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.

John Conway defined the term n-orthoplex from orthant complex as a regular polytope in n-dimensions with 2ⁿ simplex facets, one per orthant.^[3]

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", MathWorld.
3. J. H. Conway, N. J. A. Sloane, The Cell Structures of Certain Lattices (1991)

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