Face (geometry)

In solid geometry, a face is a flat (planar) surface that forms part of the boundary of a solid object;[1] 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).[2]

Polygonal face

In elementary geometry, a face is a polygon on the boundary of a polyhedron.[2][3] 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.

Regular examples by Schläfli symbol
Polyhedron Star polyhedron Euclidean tiling Hyperbolic tiling 4-polytope
{4,3} {5/2,5} {4,4} {4,5} {4,3,3}
Hexahedron.png
The cube has 3 square faces per vertex.
Small stellated dodecahedron.png
The small stellated dodecahedron has 5 pentagrammic faces per vertex.
Tile 4,4.svg
The square tiling in the Euclidean plane has 4 square faces per vertex.
H2 tiling 245-4.png
The order-5 square tiling has 5 square faces per vertex.
Hypercube.svg
The tesseract has 3 square faces per edge.

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).

Number of polygonal faces of a 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.

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