Orthant

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 two dimensions, there are 4 orthants (called quadrants)

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     · · ·     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:

  • 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 2n 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
  1. Roman, Steven (2005). Advanced Linear Algebra (2nd ed.). New York: Springer. ISBN 0-387-24766-1.
  2. Weisstein, Eric W. "Hyperoctant". MathWorld.
  3. 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. ISBN 978-3-642-76711-1.
  • The facts on file: Geometry handbook, Catherine A. Gorini, 2003, ISBN 0-8160-4875-4, p.113
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