thumb|Projections of <math>k</math>-cells onto the plane (from <math>k\in\{1,\dots{},6\}</math>). Only the edges of the higher-dimensional cells are shown.
In geometry, a hyperrectangle (also called a box, hyperbox, <math>k</math>-cell or orthotope), is the generalization of a rectangle (a plane figure) and the rectangular cuboid (a solid figure) to higher dimensions. A necessary and sufficient condition is that it is congruent to the Cartesian product of finite intervals. This means that a <math>k</math>-dimensional rectangular solid has each of its edges equal to one of the closed intervals used in the definition. Every <math>k</math>-cell is compact.
If all of the edges are equal length, it is a hypercube. A hyperrectangle is a special case of a parallelotope.
Formal definition
For every integer <math>i</math> from <math>1</math> to <math>k</math>, let <math>a_i</math> and <math>b_i</math> be real numbers such that <math>a_i < b_i</math>. The set of all points <math>x=(x_1,\dots,x_k)</math> in <math>\mathbb{R}^k</math> whose coordinates satisfy the inequalities <math>a_i\leq x_i\leq b_i</math> is a <math>k</math>-cell.
Intuition
A <math>k</math>-cell of dimension <math>k\leq 3</math> is especially simple. For example, a 1-cell is simply the interval <math>[a,b]</math> with <math>a < b</math>. A 2-cell is the rectangle formed by the Cartesian product of two closed intervals, and a 3-cell is a rectangular solid.
The sides and edges of a <math>k</math>-cell need not be equal in (Euclidean) length; although the unit cube (which has boundaries of equal Euclidean length) is a 3-cell, the set of all 3-cells with equal-length edges is a strict subset of the set of all 3-cells.
Types
A four-dimensional orthotope is likely a hypercuboid.
The special case of an -dimensional orthotope where all edges have equal length is the -cube or hypercube.