Orthants

Orthants are the regions into which the coordinate hyperplanes x1 = 0, x2 = 0, ..., xn = 0 cut the Euclidean space R^n. Each region is determined by a sign pattern (s1, ..., sn) with si ∈ {+, −}, where the i-th coordinate has the indicated sign. There are 2^n orthants in total.

Open versus closed: The open orthants require strict signs (xi > 0 for si = + and xi < 0 for si = −). Equivalently, O_s = {x ∈ R^n : si xi > 0 for all i}. The closed orthants allow zeros and are the closures of the corresponding open orthants; they include the coordinate axes and planes as boundaries.

Examples: In R^2 there are four quadrants; in R^3 there are eight octants. The “first orthant” typically refers to the region with all coordinates nonnegative (x_i ≥ 0 for all i), and its interior is the open first orthant (x_i > 0 for all i).

Geometric properties: Each open orthant is a convex cone. The closure of an open orthant is a closed convex cone. The origin lies on the boundary of every orthant and is not contained in any open orthant.

Notation and symmetry: The 2^n orthants are permuted by changing coordinate signs and by permuting coordinates, reflecting the hyperoctahedral symmetry of the arrangement of coordinate hyperplanes.

Applications: Orthants are used to partition R^n in computational geometry and optimization, and in statistics for orthant probabilities—the probability that a random vector falls into a given orthant.