Face (geometry)

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Face (geometry)

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a face of a polyhedron

Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
is any of the polygons that make up its boundaries. For example, any of the square

Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
s that bound a cube

Cube

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Encyclopedia

In geometry

Geometry

Polyhedron

Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
s that bound a cube

Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
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

Convex geometry

Convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space.Convex sets occur naturally in many areas of mathematics: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming, probability theory, etc....
, a face of a polytope P is the intersection of any supporting hyperplane

Supporting hyperplane

Supporting hyperplane is a concept in geometry. A hyperplane divides a space into two half-spaces. A hyperplane is said to support a Set in Euclidean space if it meets both of the following:...
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 R³ 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-dimensional polychoronPolychoron

In geometry, a four-dimensional polytope is sometimes called a polychoron , from the Greek language root poly, meaning "many", and choros meaning "room" or "space"....
itself
3-face - any 3-dimensional cellCell (geometry)

In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object....
2-face - any 2-dimensional polygonal face (using the common definition of face)
1-face - any 1-dimensional edgeEdge (geometry)

In geometry, an edge is a one-dimensional line segment joining two zero-dimensional vertex in a polytope. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....
0-face - any 0-dimensional vertexVertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
the empty set.

Facets

If the polytope lies in n-dimensions, a face in the (n-1)-dimension is called a facetFacet (mathematics)

A facet of a simplicial complex is a maximal simplex.In the general theory of polyhedra and polytopes, two conflicting meanings are currently jostling for acceptability:...
. 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 ridgeRidge (geometry)

In geometry, a ridge is an -dimensional element of an n-dimensional polytope. It is also sometimes called a subfacet for having one lower dimension than a Facet ....
.

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