- Fundamental domain
geometry, the fundamental domain of a symmetry groupof an object or pattern is a part of the pattern, as small as possible, which, based on the symmetry, determines the whole object or pattern. The set of orbits of the symmetry group define a partitioning of space. Each partition consists of points which, based on the symmetry, have equal properties, e.g., for a color pattern on the plane, have the same color. A fundamental domain is a set of representatives of these orbits. This is not unique, but typically a convenient connected part of space is chosen.
Hints at general definition
Given an action of a group "G" on a
topological space"X" by homeomorphisms, a fundamental domain (also called fundamental region) for this action is a set "D" of representatives for the orbits. It is usually required to be a reasonbly 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 differenceof an open set in "G" with a set of measure zero, for a certain (quasi)invariant measure on "X". A fundamental domain always contains a free regular set"U", an open setmoved around by "G" into disjointcopies, 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. If a fundamental domain is used to calculate an integralon "X"/"G", sets of measure zero do not matter.
For example, when "X" is
Euclidean spaceR"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. A fundamental domain "D" here can be taken to be [0,1)"n", which differs from the open set (0,1)"n" by a set of measure zero, or the closed unit cube [0,1]"n", whose boundary consists of the points whose orbit has more than one representative in "D".
Examples in the three-dimensional Euclidean space R3.
*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 inversion 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
translational symmetryin 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
parallelogrammatic cross section
*for discrete translational symmetry in three directions: the orbits are translates of the lattice; the fundamental domain is a
primitive cellwhich is e.g. a parallelepiped, or a Wigner-Seitz cell, also called Voronoi cell/diagram.
In the case of translational symmetry combined with other symmetries, the fundamental domain is part of the primitive cell. For example, for
wallpaper groups 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 existence and description of a fundamental domain is in general something requiring painstaking work to establish. The diagram to the right shows part of the construction of the fundamental domain for the action of the
modular groupΓ on the upper half-plane"H".
This famous diagram appears in all classical books on
elliptic modular functions. (It was probably well known to C. F. Gauss, who dealt with fundamental domains in the guise of the reduction theory of quadratic forms.) Here, each triangular region (bounded by the blue lines) is a free regular setof 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:
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