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  "body": "\n## Summary\n\nA fundamental domain for a [[group action|0WCY42EV9V]] on a space is a subset that contains one point of each [[orbit|0YQPZB5Q1G]].\nOften this is only asked up to a negligible boundary.\nIts translates by the group tile the space.\nFundamental domains turn questions about the quotient space into questions about a concrete piece of the original space.\nFor instance, they compute the volume of a quotient by a lattice.\n\n## Explanations\n\n### L1 human\n\nThe integers act on the real line by shifting, and the interval $[0, 1)$ contains exactly one point from each set $x + \\Z$.\nIts shifts $[n, n+1)$ cover the line without overlap, like tiles.\nIn the plane, the unit square plays the same role for shifts by $\\Z^2$.\nStudying the quotient, here a circle or a torus, is the same as studying one tile with its edges glued.\nThe modular group acting on the upper half-plane has a famous fundamental domain bounded by two vertical lines and an arc of the unit circle.\n",
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