Discrete geometry andcombinatorial geometry are branches ofgeometry that studycombinatorial properties and constructive methods ofdiscrete geometric objects. Most questions in discrete geometry involvefinite ordiscretesets of basic geometric objects, such aspoints,lines,planes,circles,spheres,polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how theyintersect one another, or how they may be arranged to cover a larger object.
Discrete geometry has a large overlap withconvex geometry andcomputational geometry, and is closely related to subjects such asfinite geometry,combinatorial optimization,digital geometry,discrete differential geometry,geometric graph theory,toric geometry, andcombinatorial topology.
Polyhedra andtessellations had been studied for many years by people such asKepler andCauchy, modern discrete geometry has its origins in the late 19th century. Early topics studied were: the density ofcircle packings byThue,projective configurations by Reye andSteinitz, thegeometry of numbers by Minkowski, andmap colourings by Tait, Heawood, andHadwiger.
László Fejes Tóth,H.S.M. Coxeter, andPaul Erdős laid the foundations ofdiscrete geometry.[1][2][3]
Apolytope is a geometric object with flat sides, which exists in any general number of dimensions. Apolygon is a polytope in two dimensions, apolyhedron in three dimensions, and so on in higher dimensions (such as a4-polytope in four dimensions). Some theories further generalize the idea to include such objects as unbounded polytopes (apeirotopes andtessellations), andabstract 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 (typically circles, spheres, or tiles) in a regular way on a surface ormanifold.
Asphere packing is an arrangement of non-overlappingspheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensionalEuclidean space. However, spherepacking problems can be generalised to consider unequal spheres,n-dimensional Euclidean space (where the problem becomescircle packing in two dimensions, orhypersphere packing in higher dimensions) or tonon-Euclidean spaces such ashyperbolic space.
Atessellation of a flat surface is the tiling of aplane using one or more geometric shapes, called tiles, with no overlaps and no gaps. Inmathematics, tessellations can be generalized to higher dimensions.
Specific topics in this area include:
Structural rigidity is acombinatorial theory for predicting the flexibility of ensembles formed byrigid bodies connected by flexiblelinkages orhinges.
Topics in this area include:
Incidence structures generalize planes (such asaffine,projective, andMöbius planes) as can be seen from their axiomatic definitions. Incidence structures also generalize the higher-dimensional analogs and the finite structures are sometimes calledfinite geometries.
Formally, anincidence structure is a triple
whereP is a set of "points",L is a set of "lines" and is theincidence relation. The elements of are calledflags. If
we say that pointp "lies on" line.
Topics in this area include:
Anoriented matroid is amathematical structure that abstracts the properties ofdirected graphs and of arrangements of vectors in avector space over anordered field (particularly forpartially ordered vector spaces).[4] In comparison, an ordinary (i.e., non-oriented)matroid abstracts thedependence properties that are common both tographs, which are not necessarilydirected, and to arrangements of vectors overfields, which are not necessarilyordered.[5][6]
Ageometric graph is agraph in which thevertices oredges are associated withgeometric objects. Examples include Euclidean graphs, the 1-skeleton of apolyhedron orpolytope,unit disk graphs, andvisibility graphs.
Topics in this area include:
Asimplicial complex is atopological space of a certain kind, constructed by "gluing together"points,line segments,triangles, and theirn-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of asimplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is anabstract simplicial complex. See alsorandom geometric complexes.
The discipline of combinatorial topology used combinatorial concepts intopology and in the early 20th century this turned into the field ofalgebraic topology.
In 1978, the situation was reversed – methods from algebraic topology were used to solve a problem incombinatorics – whenLászló Lovász proved theKneser conjecture, thus beginning the new study oftopological combinatorics. Lovász's proof used theBorsuk-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 offair division problems.
Topics in this area include:
Adiscrete group is agroupG equipped with thediscrete topology. With this topology,G becomes atopological group. Adiscrete subgroup of a topological groupG is asubgroupH whoserelative topology is the discrete one. For example, theintegers,Z, form a discrete subgroup of thereals,R (with the standardmetric topology), but therational numbers,Q, do not.
Alattice in alocally compacttopological group is adiscrete subgroup with the property that thequotient space has finiteinvariant measure. In the special case of subgroups ofRn, this amounts to the usual geometric notion of alattice, and both the algebraic structure of lattices and the geometry of the totality of all lattices are relatively well understood. Deep results ofBorel,Harish-Chandra,Mostow,Tamagawa,M. S. Raghunathan,Margulis,Zimmer obtained from the 1950s through the 1970s provided examples and generalized much of the theory to the setting ofnilpotentLie groups andsemisimple algebraic groups over alocal field. In the 1990s,Bass andLubotzky initiated the study oftree lattices, which remains an active research area.
Topics in this area include:
Digital geometry deals withdiscrete sets (usually discretepoint sets) considered to bedigitizedmodels orimages of objects of the 2D or 3DEuclidean space.
Simply put,digitizing is replacing an object by a discrete set of its points. The images we see on the TV screen, theraster display of a computer, or in newspapers are in factdigital images.
Its main application areas arecomputer graphics andimage analysis.[7]
Discrete differential geometry is the study of discrete counterparts of notions indifferential geometry. Instead of smooth curves and surfaces, there arepolygons,meshes, andsimplicial complexes. It is used in the study ofcomputer graphics andtopological combinatorics.
Topics in this area include:
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