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Concept
Face (geometry)
Summary
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).
In elementary geometry, a face is a polygon 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.
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 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 (polygonal 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, a polytope is by definition convex.
Official source
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