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:
“POLITICIAN, n. An eel in the fundamental mud upon which the superstructure of organized society is reared. When he wriggles he mistakes the agitation of his tail for the trembling of the edifice. As compared with the statesman, he suffers the disadvantage of being alive.”
—Ambrose Bierce (1842–1914?)
“You are the harvest and not the reaper
And of your domain another is the keeper.”
—John Ashbery (b. 1927)