A polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions. Some theories further generalize the idea to include such objects as unbounded polytopes, and abstract polytopes. The following are some of the aspects of polytopes studied in discrete geometry:
Packings, coverings, and tilings are all ways of arranging uniform objects in a regular way on a surface or manifold. A sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, n-dimensional Euclidean space or to non-Euclidean spaces such as hyperbolic space. A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions. Specific topics in this area include:
Structural rigidity is a combinatorial theory for predicting the flexibility of ensembles formed by rigid bodies connected by flexible linkages or hinges. Topics in this area include:
Cauchy's theorem
Flexible polyhedra
Incidence structures
Incidence structures generalize planes as can be seen from their axiomatic definitions. Incidence structures also generalize the higher-dimensional analogs and the finite structures are sometimes called finite geometries. Formally, an incidencestructure is a triple where P is a set of "points", L is a set of "lines" and is the incidence relation. The elements of are called flags. If we say that pointp "lies on" line. Topics in this area include:
Configurations
Line arrangements
Hyperplane arrangements
Buildings
Oriented matroids
An oriented matroid is a mathematical structure that abstracts the properties of directed graphs and of arrangements of vectors in a vector space over an ordered field. In comparison, an ordinary matroid abstracts the dependence properties that are common both to graphs, which are not necessarily directed, and to arrangements of vectors over fields, which are not necessarily ordered.
Geometric graph theory
A geometric graph is a graph in which the vertices or edges are associated with geometric objects. Examples include Euclidean graphs, the 1-skeleton of a polyhedron or polytope, intersection graphs, and visibility graphs. Topics in this area include:
The discipline of combinatorial topology used combinatorial concepts in topology and in the early 20th century this turned into the field of algebraic topology. In 1978, the situation was reversed – methods from algebraic topology were used to solve a problem in combinatorics – when László Lovász proved the Kneser conjecture, thus beginning the new study of topological combinatorics. Lovász's proof used the Borsuk-Ulam theorem and this theorem retains a prominent role in this new field. This theorem has many equivalent versions and analogs and has been used in the study of fair division problems. Topics in this area include:
Digital geometry deals with discrete sets considered to be digitizedmodels or images of objects of the 2D or 3D Euclidean space. Simply put, digitizing is replacing an object by a discrete set of its points. The images we see on the TV screen, the raster display of a computer, or in newspapers are in fact digital images. Its main application areas are computer graphics and image analysis.
Discrete differential geometry is the study of discrete counterparts of notions in differential geometry. Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes. It is used in the study of computer graphics and topological combinatorics. Topics in this area include: