Hyperrectangle

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title: "Hyperrectangle" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["polytopes", "prismatoid-polyhedra", "multi-dimensional-geometry"] description: "Generalization of a rectangle for higher dimensions" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Hyperrectangle" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Generalization of a rectangle for higher dimensions ::

::data[format=table title="Infobox polyhedron"]

Field	Value
name	Hyperrectangle
Orthotope
image	Cuboid no label.svg
caption	A rectangular cuboid is a 3-orthotope
type	Prism
faces	2n
edges	n × 2n−1
vertices	2n
vertex_config
schläfli
wythoff
conway
coxeter	···
symmetry	[2n−1], order 2n
rotation_group
surface_area
volume
angle
dual	Rectangular n-fusil
properties	convex, zonohedron, isogonal
::

| name = Hyperrectangle Orthotope | image = Cuboid no label.svg | caption = A rectangular cuboid is a 3-orthotope | type = Prism | faces = 2n | edges = n × 2n−1 | vertices = 2n | vertex_config = | schläfli = | wythoff = | conway = | coxeter = ··· | symmetry = [2n−1], order 2n | rotation_group = | surface_area = | volume = | angle = | dual = Rectangular n-fusil | properties = convex, zonohedron, isogonal ::figure[src="https://upload.wikimedia.org/wikipedia/commons/3/3d/N-wymiarowe_sześciany.svg" caption="Projections of K-cells onto the plane (from k=1 to 6). Only the edges of the higher-dimensional cells are shown."] ::

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.

Formal definition

For every integer i from 1 to k, let a_i and b_i be real numbers such that a_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.

Intuition

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

Types

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.See e.g. {{citation | last1 = Zhang | first1 = Yi | last2 = Munagala | first2 = Kamesh | last3 = Yang | first3 = Jun | issue = 11 | journal = Proc. VLDB | pages = 1075–1086 | title = Storing matrices on disk: Theory and practice revisited | url = http://www.vldb.org/pvldb/vol4/p1075-zhang.pdf | volume = 4 | year = 2011| doi = 10.14778/3402707.3402743

Dual polytope

| name = n-fusil | image = File:Rhombic 3-orthoplex.svg | caption = Example: 3-fusil | type = Prism | faces = 2n | edges = | vertices = 2n | vertex_config = | schläfli = | wythoff = | conway = | coxeter = ... | symmetry = [2n−1], order 2n | rotation_group = | surface_area = | volume = | angle = | dual = n-orthotope | properties = convex, 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.

::data[format=table]

n	Example image	1	2	3
[[File:Cross graph 1.svg	160px]]
Line segment
{ }
[[File:Rhombus (polygon).png	160px]]
Rhombus

| | | | | | [[File:Dual orthotope-orthoplex.svg|160px]] Rhombic 3-orthoplex inside 3-orthotope

| | | | | ::

Notes

References

• {{cite book |last = Coxeter |first = Harold Scott MacDonald |title = Regular Polytopes |edition = 3rd |location = New York |publisher = Dover |year = 1973 |pages = 122–123 |isbn = 0-486-61480-8

References

1. [[Norman Johnson (mathematician). N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN. 978-1-107-10340-5 Chapter 11: ''Finite symmetry groups'', 11.5 Spherical Coxeter groups, p.251
2. Coxeter, 1973
3. {{harvcoltxt. Foran. 1991
4. {{harvcoltxt. Rudin. 1976
5. {{harvcoltxt. Foran. 1991
6. {{harvcoltxt. Rudin. 1976
7. (2022). "Normal-sized hypercuboids in a given hypercube".

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