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| Apolyhedron is a 3-dimensional polytope | |||||
In elementarygeometry, apolytope is a geometric object withflat sides (faces). Polytopes are the generalization of three-dimensionalpolyhedra to any number of dimensions. Polytopes may exist in any general number of dimensionsn as ann-dimensional polytope orn-polytope. For example, a two-dimensionalpolygon is a 2-polytope and a three-dimensionalpolyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a(k + 1)-polytope consist ofk-polytopes that may have(k – 1)-polytopes in common.
Some theories further generalize the idea to include such objects as unboundedapeirotopes andtessellations, decompositions or tilings of curvedmanifolds includingspherical polyhedra, and set-theoreticabstract polytopes.
Polytopes of more than three dimensions were first discovered byLudwig Schläfli before 1853, who called such a figure apolyschem.[1] TheGerman termpolytop was coined by the mathematicianReinhold Hoppe, and was introduced to English mathematicians aspolytope byAlicia Boole Stott.
Nowadays, the termpolytope is a broad term that covers a wide class of objects, and various definitions appear in the mathematical literature. Many of these definitions are not equivalent to each other, resulting in different overlapping sets of objects being calledpolytopes. They represent different approaches to generalizing theconvex polytopes to include other objects with similar properties.
The original approach broadly followed byLudwig Schläfli,Thorold Gosset and others begins with the extension by analogy into four or more dimensions, of the idea of a polygon and polyhedron respectively in two and three dimensions.[2]
Attempts to generalise theEuler characteristic of polyhedra to higher-dimensional polytopes led to the development oftopology and the treatment of a decomposition orCW-complex as analogous to a polytope.[3] In this approach, a polytope may be regarded as atessellation or decomposition of some givenmanifold. An example of this approach defines a polytope as a set of points that admits asimplicial decomposition. In this definition, a polytope is the union of finitely manysimplices, with the additional property that, for any two simplices that have a nonempty intersection, their intersection is a vertex, edge, or higher dimensional face of the two.[4] However this definition does not allowstar polytopes with interior structures, and so is restricted to certain areas of mathematics.
The discovery ofstar polyhedra and other unusual constructions led to the idea of a polyhedron as a bounding surface, ignoring its interior.[5] In this light convex polytopes inp-space are equivalent totilings of the (p−1)-sphere, while others may be tilings of otherelliptic, flat ortoroidal (p−1)-surfaces – seeelliptic tiling andtoroidal polyhedron. Apolyhedron is understood as a surface whosefaces arepolygons, a4-polytope as a hypersurface whose facets (cells) are polyhedra, and so forth.
The idea of constructing a higher polytope from those of lower dimension is also sometimes extended downwards in dimension, with an (edge) seen as a1-polytope bounded by a point pair, and a point orvertex as a 0-polytope. This approach is used for example in the theory ofabstract polytopes.
In certain fields of mathematics, the terms "polytope" and "polyhedron" are used in a different sense: apolyhedron is the generic object in any dimension (referred to aspolytope in this article) andpolytope means abounded polyhedron.[6] This terminology is typically confined to polytopes and polyhedra that areconvex. With this terminology, a convex polyhedron is the intersection of a finite number ofhalfspaces and is defined by its sides while a convex polytope is theconvex hull of a finite number of points and is defined by its vertices.
Polytopes in lower numbers of dimensions have standard names:
| Dimension of polytope | Description[7] |
|---|---|
| −1 | Nullitope |
| 0 | Monon |
| 1 | Dion |
| 2 | Polygon |
| 3 | Polyhedron |
| 4 | Polychoron[7] |
A polytope comprises elements of different dimensionality such as vertices, edges, faces, cells and so on. Terminology for these is not fully consistent across different authors. For example, some authors useface to refer to an (n − 1)-dimensional element while others useface to denote a 2-face specifically. Authors may usej-face orj-facet to indicate an element ofj dimensions. Some useedge to refer to a ridge, whileH. S. M. Coxeter usescell to denote an (n − 1)-dimensional element.[8][citation needed]
The terms adopted in this article are given in the table below:
| Dimension of element | Term (in ann-polytope) |
|---|---|
| −1 | Nullity (necessary inabstract theory)[7] |
| 0 | Vertex |
| 1 | Edge |
| 2 | Face |
| 3 | Cell |
| j | j-face – element of rankj = −1, 0, 1, 2, 3, ...,n |
| n − 3 | Peak – (n − 3)-face |
| n − 2 | Ridge or subfacet – (n − 2)-face |
| n − 1 | Facet – (n − 1)-face |
| n | The polytope itself |
Ann-dimensional polytope is bounded by a number of (n − 1)-dimensionalfacets. These facets are themselves polytopes, whose facets are (n − 2)-dimensionalridges 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 asfaces, or specificallyj-dimensional faces orj-faces. A 0-dimensional face is called avertex, and consists of a single point. A 1-dimensional face is called anedge, and consists of a line segment. A 2-dimensional face consists of apolygon, and a 3-dimensional face, sometimes called acell, consists of apolyhedron.
A polytope may beconvex. The convex polytopes are the simplest kind of polytopes, and form the basis for several different generalizations of the concept of polytopes. A convex polytope is sometimes defined as the intersection of a set ofhalf-spaces. This definition allows a polytope to be neither bounded nor finite. Polytopes are defined in this way, e.g., inlinear programming. A polytope isbounded if there is a ball of finite radius that contains it. A polytope is said to bepointed if it contains at least one vertex. Every bounded nonempty polytope is pointed. An example of a non-pointed polytope is the set. A polytope isfinite if it is defined in terms of a finite number of objects, e.g., as an intersection of a finite number of half-planes.It is anintegral polytope if all of its vertices have integer coordinates.
A certain class of convex polytopes arereflexive polytopes. An integral-polytope is reflexive if for someintegral matrix,, where denotes a vector of all ones, and the inequality is component-wise. It follows from this definition that is reflexive if and only if for all. In other words, a-dilate of differs, in terms of integer lattice points, from a-dilate of only by lattice points gained on the boundary. Equivalently, is reflexive if and only if itsdual polytope is an integral polytope.[9]
Regular polytopes have the highest degree of symmetry of all polytopes. The symmetry group of a regular polytope acts transitively on itsflags; hence, thedual polytope of a regular polytope is also regular.
There are three main classes of regular polytope which occur in any number of dimensions:
Dimensions two, three and four include regular figures which have fivefold symmetries and some of which are non-convex stars, and in two dimensions there are infinitely manyregular polygons ofn-fold symmetry, both convex and (forn ≥ 5) star. But in higher dimensions there are no other regular polytopes.[2]
In three dimensions the convexPlatonic solids include the fivefold-symmetricdodecahedron andicosahedron, and there are also four starKepler-Poinsot polyhedra with fivefold symmetry, bringing the total to nine regular polyhedra.
In four dimensions theregular 4-polytopes include one additional convex solid with fourfold symmetry and two with fivefold symmetry. There are ten starSchläfli-Hess 4-polytopes, all with fivefold symmetry, giving in all sixteen regular 4-polytopes.
A non-convex polytope may be self-intersecting; this class of polytopes include thestar polytopes. Some regular polytopes are stars.[2]
Since a (filled) convex polytopeP in dimensions iscontractible to a point, theEuler characteristic of its boundary ∂P is given by the alternating sum:
This generalizesEuler's formula for polyhedra.[10]
TheGram–Euler theorem similarly generalizes the alternating sum ofinternal angles for convex polyhedra to higher-dimensional polytopes:[10]
Not all manifolds are finite. Where a polytope is understood as a tiling or decomposition of a manifold, this idea may be extended to infinite manifolds.plane tilings, space-filling (honeycombs) andhyperbolic tilings are in this sense polytopes, and are sometimes calledapeirotopes because they have infinitely many cells.
Among these, there are regular forms including theregular skew polyhedra and the infinite series of tilings represented by the regularapeirogon, square tiling, cubic honeycomb, and so on.
The theory ofabstract 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 an intuitive underlying space, such as the11-cell.
An abstract polytope is apartially ordered set of elements or members, which obeys certain rules. It is a purely algebraic structure, and the theory was developed in order to avoid some of the issues which make it difficult to reconcile the various geometric classes within a consistent mathematical framework. A geometric polytope is said to be a realization in some real space of the associated abstract polytope.[11]
Structures analogous to polytopes exist in complexHilbert spaces wheren real dimensions are accompanied bynimaginary ones.Regular complex polytopes are more appropriately treated asconfigurations.[12]
Everyn-polytope has a dual structure, obtained by interchanging its vertices for facets, edges for ridges, and so on generally interchanging its (j − 1)-dimensional elements for (n − j)-dimensional elements (forj = 1 ton − 1), while retaining the connectivity or incidence between elements.
For an abstract polytope, this simply reverses the ordering of the set. This reversal is seen in theSchläfli symbols for regular polytopes, where the symbol for the dual polytope is simply the reverse of the original. For example, {4, 3, 3} is dual to {3, 3, 4}.
In the case of a geometric polytope, some geometric rule for dualising is necessary, see for example the rules described fordual polyhedra. Depending on circumstance, the dual figure may or may not be another geometric polytope.[13]
If the dual is reversed, then the original polytope is recovered. Thus, polytopes exist in dual pairs.
If a polytope has the same number of vertices as facets, of edges as ridges, and so forth, and the same connectivities, then the dual figure will be similar to the original and the polytope is self-dual.
Some common self-dual polytopes include:
Polygons and polyhedra have been known since ancient times.
An early hint of higher dimensions came in 1827 whenAugust Ferdinand Möbius discovered that two mirror-image solids can be superimposed by rotating one of them through a fourth mathematical dimension. By the 1850s, a handful of other mathematicians such asArthur Cayley andHermann Grassmann had also considered higher dimensions.
Ludwig Schläfli was the first to consider analogues of polygons and polyhedra in these higher spaces. He described the sixconvex regular 4-polytopes in 1852 but his work was not published until 1901, six years after his death. By 1854,Bernhard Riemann'sHabilitationsschrift had firmly established the geometry of higher dimensions, and thus the concept ofn-dimensional polytopes was made acceptable. Schläfli's polytopes were rediscovered many times in the following decades, even during his lifetime.
In 1882Reinhold Hoppe, writing in German, coined the wordpolytop to refer to this more general concept of polygons and polyhedra. In due courseAlicia Boole Stott, daughter of logicianGeorge Boole, introduced the anglicisedpolytope into the English language.[2]: vi
In 1895,Thorold Gosset not only rediscovered Schläfli's regular polytopes but also investigated the ideas ofsemiregular polytopes and space-fillingtessellations in higher dimensions. Polytopes also began to be studied in non-Euclidean spaces such as hyperbolic space.
An important milestone was reached in 1948 withH. S. M. Coxeter's bookRegular Polytopes, summarizing work to date and adding new findings of his own.
Meanwhile, the French mathematicianHenri Poincaré had developed thetopological idea of a polytope as the piecewise decomposition (e.g.CW-complex) of amanifold.Branko Grünbaum published his influential work onConvex Polytopes in 1967.
In 1952Geoffrey Colin Shephard generalised the idea ascomplex polytopes in complex space, where each real dimension has an imaginary one associated with it. Coxeter developed the theory further.
The conceptual issues raised by complex polytopes, non-convexity, duality and other phenomena led Grünbaum and others to the more general study of abstract combinatorial properties relating vertices, edges, faces and so on. A related idea was that of incidence complexes, which studied the incidence or connection of the various elements with one another. These developments led eventually to the theory ofabstract polytopes as partially ordered sets, or posets, of such elements.Peter McMullen and Egon Schulte published their bookAbstract Regular Polytopes in 2002.
Enumerating theuniform polytopes, convex and nonconvex, in four or more dimensions remains an outstanding problem. The convex uniform 4-polytopes were fully enumerated byJohn Conway andMichael Guy using a computer in 1965;[14][15] in higher dimensions this problem was still open as of 1997.[16] The full enumeration for nonconvex uniform polytopes is not known in dimensions four and higher as of 2008.[17]
In modern times, polytopes and related concepts have found many important applications in fields as diverse ascomputer graphics,optimization,search engines,cosmology,quantum mechanics and numerous other fields. In 2013 theamplituhedron was discovered as a simplifying construct in certain calculations of theoretical physics.
In the field ofoptimization,linear programming studies themaxima and minima oflinear functions; these maxima and minima occur on theboundary of ann-dimensional polytope. In linear programming, polytopes occur in the use ofgeneralized barycentric coordinates andslack variables.
Intwistor theory, a branch oftheoretical physics, a polytope called theamplituhedron is used in to calculate the scattering amplitudes of subatomic particles when they collide. The construct is purely theoretical with no known physical manifestation, but is said to greatly simplify certain calculations.[18]
