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Given a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
and a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
acting
Acting is an activity in which a story is told by means of its enactment by an actor or actress who adopts a character—in theatre, television, film, radio, or any other medium that makes use of the mimetic mode.
Acting involves a bro ...
on it, the images of a single point under the group action form an
orbit
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such a ...
of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each of these orbits. It serves as a geometric realization for the abstract set of representatives of the orbits.
There are many ways to choose a fundamental domain. Typically, a fundamental domain is required to be a
connected subset with some restrictions on its boundary, for example, smooth or polyhedral. The images of a chosen fundamental domain under the group action then
tile
Tiles are usually thin, square or rectangular coverings manufactured from hard-wearing material such as ceramic, Rock (geology), stone, metal, baked clay, or even glass. They are generally fixed in place in an array to cover roofs, floors, wa ...
the space. One general construction of fundamental domains uses
Voronoi cell
In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed ...
s.
Hints at a general definition
Given an
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
of a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
''G'' on a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
''X'' by
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
s, a fundamental domain for this action is a set ''D'' of representatives for the orbits. It is usually required to be a reasonably nice set topologically, in one of several precisely defined ways. One typical condition is that ''D'' is ''almost'' an open set, in the sense that ''D'' is the
symmetric difference
In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets \ and \ is \.
T ...
of an open set in ''X'' with a set of
measure zero
In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.
The notion of null ...
, for a certain (quasi)invariant
measure on ''X''. A fundamental domain always contains a
free regular set ''U'', an
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that a ...
moved around by ''G'' into
disjoint copies, and nearly as good as ''D'' in representing the orbits. Frequently ''D'' is required to be a complete set of coset representatives with some repetitions, but the repeated part has measure zero. This is a typical situation in
ergodic theory
Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...
. If a fundamental domain is used to calculate an
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
on ''X''/''G'', sets of measure zero do not matter.
For example, when ''X'' is
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
R
''n'' of dimension ''n'', and ''G'' is the
lattice Z
''n'' acting on it by translations, the quotient ''X''/''G'' is the ''n''-dimensional
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not ...
. A fundamental domain ''D'' here can be taken to be
closed unit cube [0,1">closed set">closed unit cube [0,1/nowiki>''n'', whose boundary (topology)">boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
* ''Boundaries'' (2016 film), a 2016 Canadian film
* ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film
*Boundary (cricket), the edge of the pla ...
consists of the points whose orbit has more than one representative in ''D''.
Examples
Examples in the three-dimensional Euclidean space R
3.
*for ''n''-fold rotation: an orbit is either a set of ''n'' points around the axis, or a single point on the axis; the fundamental domain is a sector
*for reflection in a plane: an orbit is either a set of 2 points, one on each side of the plane, or a single point in the plane; the fundamental domain is a half-space bounded by that plane
*for reflection in a point: an orbit is a set of 2 points, one on each side of the center, except for one orbit, consisting of the center only; the fundamental domain is a half-space bounded by any plane through the center
*for 180° rotation about a line: an orbit is either a set of 2 points opposite to each other with respect to the axis, or a single point on the axis; the fundamental domain is a half-space bounded by any plane through the line
*for discrete translational symmetry in one direction: the orbits are translates of a 1D lattice in the direction of the translation vector; the fundamental domain is an infinite slab
*for discrete translational symmetry in two directions: the orbits are translates of a 2D lattice in the plane through the translation vectors; the fundamental domain is an infinite bar with
parallelogram
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of eq ...
matic cross section
*for discrete translational symmetry in three directions: the orbits are translates of the lattice; the fundamental domain is a
primitive cell
In geometry, biology, mineralogy and solid state physics, a unit cell is a repeating unit formed by the vectors spanning the points of a lattice. Despite its suggestive name, the unit cell (unlike a unit vector, for example) does not necessaril ...
which is e.g. a
parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term '' rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclid ...
, or a
Wigner-Seitz cell, also called
Voronoi cell
In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed ...
/diagram.
In the case of translational symmetry combined with other symmetries, the fundamental domain is part of the primitive cell. For example, for
wallpaper group
A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformat ...
s the fundamental domain is a factor 1, 2, 3, 4, 6, 8, or 12 smaller than the primitive cell.
Fundamental domain for the modular group
The diagram to the right shows part of the construction of the fundamental domain for the action of the
modular group
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fraction ...
Γ on the
upper half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to ...
''H''.
This famous diagram appears in all classical books on
modular function
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of ...
s. (It was probably well known to
C. F. Gauss, who dealt with fundamental domains in the guise of the
reduction theory of
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
s.) Here, each triangular region (bounded by the blue lines) is a
free regular set of the action of Γ on ''H''. The boundaries (the blue lines) are not a part of the free regular sets. To construct a fundamental domain of ''H''/Γ, one must also consider how to assign points on the boundary, being careful not to double-count such points. Thus, the free regular set in this example is
:
The fundamental domain is built by adding the boundary on the left plus half the arc on the bottom including the point in the middle:
:
The choice of which points of the boundary to include as a part of the fundamental domain is arbitrary, and varies from author to author.
The core difficulty of defining the fundamental domain lies not so much with the definition of the set ''per se'', but rather with how to treat integrals over the fundamental domain, when integrating functions with poles and zeros on the boundary of the domain.
See also
*
Free regular set
*
Fundamental polygon
*
Brillouin zone
In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice ...
*
Fundamental pair of periods In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane. This type of lattice is the underlying object with which elliptic functions and modular forms are defined.
Definitio ...
*
Petersson inner product In mathematics the Petersson inner product is an inner product defined on the space
of entire modular forms. It was introduced by the German mathematician Hans Petersson.
Definition
Let \mathbb_k be the space of entire modular forms of weight k ...
*
Cusp neighborhood
External links
* {{MathWorld , urlname=FundamentalDomain , title=Fundamental domain
Topological groups
Ergodic theory
Riemann surfaces
Group actions (mathematics)