Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain 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 the space. One general construction of fundamental domains uses Voronoi cells.
Read more about Fundamental Domain: Hints At General Definition, Examples, Fundamental Domain For The Modular Group
Famous quotes containing the words fundamental and/or domain:
“This is the fundamental idea of culture, insofar as it sets but one task for each of us: to further the production of the philosopher, of the artist, and of the saint within us and outside us, and thereby to work at the consummation of nature.”
—Friedrich Nietzsche (1844–1900)
“The vice named surrealism is the immoderate and impassioned use of the stupefacient image or rather of the uncontrolled provocation of the image for its own sake and for the element of unpredictable perturbation and of metamorphosis which it introduces into the domain of representation; for each image on each occasion forces you to revise the entire Universe.”
—Louis Aragon (1897–1982)