six faces in this sense.
In more modern treatments of the geometry of polyhedra and higher-dimensional polytopes, a "face" is defined in such a way that it may have any dimension. The vertices, edges, and (2-dimensional) faces of a polyhedron are all faces in this more general sense.
In elementary geometry, a face is a polygon on the boundary of a polyhedron. (Here a "polygon" should be viewed as including the 2-dimensional region inside it.) 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.
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 sometimes called a k-face. For example, the polygonal faces of an ordinary polyhedron are 2-faces. The word "face" is defined differently in different areas of mathematics. For example, many but not all authors allow the polytope itself and the empty set as faces of a polytope, where the empty set is for consistency given a "dimension" of −1. For any n-dimensional polytope, faces have dimension with .
For example, with this meaning, the faces of a cube comprise the cube itself (a 3-face), its (square) facets (2-faces), its (line segment) edges (1-faces), its (point) vertices (0-faces), and the empty set.
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Face (geometry) AI simulator
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Face (geometry)
In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object. For example, a cube has six faces in this sense.
In more modern treatments of the geometry of polyhedra and higher-dimensional polytopes, a "face" is defined in such a way that it may have any dimension. The vertices, edges, and (2-dimensional) faces of a polyhedron are all faces in this more general sense.
In elementary geometry, a face is a polygon on the boundary of a polyhedron. (Here a "polygon" should be viewed as including the 2-dimensional region inside it.) 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.
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 sometimes called a k-face. For example, the polygonal faces of an ordinary polyhedron are 2-faces. The word "face" is defined differently in different areas of mathematics. For example, many but not all authors allow the polytope itself and the empty set as faces of a polytope, where the empty set is for consistency given a "dimension" of −1. For any n-dimensional polytope, faces have dimension with .
For example, with this meaning, the faces of a cube comprise the cube itself (a 3-face), its (square) facets (2-faces), its (line segment) edges (1-faces), its (point) vertices (0-faces), and the empty set.