In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

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

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

 of a quadrant in the plane or an octant

Octant

An octant is one of eight divisions.-Octant in the plane :Traditionally wind direction is given as one of the 8 octants because that is more accurate than merely giving one of the 4 quadrants, and the wind vane typically does not have enough accuracy to bother with more precise indication.-Octant...

 in three dimensions.

In general an orthant in n-dimensions can be considered the intersection of n mutually orthogonal half-space

Half-space

In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional euclidean space. More generally, a half-space is either of the two parts into which a hyperplane divides an affine space...

s. By permutation

Permutation

In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...

s 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.

Standard space

The n-orthant is a standard space in two ways: every polytope with n faces maps into the n-orthant via slack variable

Slack variable

In an optimization problem, a slack variable is a variable that is added to an inequality constraint to transform it to an equality. Introducing a slack variable replaces an inequality constraint with an equality constraint and a nonnegativity constraint....

s, and conversely every polygonal cone

Cone (linear algebra)

In linear algebra, a cone is a subset of a vector space that is closed under multiplication by positive scalars. In other words, a subset C of a real vector space V is a cone if and only if λx belongs to C for any x in C and any positive scalar λ of V .A cone is said...

 on n vertices is the image of (maps from) the n-orthant. Compare the n-simplex, which maps to every polytope with n-vertices.

See also

• Cross polytope (or orthoplex) - a family of regular polytope
  Regular polytope
  In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of...
  
  s in n-dimensions which can be constructed with one simplex
  Simplex
  In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
  
   facets in each orthant space.
• Measure polytope (or hypercube) - a family of regular polytopes in n-dimensions which can be constructed with one vertex
  Vertex (geometry)
  In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
  
   in each orthant space.
• Orthotope - Generalization of a rectangle in n-dimensions, with one vertex in each orthant.