In geometry, a hyperrectangle (also called a box, hyperbox,

-cell or orthotope[2]), 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.

-dimensional rectangular solid has each of its edges equal to one of the closed intervals used in the definition.

[4][5] If all of the edges are equal length, it is a hypercube.

A hyperrectangle is a special case of a parallelotope.

The set of all points

whose coordinates satisfy the inequalities

A 2-cell is the rectangle formed by the Cartesian product of two closed intervals, and a 3-cell is a rectangular solid.

-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.

A four-dimensional orthotope is likely a hypercuboid.

[7] The special case of an n-dimensional orthotope where all edges have equal length is the n-cube or hypercube.

[2] By analogy, the term "hyperrectangle" can refer to Cartesian products of orthogonal intervals of other kinds, such as ranges of keys in database theory or ranges of integers, rather than real numbers.

[8] The dual polytope of an n-orthotope has been variously called a rectangular n-orthoplex, rhombic n-fusil, or n-lozenge.

It is constructed by 2n points located in the center of the orthotope rectangular faces.

An n-fusil's Schläfli symbol can be represented by a sum of n orthogonal line segments: { } + { } + ... + { } or n{ }.

A 1-fusil is a line segment.

Its plane cross selections in all pairs of axes are rhombi.