Hyperrectangle

$n$ -fusil
n-fusil
	Example: 3-fusil
Type	Prism
Faces	2n
Vertices	2n
Schläfli symbol	{}+{}+···+{} =n{}
Coxeter diagram	...
Symmetry group	[2n−1],order2n
Dual polyhedron	n-orthotope
Properties	convex,isotopal

Hyperrectangle
Orthotope
Hyperrectangle; Orthotope
	A rectangularcuboidis a 3-orthotope
Type	Prism
Faces	2n
Edges	n× 2n−1
Vertices	2n
Schläfli symbol	{}×{}×···×{} = {}n
Coxeter diagram	···
Symmetry group	[2n−1],order2n
Dual polyhedron	Rectangularn-fusil
Properties	convex,zonohedron,isogonal

Ingeometry,ahyperrectangle(also called abox,hyperbox,ororthotope^[2]), is the generalization of arectangle(aplane figure) and therectangular cuboid(asolid figure) tohigher dimensions. Anecessary and sufficient conditionis that it iscongruentto theCartesian productof finiteintervals.If all of the edges are equal length, it is ahypercube. A hyperrectangle is a special case of aparallelotope.

Types

A four-dimensional orthotope is likely a hypercuboid.^[3]

The special case of an $n$ -dimensional orthotope where all edges have equal length is the $n$ -cubeor hypercube.^[2]

By analogy, the term "hyperrectangle" can refer to Cartesian products oforthogonalintervals of other kinds, such as ranges of keys indatabase theoryor ranges ofintegers,rather thanreal numbers.^[4]

Dual polytope

Thedual polytopeof 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'sSchläfli symbolcan be represented by a sum of $n$ orthogonal line segments: ${ } + { } +... + { }$ or $n { }.$

A 1-fusil is aline segment.A 2-fusil is arhombus.Its plane cross selections in all pairs of axes arerhombi.

$n$	Example image
1	Line segment ${ }$
2	Rhombus ${ } + { } = 2{ }$
3	Rhombic 3-orthoplex inside3-orthotope ${ } + { } + { } = 3{ }$

Notes

^^a ^bN.W. Johnson:Geometries and Transformations,(2018)ISBN 978-1-107-10340-5Chapter 11:Finite symmetry groups,11.5 Spherical Coxeter groups, p.251
^^a ^bCoxeter, 1973
^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.

[johnson-1] N.W. Johnson:Geometries and Transformations,(2018)ISBN 978-1-107-10340-5Chapter 11:Finite symmetry groups,11.5 Spherical Coxeter groups, p.251

[regpoly-2] Coxeter, 1973

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

[4] 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.

[1]

[2]

[3]

[4]

v t e Dimension
Dimensional spaces	Vector space Euclidean space Affine space Projective space Free module Manifold Algebraic variety Spacetime
Other dimensions	Krull Lebesgue covering Inductive Hausdorff Minkowski Fractal Degrees of freedom
Polytopesandshapes	Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Hypercomplex numbers	Real numbers( $\mathbb {R}$ ) Complex numbers( $\mathbb {C}$ ) Quaternions( $\mathbb {H}$ ) Octonions( $\mathbb {O}$ ) Sedenions( $\mathbb {S}$ ) Cayley–Dickson construction
Dimensions by number	Zero One Two Three Four Five Six Seven Eight n-dimensions
See also	Hyperspace Codimension
Category

Types

Dual polytope

See also

Notes

References

External links