Fundamental domain
In geometry, the fundamental domain of a symmetry group of 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 2D color pattern, 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.
Examples in 3D:
- 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
- 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 parallelogrammatic cross section
- for discrete translational symmetry in three directions: the orbits are translates of the lattice; the fundamental domain is a primitive cell which is e.g. a parallelepiped, or a Wigner-Seitz cell, also called Voronoi cell.
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.
More generally, in mathematics, given a lattice Γ in a Lie group G, a fundamental domain is a set D of representatives for the cosets of Γ in G, that is also a well-behaved set topologically, in a sense that can be made precise in one of several ways. A fundamental domain always contains a free regular set U, an open set moved around by G into disjoint copies, and nearly as good as D in representing the cosets. One typical condition is that D is almost an open set, in the sense that D is the symmetric difference of an open set in G with a set of measure zero, for the Haar measure on G.
For example, when G is Euclidean space of dimension n, and Γ is Zn, the quotient G/Γ is the n-torus. A fundamental domain (also called fundamental region) here can be taken to be [0,1)n, which is the open set (0,1)n up to a set of measure zero. In practice the main use of a fundamental domain may be to compute integrals on G/Γ, in which case the set of measure zero is mentioned only to keep straight the pedantic assertion that D is exactly a set of coset representatives, and may quickly be forgotten. Other uses, for example in ergodic theory, are similarly based on having a reasonable set D up to sets of measure zero.
Example
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 group Γ 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 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
- <math>U = \left\{ z \in H: \left| z \right| > 1,\, \left| \,\mbox{Re}(z) \,\right| < \frac{1}{2} \right\}.</math>
The fundamental domain is built by adding the boundary on the left plus half the arc on the bottom:
- <math>D=U\cup\left\{ z \in H: \left| z \right| \geq 1,\, \mbox{Re}(z)=\frac{-1}{2} \right\} \cup \left\{ z \in H: \left| z \right| = 1,\, \mbox{Re}(z)<0 \right\}.</math>
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
- Voronoi cell
- Free regular set
- Fundamental polygon
- Brillouin zone
- Fundamental pair of periods
- Petersson inner product
- Cusp neighborhood
Categories
Topological groups | Ergodic theory | Riemann surfaces | Group actions