In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, polyhedron in three dimensions, and polychoron in four dimensions. Beyond that, the term is used for a variety of related mathematical concepts. This is analogous to the way the term square may be used to refer to a square-shaped region of the plane, or just to its boundary, or even to a mere list of its vertices and edges along with some information about the way they are connected. The term was coined by Alicia Boole, the daughter of logician George Boole. For other uses, see Geometry (disambiguation). ... Look up polygon in Wiktionary, the free dictionary. ... For the game magazine, see Polyhedron (magazine). ... In geometry, a four-dimensional polytope is sometimes called a polychoron (plural: polychora) (from Greek poly meaning many and choros meaning room or space), 4-polytope, or polyhedroid. ... Alicia Boole Stott (June 8, 1860 - December 17, 1940) was the third daughter of George Boole. ... Not to be confused with George Boolos. ...

The Platonic solids, or regular polytopes in three dimensions, were a major focus of study of ancient Greek mathematicians (most notably Euclid's Elements), probably because of their intrinsic aesthetic qualities. In modern times, polytopes and related concepts have found important applications in computer graphics, optimization, search engines and numerous other fields. In geometry, a Platonic solid is a convex regular polyhedron. ... For other uses, see Euclid (disambiguation). ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems... This article is about the scientific discipline of computer graphics. ... In mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integer variables from within an allowed set. ... A search engine is an information retrieval system designed to help find information stored on a computer system. ...

Convex polytopes

One special kind of polytope is a convex polytope, which is the convex hull of a finite set of points. A convex polytope can also be represented as the intersection of half-spaces. This intersection can be written as the matrix inequality: Convex hull: elastic band analogy In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X. // For planar objects, i. ... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ... In geometry, a half-space is any of the two parts into which a hyperplane divides an affine space. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ...

where A is an m by n matrix, m being the number of bounding half-spaces and n being the number of dimensions of the affine space Rⁿ in which the polytope is contained; and b is an m by 1 column vector. The coefficients of each row of A and b correspond with the coefficients of the linear inequality defining the respective half-space (see hyperplane for an explanation). Hence, each row in the matrix corresponds with a supporting hyperplane of the polytope, which is any hyperplane one of whose closed half-spaces contains the polytope; and some of the rows correspond with a bounding hyperplane of the polytope, which is a hyperplane that is spanned by its intersection with the polytope. This definition assumes that the polytope is n-dimensional; if it is not, then the solution of Ax ≤ b lies in a proper affine subspace of Rⁿ and the preceding discussion should be applied to that subspace. (Note that the intersection of arbitrary half-spaces need not be bounded; it is a convex polytope if and only if it is bounded.) In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. ... A hyperplane is a concept in geometry. ...

An n-dimensional convex polytope is bounded by a number of (n−1)-dimensional facets. These facets are themselves polytopes, whose facets are (n−2)-dimensional ridges of the original polytope. Every ridge arises as the intersection of two facets (but the intersection of two facets need not be a ridge). Ridges are once again polytopes whose facets give rise to (n−3)-dimensional boundaries of the original polytope, and so on. These bounding sub-polytopes may be referred to as faces, or specifically k-dimensional faces or k-faces (although the term "face" may also refer specifically to a 2-dimensional face). A 0-dimensional face is called a vertex; and a 1-dimensional face is called an edge. A 3-dimensional face is sometimes called a cell. A facet of an n-dimensional simplex is its (n-1)-dimensional face. ... In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. ... A cell is a three-dimensional object that is part of a higher-dimensional object, such as a polychoron. ...

Assume in the matrix definition of a convex polytope P, Ax ≤ b, that the matrix A has the smallest possible number of rows of any matrix that defines P. Then there is one row for each facet, and the facet consists of the points on the polytope that satisfiy equality in the one row of the matrix that corresponds to that facet (it does not matter whether they also satisfy equality in other rows). Similarly, a ridge satisfies equality in two rows the matrix representation. In general, an (n−j)-dimensional face satisfies equality in j specific rows of A. These rows form a basis of the face. (They are not arbitrary; the set must be j-dimensional so the rows must be linearly independent in the augmented matrix [A | b]. The choice of the j rows may not be unique.) Geometrically speaking, this means that the face is the set of points on the polytope that lie in the intersection of j of the polytope's bounding hyperplanes. In geometry, a vertex (plural vertices) is a special kind of point, usually a corner of a polygon, polyhedron, or higher dimensional polytope. ... An edge between two vertices For edge in graph theory, see Edge (graph theory) In geometry, an edge is a one-dimensional line segment joining two zero-dimensional vertices in a polytope. ... In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. ... Cubic honeycomb - four cubic cells per edge hypercube - three cubic cells per edge In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object. ... In geometry, a peak is an (n-3)-dimensional element of a polytope. ... A facet of an n-dimensional simplex is its (n-1)-dimensional face. ...

The face lattice of a square pyramid, drawn as a Hasse diagram; each face in the lattice is labeled by its vertex set.

The faces of a convex polytope thus form a lattice called its face lattice, where the partial ordering is by set containment of faces. (To ensure that every pair of faces has a join and a meet in the face lattice, the polytope itself and the empty set are considered 'faces'. The whole polytope is the unique maximum element of the lattice. The empty set, considered to be a −1-dimensional face of every polytope, is the unique minimum element of the lattice.) In geometry, the square pyramid, a pyramid with a square base and equilateral sides, is one of the Johnson solids (J1). ... In the mathematical discipline known as order theory, a Hasse diagram (pronounced HAHS uh, named after Helmut Hasse (1898–1979)) is a simple picture of a finite partially ordered set, forming a drawing of the transitive reduction of the partial order. ... The name lattice is suggested by the form of the Hasse diagram depicting it. ...

This terminology is not fully standardized. The term face is sometimes used to refer only to 2-dimensional faces, and other times used in place of facet. The word edge is sometimes used to refer to a ridge.

Simplicial decomposition

Now given any convex hull in r-dimensional space (but not in any (r-1)-plane, say) we can take linearly independent subsets of the vertices, and define r-simplices with them. In fact, you can choose several simplices in this way such that their union as sets is the original hull, and the intersection of any two is either empty or an s-simplex (for some s < r). A 3-simplex or tetrahedron In geometry, a simplex (plural simplexes or simplices) or n-simplex is an n-dimensional analogue of a triangle. ...

For example, in the plane a square (convex hull of its corners) is the union of the two triangles (2-simplices), defined by a diagonal 1-simplex which is their intersection.

In general, the definition (attributed to Alexandrov) is that an r-polytope is defined as a set with an r-simplicial decomposition with some additional properties. If a set has an r-simplicial decomposition this means it is a union of s-simplices for values of s with s at most r, that is closed under intersection, and such that the only time that one of simplices is contained in another is as a face. (For a more abstract treatment, see simplicial complex). Pavel Sergeevich Alexandrov (Па́вел Серге́евич Алекса́ндров, sometimes romanized Alexandroff or Aleksandrov) (born May 7, 1896 - died November 16, 1982) was a Russian mathematician. ... A 3-simplex or tetrahedron In geometry, a simplex (plural simplexes or simplices) or n-simplex is an n-dimensional analogue of a triangle. ... In mathematics, a simplicial complex is a topological space of a particular kind, built up of points, line segments, triangles, and their n-dimensional counterparts. ...

What does this let us build? Let's start with the 1-simplex, or line segment. Then we have the line segment, of course, and anything that you can get by joining line segments end-to-end:

If two segments meet at each vertex (so not the case for the final one), we get a topological curve, called a polygonal curve. One may categorize these as open or closed, depending on whether the ends match up, and as simple or complex, depending on whether they intersect themselves. Closed polygonal curves are called polygons. Image File history File links r-polytopes built with a set of line segments (1-simplices) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Look up polygon in Wiktionary, the free dictionary. ...

Simple polygons in the plane are Jordan curves: they have an interior that is a topological disk. So does a 2-polytope (as you can see in the third example above), and these are often treated interchangeably with their boundary, the word polygon referring to either. In topology, the Jordan curve theorem states that every non-self-intersecting loop in the plane divides the plane into an inside and an outside. It was proved by Oswald Veblen in 1905. ... Look up polygon in Wiktionary, the free dictionary. ...

Now the process can be repeated. Joining polygons along edges (1-faces) gives a polyhedral surface, called a skew polygon when open and a polyhedron when closed. Simple polyhedra are interchangeable with their interiors, which are 3-polytopes that can be used to build 4-dimensional forms (sometimes called polychora), and so on to higher polytopes. For the game magazine, see Polyhedron (magazine). ... In geometry, a four-dimensional polytope is sometimes called a polychoron (plural: polychora) (from Greek poly meaning many and choros meaning room or space), 4-polytope, or polyhedroid. ...

Other definitions (equivalent and otherwise) are possible and occur in the mathematical literature. Polytopes may be regarded as a tessellation of some sort of the manifold of their surface. A tessellated plane seen in street pavement. ... On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ...

The theory of abstract polytopes attempts to detach polytopes from the space containing them, considering their purely combinatorial properties. This allows the definition of the term to be extended to include objects for which it is difficult to define clearly a natural underlying space. The hemicube is constructed from the cube by treating opposite edges (likewise faces and corners) as really the same edge. ...

Uses

In the study of optimization, linear programming studies the maxima and minima of linear functions constricted to the boundary of an n-dimensional polytope. In mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integer variables from within an allowed set. ... In mathematics, linear programming (LP) problems involve the optimization of a linear objective function, subject to linear equality and inequality constraints. ... Local and global maxima and minima for cos(3Ï€x)/x, 0. ... For other uses, see Linear (disambiguation). ... In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of...

References

• Coxeter, Harold Scott MacDonald (1973), Regular Polytopes, New York: Dover Publications, ISBN 978-0-486-61480-9 .
• Grünbaum, Branko (2003), Kaibel, Volker; Klee, Victor & Ziegler, Günter M., eds., Convex polytopes (2nd ed.), New York & London: Springer-Verlag, ISBN 0-387-00424-6 .
• Ziegler, Günter M. (1995), Lectures on Polytopes, vol. 152, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag .

H.S.M. Coxeter. ... Stereographic projection of the 120-cell, a 4-dimensional regular polytope. ... Dover Publications is a book publisher founded in 1941. ... Branko Grünbaum is a mathematician who works mainly in geometry and is considered a founder of discrete geometry. ... Victor L. Klee, Jr. ... The Springer-Verlag (pronounced SHPRING er FAIR lahk) was a worldwide publishing company base in Germany. ... The Springer-Verlag (pronounced SHPRING er FAIR lahk) was a worldwide publishing company base in Germany. ...

See also

This page lists the regular polytopes in Euclidean space. ... A dodecahedron, one of the five Platonic solids. ... The hemicube is constructed from the cube by treating opposite edges (likewise faces and corners) as really the same edge. ... A bounding box for a three dimensional model For code compliance, see Bounding. ... A 3-simplex or tetrahedron In geometry, a simplex (plural simplexes or simplices) or n-simplex is an n-dimensional analogue of a triangle. ... A square A projection of a cube (into a two-dimensional image) A projection of a hypercube (into a two-dimensional image) In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). ... In geometry, a cross-polytope, or orthoplex, is a regular, convex polytope that exists in any number of dimensions. ... In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. ... Look up polygon in Wiktionary, the free dictionary. ... For the game magazine, see Polyhedron (magazine). ... In geometry, a four-dimensional polytope is sometimes called a polychoron (plural: polychora) (from Greek poly meaning many and choros meaning room or space), 4-polytope, or polyhedroid. ... In geometry, a five-dimensional polytope, or 5-polytope, is a polytope in 5-dimensional space. ... In geometry, a six-dimensional polytope, or 6-polytope, is a polytope in 6-dimensional space. ... In geometry, a seven-dimensional polytope, or 7-polytope, is a polytope in 7-dimensional space. ... In geometry, an eight-dimensional polytope, or 8-polytope, is a polytope in 8-dimensional space. ... In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, polyhedron in three dimensions, and polychoron in four dimensions. ... In recreational mathematics, a polyform is a plane figure constructed by joining together identical basic polygons. ... In mathematics, the Schläfli symbol is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations. ... In geometry, a honeycomb is a name for a space-filling tessellation, just as a tiling is a tessellation of a plane or 2-dimensional surface. ...

External links

• Eric W. Weisstein, Polytope at MathWorld.
• "Math will rock your world" - application of polytopes to a database of articles used to support custom news feeds via the Internet - (Business Week Online)
• Regular and semi-regular convex polytopes a short historical overview: