Discrete geometry

Discrete geometry and combinatorial geometry are branches of

László Fejes Tóth, H.S.M. Coxeter, and Paul Erdős laid the foundations of discrete geometry.^[1][2][3]

Topics

Polyhedra and polytopes

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 (such as a 4-polytope in four dimensions). Some theories further generalize the idea to include such objects as unbounded polytopes (apeirotopes and tessellations), and abstract polytopes.

The following are some of the aspects of polytopes studied in discrete geometry:

• Polyhedral combinatorics
• Lattice polytopes
• Ehrhart polynomials
• Pick's theorem
• Hirsch conjecture
• Opaque set

Packings, coverings and tilings

Packings, coverings, and tilings are all ways of arranging uniform objects (typically circles, spheres, or tiles) in a regular way on a surface or manifold.

A sphere packing is an arrangement of non-overlapping

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:

• Circle packings
• Sphere packings
• Kepler conjecture
• Quasicrystals
• Aperiodic tilings
• Periodic graph
• Finite subdivision rules

Structural rigidity and flexibility

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 (such as affine, projective, and Möbius 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 incidence structure is a triple

${\displaystyle C=(P,L,I).\,}$

where P is a set of "points", L is a set of "lines" and ${\displaystyle I\subseteq P\times L}$ is the incidence relation. The elements of ${\displaystyle I}$ are called flags. If

${\displaystyle (p,l)\in I,}$

we say that point p "lies on" line ${\displaystyle l}$.

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 (particularly for partially ordered vector spaces).^[4] In comparison, an ordinary (i.e., non-oriented) 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.^[5][6]

Geometric graph theory

A geometric graph is a

Topics in this area include:

• Graph drawing
• Polyhedral graphs
• Random geometric graphs
• Voronoi diagrams and Delaunay triangulations

Simplicial complexes

A simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. See also random geometric complexes.

Topological combinatorics

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

• Sperner's lemma
• Regular maps

Lattices and discrete groups

A discrete group is a

• Reflection groups
• Triangle groups

Digital geometry

Digital geometry deals with

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.^[7]

Discrete differential geometry

Discrete differential geometry is the study of discrete counterparts of notions in

• Discrete Laplace operator
• Discrete exterior calculus
• Discrete calculus
• Discrete Morse theory
• Topological combinatorics
• Spectral shape analysis
• Abstract differential geometry
• Analysis on fractals

See also

• Discrete and Computational Geometry
  (journal)
• Discrete mathematics
• Paul Erdős

Notes