Polytope
In elementary geometry, a polytope is a geometric object with flat sides (faces).
Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes.
Polytopes of more than three dimensions were first discovered by Ludwig Schläfli before 1853, who called such a figure a polyschem.
Many of these definitions are not equivalent to each other, resulting in different overlapping sets of objects being called polytopes.
They represent different approaches to generalizing the convex polytopes to include other objects with similar properties.
An example of this approach defines a polytope as a set of points that admits a simplicial decomposition.
[4] However this definition does not allow star polytopes with interior structures, and so is restricted to certain areas of mathematics.
The discovery of star polyhedra and other unusual constructions led to the idea of a polyhedron as a bounding surface, ignoring its interior.
A polyhedron is understood as a surface whose faces are polygons, a 4-polytope as a hypersurface whose facets (cells) are polyhedra, and so forth.
[8][citation needed] 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.
There are ten star Schläfli-Hess 4-polytopes, all with fivefold symmetry, giving in all sixteen regular 4-polytopes.
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.
Some common self-dual polytopes include: Polygons and polyhedra have been known since ancient times.
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.
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.
In 1882 Reinhold Hoppe, writing in German, coined the word polytop to refer to this more general concept of polygons and polyhedra.
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.
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.
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;[14][15] in higher dimensions this problem was still open as of 1997.
[17] 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 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.
