Hyperrectangle

$n$ -fusil
n-fusil
	Example: 3-fusil
Type	Prism
Faces	2n
Vertices	2n
Schläfli symbol	{}+{}+···+{} = n{}
Coxeter diagram	...
Symmetry group	[2n−1], order 2n
Dual polyhedron	n-orthotope
Properties	convex, isotopal

Hyperrectangle
Orthotope
Hyperrectangle; Orthotope
	A rectangular cuboid is a 3-orthotope
Type	Prism
Faces	2n
Edges	n × 2n−1
Vertices	2n
Schläfli symbol	{}×{}×···×{} = {}n
Coxeter diagram	···
Symmetry group	[2n−1], order 2n
Dual polyhedron	Rectangular n-fusil
Properties	convex, zonohedron, isogonal

In geometry, a hyperrectangle (also called a box, hyperbox, $k$ -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.^[3] 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.^[4]^[5]

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 $a_{i}<b_{i}$ . 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.^[6]

Intuition

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

Types

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]

Dual polytope

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 $2 n$ 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 ${ } + { } + { } = 3{ }$

Notes

^ ^a ^b 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
^ ^a ^b Coxeter, 1973
^ Foran (1991)
^ Rudin (1976:39)
^ Foran (1991:24)
^ Rudin (1976:31)
^ Hirotsu, Takashi (2022). "Normal-sized hypercuboids in a given hypercube". arXiv:2211.15342.
^ See e.g. Zhang, Yi; Munagala, Kamesh; Yang, Jun (2011), "Storing matrices on disk: Theory and practice revisited" (PDF), Proc. VLDB, 4 (11): 1075–1086, doi:10.14778/3402707.3402743.

References

Coxeter, Harold Scott MacDonald (1973). Regular Polytopes (3rd ed.). New York: Dover. pp. 122–123. ISBN 0-486-61480-8.

External links

Weisstein, Eric W. "Orthotope". MathWorld.
Foran, James (1991-01-07). Fundamentals of Real Analysis. CRC Press. ISBN 9780824784539. Retrieved 23 May 2014.
Rudin, Walter (1976). Principles of Mathematical Analysis. McGraw-Hill.

[johnson-1] 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

[regpoly-2] Coxeter, 1973

[3] Foran (1991)

[4] Rudin (1976:39)

[5] Foran (1991:24)

[6] Rudin (1976:31)

[7] Hirotsu, Takashi (2022). "Normal-sized hypercuboids in a given hypercube". arXiv:2211.15342.

[8] See e.g. Zhang, Yi; Munagala, Kamesh; Yang, Jun (2011), "Storing matrices on disk: Theory and practice revisited" (PDF), Proc. VLDB, 4 (11): 1075–1086, doi:10.14778/3402707.3402743.

[2]