Face (geometry)

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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 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[edit]

In elementary geometry, a face is a polygon[note 1] 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. 		H2-5-4-primal.svg
The order-5 square tiling has 5 square faces per vertex. 		Hypercube.svg
The tesseract has 3 square faces per edge.

Number of polygonal faces of a polyhedron[edit]

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.

k-face[edit]

In higher-dimensional geometry, the faces of a polytope are features of all dimensions.[2][4][5] 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:

In some areas of mathematics, such as polyhedral combinatorics, a polytope is by definition convex. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P.[6] From this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set.[4][5]

In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, the requirement for convexity is relaxed. Abstract theory still requires that the set of faces include the polytope itself and the empty set.

Cell or 3-face[edit]

A cell is a polyhedral element (3-face) of a 4-dimensional polytope or 3-dimensional tessellation, or higher. Cells are facets for 4-polytopes and 3-honeycombs.

Examples:

Regular examples by Schläfli symbol
4-polytopes 3-honeycombs
{4,3,3} {5,3,3} {4,3,4} {5,3,4}
Hypercube.svg
The tesseract has 3 cubic cells (3-faces) per edge. 		Schlegel wireframe 120-cell.png
The 120-cell has 3 dodecahedral cells (3-faces) per edge. 		Partial cubic honeycomb.png
The cubic honeycomb fills Euclidean 3-space with cubes, with 4 cells (3-faces) per edge. 		Hyperbolic orthogonal dodecahedral honeycomb.png
The order-4 dodecahedral honeycomb fills 3-dimensional hyperbolic space with dodecahedra, 4 cells (3-faces) per edge.

Facet or (n-1)-face[edit]

In higher-dimensional geometry, the facets (also called hyperfaces)[7] of a n-polytope are the (n-1)-faces (faces of dimension one less than the polytope itself).[8] A polytope is bounded by its facets.

For example:

Ridge or (n-2)-face[edit]

In related terminology, the (n − 2)-faces of an n-polytope are called ridges (also subfacets).[9] A ridge is seen as the boundary between exactly two facets of a polytope or honeycomb.

For example:

Peak or (n-3)-face[edit]

The (n − 3)-faces of an n-polytope are called peaks. A peak contains a rotational axis of facets and ridges in a regular polytope or honeycomb.

For example:

See also[edit]

Notes[edit]

  1. ^ Some other polygons, which are not faces, are also important for polyhedra and tilings. These include Petrie polygons, vertex figures and facets (flat polygons formed by coplanar vertices that do not lie in the same face of the polyhedron).

References[edit]

  1. ^ Merriam-Webster's Collegiate Dictionary (Eleventh ed.). Springfield, MA: Merriam-Webster. 2004.
  2. ^ a b c Matoušek, Jiří (2002), Lectures in Discrete Geometry, Graduate Texts in Mathematics, 212, Springer, 5.3 Faces of a Convex Polytope, p. 86.
  3. ^ Cromwell, Peter R. (1999), Polyhedra, Cambridge University Press, p. 13.
  4. ^ a b Grünbaum, Branko (2003), Convex Polytopes, Graduate Texts in Mathematics, 221 (2nd ed.), Springer, p. 17.
  5. ^ a b Ziegler, Günter M. (1995), Lectures on Polytopes, Graduate Texts in Mathematics, 152, Springer, Definition 2.1, p. 51.
  6. ^ Matoušek (2002) and Ziegler (1995) use a slightly different but equivalent definition, which amounts to intersecting P with either a hyperplane disjoint from the interior of P or the whole space.
  7. ^ N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite symmetry groups, 11.1 Polytopes and Honeycombs, p.225
  8. ^ Matoušek (2002), p. 87; Grünbaum (2003), p. 27; Ziegler (1995), p. 17.
  9. ^ Matoušek (2002), p. 87; Ziegler (1995), p. 71.

External links[edit]

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