Polytope

Elements

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 use face to refer to an (n − 1)-dimensional element while others use face to denote a 2-face specifically. Authors may use j-face or j-facet to indicate an element of j dimensions. Some use edge to refer to a ridge, while H. S. M. Coxeter uses cell to denote an (n − 1)-dimensional element.

The terms adopted in this article are given in the table below:

An n-dimensional 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 j-dimensional faces or j-faces. A 0-dimensional face is called a vertex, and consists of a single point. A 1-dimensional face is called an edge, and consists of a line segment. A 2-dimensional face consists of a polygon, and a 3-dimensional face, sometimes called a cell, consists of a polyhedron.

Important classes of polytopes

Convex polytopes

A polytope may be convex. 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 of half-spaces. This definition allows a polytope to be neither bounded nor finite. Polytopes are defined in this way, e.g., in linear programming. A polytope is bounded if there is a ball of finite radius that contains it. A polytope is said to be pointed if it contains at least one vertex. Every bounded nonempty polytope is pointed. An example of a non-pointed polytope is the set ${\displaystyle \{(x,y)\in {\mathbb{R}}^2\mid x\ge 0\}}$. A polytope is finite 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 an integral polytope if all of its vertices have integer coordinates.

A certain class of convex polytopes are reflexive polytopes. An integral ${\displaystyle d}$-polytope ${\displaystyle \mathcal{P}}$ is reflexive if for some integral matrix ${\displaystyle \mathbf{A}}$, ${\displaystyle \mathcal{P}=\{\mathbf{x}\in {\mathbb{R}}^d:\mathbf{A}\mathbf{x}\le \mathbf{1}\}}$, where ${\displaystyle \mathbf{1}}$ denotes a vector of all ones, and the inequality is component-wise. It follows from this definition that ${\displaystyle \mathcal{P}}$ is reflexive if and only if ${\displaystyle (t+1){\mathcal{P}}^{\circ }\cap {\mathbb{Z}}^d=t\mathcal{P}\cap {\mathbb{Z}}^d}$ for all ${\displaystyle t\in {\mathbb{Z}}_{\ge 0}}$. In other words, a ${\displaystyle (t+1)}$-dilate of ${\displaystyle \mathcal{P}}$ differs, in terms of integer lattice points, from a ${\displaystyle t}$-dilate of ${\displaystyle \mathcal{P}}$ only by lattice points gained on the boundary. Equivalently, ${\displaystyle \mathcal{P}}$ is reflexive if and only if its dual polytope ${\displaystyle {\mathcal{P}}^{\ast }}$ is an integral polytope.

Regular polytopes

Regular polytopes have the highest degree of symmetry of all polytopes. The symmetry group of a regular polytope acts transitively on its flags; hence, the dual polytope of a regular polytope is also regular.

There are three main classes of regular polytope which occur in any number of dimensions:

• Simplices, including the equilateral triangle and the regular tetrahedron.
• Hypercubes or measure polytopes, including the square and the cube.
• Orthoplexes or cross polytopes, including the square and regular octahedron.

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 many regular polygons of n-fold symmetry, both convex and (for n ≥ 5) star. But in higher dimensions there are no other regular polytopes.

In three dimensions the convex Platonic solids include the fivefold-symmetric dodecahedron and icosahedron, and there are also four star Kepler–Poinsot polyhedra with fivefold symmetry, bringing the total to nine regular polyhedra.

In four dimensions the regular 4-polytopes include one additional convex solid with fourfold symmetry and two with fivefold symmetry. There are ten star Schläfli-Hess 4-polytopes, all with fivefold symmetry, giving in all sixteen regular 4-polytopes.

Star polytopes

A non-convex polytope may be self-intersecting; this class of polytopes include the star polytopes. Some regular polytopes are stars.

Properties

Euler characteristic

Since a (filled) convex polytope P in ${\displaystyle d}$ dimensions is contractible to a point, the Euler characteristic ${\displaystyle \chi }$ of its boundary ∂P is given by the alternating sum:

${\displaystyle \chi =n_0-n_1+n_2-\cdots \pm n_{d-1}=1+(-1{)}^{d-1}}$, where ${\displaystyle n_j}$ is the number of ${\displaystyle j}$-dimensional faces.

This generalizes Euler's formula for polyhedra.

Internal angles

The Gram–Euler theorem similarly generalizes the alternating sum of internal angles $\sum \phi$ for convex polyhedra to higher-dimensional polytopes:

${\displaystyle \sum \phi =(-1{)}^{d-1}}$

Generalisations of a polytope

Infinite polytopes

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) and hyperbolic tilings are in this sense polytopes, and are sometimes called apeirotopes because they have infinitely many cells.

Among these, there are regular forms including the regular skew polyhedra and the infinite series of tilings represented by the regular apeirogon, square tiling, cubic honeycomb, and so on.

Abstract polytopes

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 an intuitive underlying space, such as the 11-cell.

An abstract polytope is a partially 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.

Complex polytopes

Structures analogous to polytopes exist in complex Hilbert spaces ${\displaystyle {\mathbb{C}}^n}$ where n real dimensions are accompanied by n imaginary ones. Regular complex polytopes are more appropriately treated as configurations.

Duality

Every n-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 (for j = 1 to n − 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 the Schläfli 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 for dual polyhedra. Depending on circumstance, the dual figure may or may not be another geometric polytope.

If the dual is reversed, then the original polytope is recovered. Thus, polytopes exist in dual pairs.

Self-dual polytopes

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:

• Every regular n-simplex, in any number of dimensions, with Schläfli symbol {3ⁿ}. These include the equilateral triangle {3}, regular tetrahedron {3,3}, and 5-cell {3,3,3}.
• Every hypercubic honeycomb, in any number of dimensions. These include the apeirogon {∞}, square tiling {4,4} and cubic honeycomb {4,3,4}.
• Numerous compact, paracompact and noncompact hyperbolic tilings, such as the icosahedral honeycomb {3,5,3}, and order-5 pentagonal tiling {5,5}.
• In 2 dimensions, all regular polygons (regular 2-polytopes)
• In 3 dimensions, the canonical polygonal pyramids and elongated pyramids, and tetrahedrally diminished dodecahedron.
• In 4 dimensions, the 24-cell, with Schläfli symbol {3,4,3}. Also the great 120-cell {5,5/2,5} and grand stellated 120-cell {5/2,5,5/2}.

History

Polygons and polyhedra have been known since ancient times.

An early hint of higher dimensions came in 1827 when August 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 as Arthur Cayley and Hermann 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 six convex regular 4-polytopes in 1852 but his work was not published until 1901, six years after his death. By 1854, Bernhard Riemann's Habilitationsschrift had firmly established the geometry of higher dimensions, and thus the concept of n-dimensional polytopes was made acceptable. Schläfli's polytopes were rediscovered many times in the following decades, even during his lifetime.

In 1882 Reinhold Hoppe, writing in German, coined the word polytop to refer to this more general concept of polygons and polyhedra. In due course Alicia Boole Stott, daughter of logician George Boole, introduced the anglicised polytope into the English language.

In 1895, Thorold Gosset not only rediscovered Schläfli's regular polytopes but also investigated the ideas of semiregular polytopes and space-filling tessellations in higher dimensions. Polytopes also began to be studied in non-Euclidean spaces such as hyperbolic space.

An important milestone was reached in 1948 with H. S. M. Coxeter's book Regular Polytopes, summarizing work to date and adding new findings of his own.

Meanwhile, the French mathematician Henri Poincaré had developed the topological idea of a polytope as the piecewise decomposition (e.g. CW-complex) of a manifold. Branko Grünbaum published his influential work on Convex Polytopes in 1967.

In 1952 Geoffrey Colin Shephard generalised the idea as complex 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 of abstract polytopes as partially ordered sets, or posets, of such elements. Peter McMullen and Egon Schulte published their book Abstract Regular Polytopes in 2002.

Enumerating the uniform polytopes, convex and nonconvex, in four or more dimensions remains an outstanding problem. The convex uniform 4-polytopes were fully enumerated by John Conway and Michael Guy using a computer in 1965; in higher dimensions this problem was still open as of 1997. The full enumeration for nonconvex uniform polytopes is not known in dimensions four and higher as of 2008.

In modern times, polytopes and related concepts have found many important applications in fields as diverse as computer graphics, optimization, search engines, cosmology, quantum mechanics and numerous other fields. In 2013 the amplituhedron was discovered as a simplifying construct in certain calculations of theoretical physics.

Applications

In the field of optimization, linear programming studies the maxima and minima of linear functions; these maxima and minima occur on the boundary of an n-dimensional polytope. In linear programming, polytopes occur in the use of generalized barycentric coordinates and slack variables.

In twistor theory, a branch of theoretical physics, a polytope called the amplituhedron 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.