Hyperrectangle

Hyperrectangle n-orthotope
A rectangular cuboid is a 3-orthotope
Type	Prism
Facets	2n
Vertices	2ⁿ
Schläfli symbol	{} × {} ... × {}^[1]
Coxeter-Dynkin diagram	...
Symmetry group	[2ⁿ⁻¹], order 2ⁿ
Dual	Rectangular n-fusil
Properties	convex, zonohedron, isogonal

In geometry, an n-orthotope^[2] (also called a hyperrectangle or a box) is the generalization of a rectangle for higher dimensions, formally defined as the Cartesian product of intervals.

Types

A three-dimensional orthotope is also called a right rectangular prism, rectangular cuboid, or rectangular parallelepiped.

A special case of an n-orthotope, where all edges are equal length, is the n-cube.^[2]

By analogy, the term "hyperrectangle" or "box" refers 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.^[3]

Dual polytope

n-fusil
Example: 3-fusil
Facets	2n
Vertices	2ⁿ
Schläfli symbol	{} + {} + ... + {}
Coxeter-Dynkin diagram	...
Symmetry group	[2ⁿ⁻¹], order 2ⁿ
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: { } + { } + ... + { }.

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	{ }
2	{ } + { }
3	Rhombic 3-orthoplex inside 3-orthotope { } + { } + { }

Notes

^ 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
^ 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.

References

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

External links

[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] 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.

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