**Discrete geometry** and **combinatorial geometry** are branches of
geometry that study
combinatorial properties and constructive methods of
discrete geometric objects. Most questions in discrete geometry involve
finite or
discrete
sets of basic geometric objects, such as
points,
lines,
planes,
circles,
spheres,
polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they
intersect one another, or how they may be arranged to cover a larger object.

Discrete geometry has a large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory, toric geometry, and combinatorial topology.

Although polyhedra and tessellations had been studied for many years by people such as Kepler and Cauchy, modern discrete geometry has its origins in the late 19th century. Early topics studied were: the density of circle packings by Thue, projective configurations by Reye and Steinitz, the geometry of numbers by Minkowski, and map colourings by Tait, Heawood, and Hadwiger.

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

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 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
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 (where the problem becomes
circle packing in two dimensions, or
hypersphere packing in higher dimensions) 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:

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

**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:

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

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 point *p* "lies on" line .

Topics in this area include:

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]}

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,
unit disk graphs, and
visibility graphs.

Topics in this area include:

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

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.

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:

A **discrete group** is a
group *G* equipped with the
discrete topology. With this topology, *G* becomes a
topological group. A **discrete subgroup** of a topological group *G* is a
subgroup *H* whose
relative topology is the discrete one. For example, the
integers, **Z**, form a discrete subgroup of the
reals, **R** (with the standard
metric topology), but the
rational numbers, **Q**, do not.

A **lattice** in a
locally compact
topological group is a
discrete subgroup with the property that the
quotient space has finite
invariant measure. In the special case of subgroups of **R**^{n}, this amounts to the usual geometric notion of a
lattice, and both the algebraic structure of lattices and the geometry of the totality of all lattices are relatively well understood. Deep results of
Borel,
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 of
nilpotent
Lie groups and
semisimple algebraic groups over a
local field. In the 1990s,
Bass and
Lubotzky initiated the study of *tree lattices*, which remains an active research area.

Topics in this area include:

**Digital geometry** deals with
discrete sets (usually discrete
point sets) considered to be
digitized
models 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.^{
[7]}

**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:

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

**^**Pach, János; et al. (2008),*Intuitive Geometry, in Memoriam László Fejes Tóth*, Alfréd Rényi Institute of Mathematics**^**Katona, G. O. H. (2005), "Laszlo Fejes Toth – Obituary",*Studia Scientiarum Mathematicarum Hungarica*,**42**(2): 113**^**Bárány, Imre (2010), "Discrete and convex geometry", in Horváth, János (ed.),*A Panorama of Hungarian Mathematics in the Twentieth Century, I*, New York: Springer, pp. 431–441, ISBN 9783540307211**^**Rockafellar 1969. Björner et alia, Chapters 1-3. Bokowski, Chapter 1. Ziegler, Chapter 7.**^**Björner et alia, Chapters 1-3. Bokowski, Chapters 1-4.**^**Because matroids and oriented matroids are abstractions of other mathematical abstractions, nearly all the relevant books are written for mathematical scientists rather than for the general public. For learning about oriented matroids, a good preparation is to study the textbook on linear optimization by Nering and Tucker, which is infused with oriented-matroid ideas, and then to proceed to Ziegler's lectures on polytopes.**^**See Li Chen, Digital and discrete geometry: Theory and Algorithms, Springer, 2014.

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