Polytope

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 ${\displaystyle \{(x,y)\in \mathbb {R} ^{2}\mid x\geq 0\}}$. 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 ${\displaystyle \mathbf {A} }$, ${\displaystyle {\mathcal {P}}=\{\mathbf {x} \in \mathbb {R} ^{d}:\mathbf {Ax} \leq \mathbf {1} \}}$, 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 ${\displaystyle t\in \mathbb {Z} _{\geq 0}}$. 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 ${\displaystyle {\mathcal {P}}^{*}}$ 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 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.[2]

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.

A non-convex polytope may be self-intersecting; this class of polytopes include the star polytopes. Some regular polytopes are stars.[2]

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.[10]

The Gram–Euler theorem similarly generalizes the alternating sum of internal angles ${\textstyle \sum \varphi }$ for convex polyhedra to higher-dimensional polytopes:[10]

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

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.

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.[11]

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.[12]

The 5-cell (4-simplex) is self-dual with 5 vertices and 5 tetrahedral cells.

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}.