What is "orthant"

n. (context geometry English) A n-dimensional generalization of the terms ''quadrant'' (in two-dimensional Cartesian space) and ''octant'' (in three-dimensional space).

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 2 orthants in n-dimensional space.

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

εx ≥ 0 εx ≥ 0 · · · εx ≥ 0,

where each ε is +1 or −1.

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

εx > 0 εx > 0 · · · εx > 0,

where each ε 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 2 simplex facets, one per orthant.