Reference articles on history, science, culture and more
Surprise meSurprise meSubscribe
Encyclopedia

Hyperrectangle

Generalization of a rectangle for higher dimensions

Wikipedia contributorsAdapted for Fathomly · View authors
2 min read
Source updated May 30, 2026
Image credit is listed at the end of this article.

In geometry, a hyperrectangle (also called a box, hyperbox, k-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 k-dimensional rectangular solid has each of its edges equal to one of the closed intervals used in the definition. Every k-cell is compact.

If all of the edges are equal length, it is a hypercube. A hyperrectangle is a special case of a parallelotope.

01Formal definition

For every integer i from 1 to k, let a_{i} and b_{i} be real numbers such that a_{i}<b_{i}. The set of all points x=(x_{1},\dots ,x_{k}) in \mathbb {R} ^{k} whose coordinates satisfy the inequalities a_{i}\leq x_{i}\leq b_{i} is a k-cell.

Projections of -cells onto the plane (from ). Only the edges of the higher-dimensional cells are shown.
Projections of -cells onto the plane (from ). Only the edges of the higher-dimensional cells are shown.

02Intuition

A k-cell of dimension k\leq 3 is especially simple. For example, a 1-cell is simply the interval [a,b] with a<b. 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 k-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.

03Types

A four-dimensional orthotope is likely a hypercuboid.

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

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.

04Dual polytope

n-fusil
Example: 3-fusil
TypePrism
Faces2n
Vertices2n
Schläfli symbol{}+{}+···+{} = n{}
Coxeter diagram ...
Symmetry group[2n−1], order 2n
Dual polyhedronn-orthotope
Propertiesconvex, isotopal

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. A 2-fusil is a rhombus. Its plane cross selections in all pairs of axes are rhombi.

n Example image
1
Line segment
{ }
2
Rhombus
{ } + { } = 2{ }
3
Rhombic 3-orthoplex inside 3-orthotope
{ } + { } + { } = 3{ }
Watch videos about HyperrectangleExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Hyperrectangle”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Images, from Wikimedia Commons:

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.

Continue exploring

Related topics

Minimum bounding rectangle

In computational geometry, the minimum bounding rectangle, also known as bounding box (BBOX) or envelope, is an expression of the maximum extents of a two-dimensional object (e.g. point, line, polygon) or set of objects within its x-y coordinate system; in other words min(x), max(x), min(y), max(y).

Cuboid

In geometry, a cuboid is a hexahedron with quadrilateral faces, meaning it is a polyhedron with six faces; it has eight vertices and twelve edges. A rectangular cuboid has all right angles and equal opposite rectangular faces.

Hilbert cube

In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, can be viewed as subspaces of the Hilbert cube.