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In geometry, a hyperrectangle (also called a box, hyperbox, k {\displaystyle k} -cell or orthotopecite-ref-regpoly-2-0[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.cite-ref-3[3] This means that a k {\displaystyle k} -dimensional rectangular solid has each of its edges equal to one of the closed intervals used in the definition. Every k {\displaystyle k} -cell is compact.cite-ref-4[4]cite-ref-5[5]

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

• Formal definition

• Intuition

• Types

• Dual polytope

• See also

• Notes

• References

• External links

For every integer i {\displaystyle i} from 1 {\displaystyle 1} to k {\displaystyle k} , let a i {\displaystyle a_{i}} and b i {\displaystyle b_{i}} be real numbers such that a i < b i {\displaystyle a_{i}<b_{i}} . The set of all points x = ( x 1 , … … , x k ) {\displaystyle x=(x_{1},\dots ,x_{k})} in R k {\displaystyle \mathbb {R} ^{k}} whose coordinates satisfy the inequalities a i ≤ ≤ x i ≤ ≤ b i {\displaystyle a_{i}\leq x_{i}\leq b_{i}} is a k {\displaystyle k} -cell.cite-ref-6[6]

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

A four-dimensional orthotope is likely a hypercuboid.cite-ref-7[7]

The special case of an n-dimensional orthotope where all edges have equal length is the n-cube or hypercube.cite-ref-regpoly-2-1[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.cite-ref-8[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. 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{ } |

• Minimum bounding rectangle

• Cuboid

• Hilbert cube

cite-note-johnson-11. 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

cite-note-regpoly-22. ↑ Coxeter, 1973

cite-note-33. ↑ Foran (1991)

cite-note-44. ↑ Rudin (1976:39)

cite-note-55. ↑ Foran (1991:24)

cite-note-66. ↑ Rudin (1976:31)

cite-note-77. ↑ citerefhirotsu2022Hirotsu, Takashi (2022). "Normal-sized hypercuboids in a given hypercube". arXiv:2211.15342 [math.CO].

cite-note-88. ↑ See e.g. citerefzhangmunagalayang2011Zhang, 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.

• citerefcoxeter1973Coxeter, Harold Scott MacDonald (1973). Regular Polytopes (3rd ed.). New York: Dover. pp. 122–123. ISBN 0-486-61480-8.

• reference-mathworld-orthotopeciterefweissteinWeisstein, Eric W. "Orthotope". MathWorld.

• citerefforan1991Foran, James (1991-01-07). Fundamentals of Real Analysis. CRC Press. ISBN 9780824784539. Retrieved 23 May 2014.

• citerefrudin1976Rudin, Walter (1976). Principles of Mathematical Analysis. McGraw-Hill.