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.

Related Topics:
Geometry - Symmetry group - Symmetry - Orbits

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Examples in 3D:

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• 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.

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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.

Related Topics:
Mathematics - Lattice - Lie group - Coset - Well-behaved - Free regular set - Open set - Disjoint - Symmetric difference - Measure zero - Haar measure

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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.

Related Topics:
Euclidean space - Torus - Integral - Ergodic theory - Up to

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Fundamental domain: Example ►