Hyperrectangle

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

In geometry, an n-orthotope^[1] (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.^[1]

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.^[2]

Dual polytope

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

See also

• Minimum bounding box

Notes

1. Coxeter, 1973
2. 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

• Weisstein, Eric W., "Rectangular parallelepiped", MathWorld.
• Weisstein, Eric W., "Orthotope", MathWorld.