geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, an orthant or hyperoctant is the analogue in ''n''-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

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: :ε₁''x''₁ ≥ 0 ε₂''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 :ε₁''x''₁ > 0 ε₂''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 and Neil Sloane defined the term ''n''-

orthoplex In geometry, a cross-polytope, hyperoctahedron, orthoplex, staurotope, or cocube is a regular polytope, regular, convex polytope that exists in ''n''-dimensions, dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensi ...

from orthant complex as a

regular polytope In mathematics, a regular polytope is a polytope whose symmetry group acts transitive group action, transitively on its flag (geometry), flags, thus giving it the highest degree of symmetry. In particular, all its elements or -faces (for all , w ...

in ''n''-dimensions with 2^''n'' simplex facets, one per orthant. The nonnegative orthant is the generalization of the first quadrant to ''n''-dimensions and is important in many

constrained optimization In mathematical optimization, constrained optimization (in some contexts called constraint optimization) is the process of optimizing an objective function with respect to some variables in the presence of constraints on those variables. The obj ...

problems.

See also

References

Further reading