Face (geometry)

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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.

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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 [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

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

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 ≤ kn.

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

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-polytopes3-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

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

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

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

Notes

  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).

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Dual polyhedron

In geometry, any 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.

Polyhedron Three-dimensional shape with flat polygonal faces, straight edges and sharp corners

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. 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. Flat sides mean that the sides of a (k+1)-polytope consist of k-polytopes that may have (k−1)-polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope.

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.

4-polytope Four-dimensional geometric object with flat sides

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.

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.

Schläfli symbol Notation that defines regular polytopes and tessellations

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

Vertex figure

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

Convex polytope

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.

Rectification (geometry)

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.

Truncation (geometry)

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.

Alternation (geometry) Operation on a polyhedron or tiling that removes alternate vertices

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

Honeycomb (geometry) Tiling of 3-or-more dimensional euclidian or hyperbolic space

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.

Expansion (geometry)

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.

5-polytope

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.

Uniform polytope

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.

Edge (geometry) Line segment joining two adjacent vertices in a polygon or polytope

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 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.

References

  1. Merriam-Webster's Collegiate Dictionary (Eleventh ed.). Springfield, MA: Merriam-Webster. 2004.
  2. 1 2 3 Matoušek, Jiří (2002), Lectures in Discrete Geometry, Graduate Texts in Mathematics, 212, Springer, 5.3 Faces of a Convex Polytope, p. 86, ISBN   9780387953748 .
  3. Cromwell, Peter R. (1999), Polyhedra, Cambridge University Press, p. 13, ISBN   9780521664059 .
  4. 1 2 Grünbaum, Branko (2003), Convex Polytopes, Graduate Texts in Mathematics, 221 (2nd ed.), Springer, p.  17 .
  5. 1 2 Ziegler, Günter M. (1995), Lectures on Polytopes, Graduate Texts in Mathematics, 152, Springer, Definition 2.1, p. 51, ISBN   9780387943657 .
  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.