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
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
is any of the polygons that make up its boundaries. For example, any of the square
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
s that bound a cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
is a face of the cube. The suffix -hedron is derived from the Greek word hedra which means face.
Sometimes, in the case of a pyramid
A pyramid is a structure whose outer surfaces are triangular and converge at a single point. The base of a pyramid can be trilateral, quadrilateral, or any polygon shape, meaning that a pyramid has at least three triangular surfaces...
, the term face is understood to exclude the base.
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 definitionIn 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,...
, a face of a polytope P is the intersection of any 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 S in Euclidean space \mathbb R^n 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 R4. 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 polytope:
- 4-face - the 4-dimensional 4-polytope 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.- 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 or honeycomb For example, a cubic honeycomb is made...
- 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 adjacent zero-dimensional vertices in a polygon. 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 that describes the corners or intersections of geometric shapes.-Of an angle:...
- the empty set.
FacetsIf the polytope lies in m-dimensions, a face in the (m-1)-dimension is called a facet
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 ridge
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.By dimension, this corresponds to:*a vertex of a polygon;...