orthant

{{Short description|Generalization of a quadrant to any dimension}}

File:Cartesian coordinates 2D.svg

In geometry, an orthant{{cite book |first=Steven |last=Roman |authorlink=Steven Roman |title=Advanced Linear Algebra |location=New York |publisher=Springer |edition=2nd |year=2005 |isbn=0-387-24766-1 |url=https://books.google.com/books?id=FV_s8W58D4UC&pg=PA394 }} or hyperoctant{{MathWorld|title=Hyperoctant|urlname=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 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:

• 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 and Neil Sloane defined the term n-orthoplex from orthant complex as a regular polytope in n-dimensions with 2ⁿ simplex facets, one per orthant.{{cite book |first1=J. H. |last1=Conway |first2=N. J. A. |last2=Sloane |chapter=The Cell Structures of Certain Lattices |title=Miscellanea Mathematica |editor-last=Hilton |editor-first=P. |editor2-last=Hirzebruch |editor2-first=F. |editor3-last=Remmert |editor3-first=R. |publisher=Springer |location=Berlin |pages=89–90 |year=1991 |doi=10.1007/978-3-642-76709-8_5 |isbn=978-3-642-76711-1 }}

The nonnegative orthant is the generalization of the first quadrant to n-dimensions and is important in many constrained optimization problems.