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
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