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     · · ·     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:

1. In one dimension, an orthant is a ray.
2. In two dimensions, an orthant is a quadrant.
3. 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.