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 2^{n} orthants in *n*-dimensional space.

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

- ε
_{1}*x*_{1}≥ 0 ε_{2}*x*_{2}≥ 0 · · · ε_{n}*x*_{n}≥ 0,

where each ε_{i} is +1 or −1.

Similarly, an **open orthant** in **R**^{n} is a subset defined by a system of strict inequalities

- ε
_{1}*x*_{1}> 0 ε_{2}*x*_{2}> 0 · · · ε_{n}*x*_{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 defined the term *n*-orthoplex from **orthant complex** as a regular polytope in n-dimensions with 2^{n} 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.

## 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

**^**Roman, Steven (2005).*Advanced Linear Algebra*(2nd ed.). New York: Springer. ISBN 0-387-24766-1.**^**Weisstein, Eric W. "Hyperoctant".*MathWorld*.**^**Conway, J. H.; Sloane, N. J. A. (1991). "The Cell Structures of Certain Lattices". In Hilton, P.; Hirzebruch, F.; Remmert, R. (eds.).*Miscellanea Mathematica*. Berlin: Springer. pp. 71–107. doi:10.1007/978-3-642-76709-8_5.

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