Orthant

In mathematics, an orthant in n-dimensional Euclidean space R^n is one of the 2^n regions formed by the coordinate hyperplanes x_i = 0 for i = 1 to n. Each orthant corresponds to a choice of signs for the coordinates of points in that region, described by a sign pattern (s1, ..., sn) with si ∈ {+1, -1}. The boundaries between orthants are portions of the coordinate hyperplanes, and the origin lies on the boundary of every orthant.

The most familiar example is the closed positive orthant R_+^n = {x ∈ R^n : x_i ≥ 0 for all i}, with its open counterpart (0, ∞)^n. Any orthant can be obtained from the closed positive orthant by applying sign changes to coordinates (x_i → ±x_i) or by permuting coordinates. There are exactly 2^n orthants in R^n.

Geometrically, an orthant is a convex cone; the closed positive orthant is a convex cone equal to the Cartesian product of n nonnegative lines. In 2D there are four quadrants; in 3D there are eight octants.

Orthants arise in various areas, including computer graphics, multivariate statistics, and optimization. They are used to describe sign patterns of vectors, to partition space for algorithms, and in problems such as linear complementarity, where cone structures play a role. The term typically refers to any of the regions determined by the coordinate hyperplanes, with open orthants requiring all coordinates to be nonzero and closed orthants allowing zeros on some coordinates.