Geometric combinatorics is a branch of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 in general and combinatorics

Combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

 in particular. It includes a number of subareas such as polyhedral combinatorics

Polyhedral combinatorics

Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher dimensional convex polytopes....

 (the study of faces

Face (geometry)

In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...

 of convex polyhedra), convex geometry

Convex geometry

Convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space.Convex sets occur naturally in many areas of mathematics: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming,...

 (the study of convex set

Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...

s, in particular combinatorics of their intersections), and discrete geometry

Discrete geometry

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,...

, which in turn has many applications to computational geometry

Computational geometry

Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational...

. Other important areas include metric geometry of polyhedra, such as the Cauchy theorem

Cauchy's theorem (geometry)

Cauchy's theorem is a theorem in geometry, named after Augustin Cauchy. It states thatconvex polytopes in three dimensions with congruent corresponding faces must be congruent to each other...

 on rigidity of convex polytopes. The study of regular polytope

Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of...

s, Archimedean solid

Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...

s, and kissing numbers is also a part of geometric combinatorics. Special polytopes are also considered, such as the permutohedron

Permutohedron

In mathematics, the permutohedron of order n is an -dimensional polytope embedded in an n-dimensional space, the vertices of which are formed by permuting the coordinates of the vector .-History:According to , permutohedra were first studied by...

, associahedron

Associahedron

In mathematics, an associahedron or Stasheff polytope Kn is a convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a word of n letters and the edges correspond to single application of the associativity rule.Initially Jim Stasheff...

 and Birkhoff polytope. Also studied are finite geometries

Finite geometry

A finite geometry is any geometric system that has only a finite number of points.Euclidean geometry, for example, is not finite, because a Euclidean line contains infinitely many points, in fact as many points as there are real numbers...

.

Further reading

• Topics in Geometric Combinatorics
• Geometric Combinatorics, Edited by: Ezra Miller and Victor Reiner
• Combinatorics of Finite Geometries