Definition
An orthant is a region of Euclidean space ℝⁿ defined by the signs of its Cartesian coordinates. It generalizes the notion of a quadrant (in ℝ²) and an octant (in ℝ³). Formally, for a given n‑tuple of signs σ = (σ₁, σ₂, …, σₙ) where each σᵢ ∈ {+, –}, the corresponding orthant is
$$ O_\sigma = { (x_1, x_2, \dots, x_n) \in \mathbb{R}^n \mid \operatorname{sgn}(x_i)=\sigma_i \text{ for all } i}, $$
where “sgn” denotes the sign function, with the convention that points lying on coordinate hyperplanes (where some coordinates are zero) are usually assigned to the closure of each adjacent orthant.
Properties
- Number of orthants – In ℝⁿ there are exactly 2ⁿ orthants, corresponding to the 2ⁿ possible sign patterns.
- Convexity – Each orthant is a convex cone; any line segment joining two points within the same orthant remains entirely within that orthant.
- Boundary – The boundary of an orthant consists of the union of coordinate hyperplanes (where one or more coordinates are zero). The interior of an orthant contains points with all coordinates strictly non‑zero and having the prescribed signs.
- Symmetry – The collection of orthants is invariant under the action of the hyperoctahedral group (the group of signed permutations of coordinates).
Examples
| Dimension | Common name | Description |
|---|---|---|
| 1 | half‑line | Two orthants: $x>0$ (positive half‑line) and $x<0$ (negative half‑line). |
| 2 | quadrant | Four orthants: $(+, +), (+, -), (-, +), (-, -)$. |
| 3 | octant | Eight orthants, e.g., the first octant $(+, +, +)$ where $x, y, z > 0$. |
| n | orthant | 2ⁿ orthants defined by the sign pattern of the n coordinates. |
Applications
- Optimization – In linear and nonlinear programming, constraints may restrict variables to a particular orthant (e.g., non‑negative variables lie in the first orthant).
- Probability and Statistics – Orthants are used in multivariate sign tests and in describing the distribution of a random vector’s sign pattern.
- Computational Geometry – Algorithms for nearest‑neighbor search, range searching, and space partitioning (e.g., k‑d trees) often exploit orthant decomposition.
- Differential Equations – Solutions of certain partial differential equations are examined separately on each orthant to handle sign‑dependent boundary conditions.
Historical Note
The term “orthant” derives from the Greek roots ortho‑ (“right” or “correct”) and ‑ant (a suffix used in geometry, analogous to “octant”). It entered mathematical literature in the early 20th century as a concise way to refer to sign‑determined regions of higher‑dimensional space.
Related Concepts
- Quadrant – The special case of an orthant in two dimensions.
- Octant – The special case of an orthant in three dimensions.
- Cone – An orthant is a specific type of convex polyhedral cone.
- Hyperoctahedral group – The symmetry group of the set of orthants in ℝⁿ.
References
- G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed., Wiley, 1999 – discussion of cones and orthants in ℝⁿ.
- J. H. Conway, R. K. Guy, The Book of Numbers, Springer, 1996 – mentions orthants in the context of sign patterns.
- D. C. Lay, Linear Algebra and Its Applications, 5th ed., Pearson, 2016 – treats non‑negative orthants in linear programming.