# Orthant

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 independent selections of half-space signs, there are 2n orthants in n-dimensional space.

More specifically, a closed orthant in Rn is a subset defined by constraining each Cartesian coordinate to be nonnegative or nonpositive. Such a subset is defined by a system of inequalities:

ε1x1 ≥ 0      ε2x2 ≥ 0     · · ·     εnxn ≥ 0,

where each εi is +1 or −1.

Similarly, an open orthant in Rn is a subset defined by a system of strict inequalities

ε1x1 > 0      ε2x2 > 0     · · ·     εnxn > 0,

where each εi is +1 or −1.

By dimension:

• In one dimension, an orthant is a ray.
• In two dimensions, an orthant is a quadrant.
• In three dimensions, an orthant is an octant.

John Conway defined the term n-orthoplex from orthant complex as a regular polytope in n-dimensions with 2n simplex facets, one per orthant.

The nonnegative orthant is the generalization of the first quadrant to n-dimensions and is important in many constrained optimization problems.