mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a polyhedral complex is a set of

polyhedra In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...

in a real

vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

that fit together in a specific way. Polyhedral complexes generalize

simplicial complexes In mathematics, a simplicial complex is a structured set composed of points, line segments, triangles, and their ''n''-dimensional counterparts, called simplices, such that all the faces and intersections of the elements are also included in ...

and arise in various areas of polyhedral geometry, such as

tropical geometry In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition: : x\oplus y=\min\, : x\otimes y=x+y. So for exampl ...

, splines and hyperplane arrangements.

Definition

A polyhedral complex

\mathcal

is a set of

that satisfies the following conditions: :1. Every

face The face is the front of the head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affect th ...

of a polyhedron from

\mathcal

is also in

\mathcal

. :2. The

intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...

of any two polyhedra

\sigma_1, \sigma_2 \in \mathcal

is a face of both

\sigma_1

and

\sigma_2

. Note that the empty set is a face of every polyhedron, and so the intersection of two polyhedra in

\mathcal

may be empty.

Examples

* Tropical varieties are polyhedral complexes satisfying a certain ''balancing condition''. *

Simplicial complexes In mathematics, a simplicial complex is a structured set composed of points, line segments, triangles, and their ''n''-dimensional counterparts, called simplices, such that all the faces and intersections of the elements are also included in ...

are polyhedral complexes in which every polyhedron is a

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

. *

Voronoi diagrams In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. It can be classified also as a tessellation. In the simplest case, these objects are just finitely many points in the plane (calle ...

. * Splines.

Fans

A fan is a polyhedral complex in which every polyhedron is a

cone In geometry, a cone is a three-dimensional figure that tapers smoothly from a flat base (typically a circle) to a point not contained in the base, called the '' apex'' or '' vertex''. A cone is formed by a set of line segments, half-lines ...

from the origin. Examples of fans include: * The

normal fan In mathematics, specifically convex geometry, the normal fan of a convex polytope ''P'' is a polyhedral fan that is dual to ''P''. Normal fans have applications to polyhedral combinatorics, linear programming, tropical geometry, toric geome ...

of a

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

. * The Gröbner fan of an ideal of a

polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...

. * A tropical variety obtained by tropicalizing an

algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, ...

over a

valued field In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of the size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of siz ...

with trivial valuation. * The ''recession fan'' of a tropical variety.

References

{{Topology Polyhedra