Orthant

In two dimensions, there are 4 orthants (called quadrants)

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 permutations 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.

Standard space

The n-orthant is a standard space in two ways: every polytope with n faces maps into the n-orthant via slack variables, and conversely every polygonal cone on n vertices is the image of (maps from) the n-orthant. Compare the n-simplex, which maps to every polytope with n-vertices.

See also

• 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.

Notes

1. ^ Advanced linear algebra By Steven Roman, Chapter 15
2. ^ Weisstein, Eric W., "Hyperoctant" from MathWorld.