# Orthant

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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     · · ·     εnxn ≥ 0,

where each εi is +1 or −1.

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

ε1x1 > 0      ε2x2 > 0     · · ·     εnxn > 0,

where each εi 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. ^
3. ^ J. H. Conway, N. J. A. Sloane, The Cell Structures of Certain Lattices (1991) [1]
• The facts on file: Geometry handbook, Catherine A. Gorini, 2003, ISBN 0-8160-4875-4, p.113