In solid geometry, a **face** is a flat surface (a planar region) that forms part of the boundary of a solid object;^{ [1] } a three-dimensional solid bounded exclusively by faces is a * polyhedron *.

- Polygonal face
- Number of polygonal faces of a polyhedron
- k-face
- Cell or 3-face
- Facet or (n − 1)-face
- Ridge or (n − 2)-face
- Peak or (n − 3)-face
- See also
- Notes
- References
- External links

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] }

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 **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.

Polyhedron | Star polyhedron | Euclidean tiling | Hyperbolic tiling | 4-polytope |
---|---|---|---|---|

{4,3} | {5/2,5} | {4,4} | {4,5} | {4,3,3} |

The cube has 3 square faces per vertex. | The small stellated dodecahedron has 5 pentagrammic faces per vertex. | The square tiling in the Euclidean plane has 4 square faces per vertex. | The order-5 square tiling has 5 square faces per vertex. | The tesseract has 3 square faces per edge. |

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.^{ [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:

- 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. 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.

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:

4-polytopes | 3-honeycombs | ||
---|---|---|---|

{4,3,3} | {5,3,3} | {4,3,4} | {5,3,4} |

The tesseract has 3 cubic cells (3-faces) per edge. | The 120-cell has 3 dodecahedral cells (3-faces) per edge. | The cubic honeycomb fills Euclidean 3-space with cubes, with 4 cells (3-faces) per edge. | The order-4 dodecahedral honeycomb fills 3-dimensional hyperbolic space with dodecahedra, 4 cells (3-faces) per edge. |

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:

- The facets of a line segment are its 0-faces or vertices.
- The facets of a polygon are its 1-faces or edges.
- The facets of a polyhedron or plane tiling are its 2-faces.
- The facets of a 4D polytope or 3-honeycomb are its 3-faces or cells.
- The facets of a 5D polytope or 4-honeycomb are its 4-faces.

In related terminology, the (*n* − 2)-*face*s 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:

- The ridges of a 2D polygon or 1D tiling are its 0-faces or vertices.
- The ridges of a 3D polyhedron or plane tiling are its 1-faces or edges.
- The ridges of a 4D polytope or 3-honeycomb are its 2-faces or simply
**faces**. - The ridges of a 5D polytope or 4-honeycomb are its 3-faces or cells.

The (*n* − 3)-*face*s 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:

- The peaks of a 3D polyhedron or plane tiling are its 0-faces or vertices.
- The peaks of a 4D polytope or 3-honeycomb are its 1-faces or edges.
- The peaks of a 5D polytope or 4-honeycomb are its 2-faces or simply
**faces**.

- ↑ 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).

In geometry, every polyhedron is associated with a second **dual** figure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron.

In geometry, a **polyhedron** is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as *poly-* + *-hedron*.

In elementary geometry, a **polytope** is a geometric object with flat sides (*faces*). It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions *n* as an *n*-dimensional polytope or ** n-polytope**. In this context, "flat sides" means that the sides of a (

A **polyhedral compound** is a figure that is composed of several polyhedra sharing a * common centre*. They are the three-dimensional analogs of polygonal compounds such as the hexagram.

In geometry, a **4-polytope** is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4-polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853.

A **regular polyhedron** is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.

In geometry, the **Schläfli symbol** is a notation of the form that defines regular polytopes and tessellations.

In geometry, a **vertex figure**, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.

A **convex polytope** is a special case of a polytope, having the additional property that it is also a convex set contained in the -dimensional Euclidean space . Most texts use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary.

In Euclidean geometry, **rectification**, also known as **critical truncation** or **complete-truncation** is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the rectified facets of the original polytope.

In geometry, a **truncation** is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids.

In geometry, an **alternation** or *partial truncation*, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.

In geometry, a **honeycomb** is a *space filling* or *close packing* of polyhedral or higher-dimensional *cells*, so that there are no gaps. It is an example of the more general mathematical *tiling* or *tessellation* in any number of dimensions. Its dimension can be clarified as *n*-honeycomb for a honeycomb of *n*-dimensional space.

In geometry, **expansion** is a polytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements. Equivalently this operation can be imagined by keeping facets in the same position but reducing their size.

In five-dimensional geometry, a **five-dimensional polytope** or **5-polytope** is a 5-dimensional polytope, bounded by (4-polytope) facets. Each polyhedral cell being shared by exactly two 4-polytope facets.

A **uniform polytope** of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.

In geometry, a **vertex**, often denoted by letters such as , , , , is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices.

In geometry, an **edge** is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a line segment on the boundary, and is often called a **polygon side**. In a polyhedron or more generally a polytope, an edge is a line segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal.

**Polyhedral combinatorics** is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes.

- ↑
*Merriam-Webster's Collegiate Dictionary*(Eleventh ed.). Springfield, MA: Merriam-Webster. 2004. - 1 2 3 Matoušek, Jiří (2002),
*Lectures in Discrete Geometry*, Graduate Texts in Mathematics, vol. 212, Springer, 5.3 Faces of a Convex Polytope, p. 86, ISBN 9780387953748 . - ↑ Cromwell, Peter R. (1999),
*Polyhedra*, Cambridge University Press, p. 13, ISBN 9780521664059 . - 1 2 Grünbaum, Branko (2003),
*Convex Polytopes*, Graduate Texts in Mathematics, vol. 221 (2nd ed.), Springer, p. 17 . - 1 2 Ziegler, Günter M. (1995),
*Lectures on Polytopes*, Graduate Texts in Mathematics, vol. 152, Springer, Definition 2.1, p. 51, ISBN 9780387943657 . - ↑ 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. - ↑ 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 - ↑ Matoušek (2002), p. 87; Grünbaum (2003), p. 27; Ziegler (1995), p. 17.
- ↑ Matoušek (2002), p. 87; Ziegler (1995), p. 71.

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