High Quality Content by WIKIPEDIA articles! In geometry, an orthant or hyperoctant is the analogue in n-dimensional Euclidean space of a quadrant in the plane or an octant in three dimensions. In general an orthant in n-dimensions can be considered the intersection of n mutually orthogonal half-spaces. By permutations of half-space signs, there are 2n orthants in n-dimensional space. The n-orthant is a standard space in two ways: every polytope with n faces maps into the n-orthant via slack variables, and conversely every polygonal cone on n vertices is the image of (maps from) the n-orthant. Compare the n-simplex, which maps to every polytope with n-vertices.

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