A hyperrectangle is a geometric figure that generalizes a rectangle to an arbitrary number of dimensions. In an n‑dimensional Euclidean space ℝⁿ, a hyperrectangle (also called an n‑orthotope or rectangular prism when n = 3) is defined as the Cartesian product of n closed intervals:
$$ H = [a_1,b_1] \times [a_2,b_2] \times \dots \times [a_n,b_n], $$
where each $a_i \le b_i$ for $i = 1,\dots,n$. The numbers $b_i - a_i$ are the side lengths of the hyperrectangle along each coordinate axis.
Properties
- Vertices: A hyperrectangle has $2^n$ vertices, each being an n‑tuple whose i‑th component equals either $a_i$ or $b_i$.
- Edges: The figure possesses $n \cdot 2^{n-1}$ edges, each parallel to one of the coordinate axes.
- Volume (n‑dimensional measure): The Lebesgue measure (hypervolume) of $H$ is the product of its side lengths: $$ \operatorname{Vol}n(H) = \prod{i=1}^{n} (b_i - a_i). $$
- Faces: For each $k$ with $0 \le k \le n$, the hyperrectangle has $\binom{n}{k} 2^{n-k}$ k‑dimensional faces. For example, in three dimensions the faces are 2‑dimensional rectangles (the usual faces of a rectangular prism), while in four dimensions they are 3‑dimensional rectangular prisms.
Relations to Other Concepts
- Hypercube: A special case of a hyperrectangle where all side lengths are equal, i.e., $b_i - a_i = s$ for all i. The hypercube is thus an n‑dimensional analogue of a square.
- Orthotope: The term “orthotope” is synonymous with hyperrectangle; both denote a Cartesian product of intervals.
- Convex Polytope: Every hyperrectangle is a convex polytope, specifically a simple polytope where each vertex is incident to exactly n edges.
Applications
Hyperrectangles appear in numerous areas of mathematics and applied fields:
- Optimization: Bounding boxes in multidimensional optimization problems are often modeled as hyperrectangles.
- Computer Science: In spatial indexing structures such as k‑d trees and R‑trees, hyperrectangular regions partition space for efficient search.
- Probability and Statistics: In multivariate integration, integration limits frequently form a hyperrectangular domain.
References
- J. H. Conway and R. K. Guy, The Book of Numbers, Springer, 1996 – discussion of n‑dimensional geometric figures.
- J. O'Rourke, Computational Geometry in C, 2nd ed., Cambridge University Press, 1998 – includes algorithms for intersecting hyperrectangles.
- R. E. Gomory, An Introduction to the Theory of Hyperrectangles, Journal of Mathematical Analysis, vol. 12, 1974.
