Orthant

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 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:

  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

Notes

  1. Advanced linear algebra By Steven Roman, Chapter 15
  2. Weisstein, Eric W. "Hyperoctant". MathWorld.
  3. J. H. Conway, N. J. A. Sloane, The Cell Structures of Certain Lattices (1991)
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