In geometry, an orthant^[1] or hyperoctant^[2] 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     · · ·     ε_nx_n ≥ 0,

where each ε_i is +1 or −1.

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

ε₁x₁ > 0      ε₂x₂ > 0     · · ·     ε_nx_n > 0,

where each ε_i 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 and Neil Sloane defined the term n-orthoplex from orthant complex as a regular polytope in n-dimensions with 2ⁿ simplex facets, one per orthant.^[3]

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

• Cross polytope (or orthoplex) – a family of regular polytopes in n-dimensions which can be constructed with one simplex facets in each orthant space.
• Measure polytope (or hypercube) – a family of regular polytopes in n-dimensions which can be constructed with one vertex in each orthant space.
• Orthotope – generalization of a rectangle in n-dimensions, with one vertex in each orthant.

• The facts on file: Geometry handbook, Catherine A. Gorini, 2003, ISBN 0-8160-4875-4, p.113