Hyperrectangle

Hyperrectangle n-orthotope
A rectangular cuboid is a 3-orthotope
Type	Prism
Facets	2n
Vertices	2n
Schläfli symbol	{} × {} ... × {}
Coxeter-Dynkin diagram	...
Symmetry group	[2n-1], order 2n
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	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 { } + { } + { }

See also

• Minimum bounding box

Notes

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. 

Dimension
Dimensional spaces	Vector space Euclidean space Affine space Projective space Free module Manifold Algebraic variety Spacetime
Other dimensions	Krull Homological Lebesgue covering Inductive Hausdorff Minkowski Fractal Degrees of freedom
Polytopes and shapes	Hyperplane Hypersurface Hypercube Hypersphere Hyperrectangle Demihypercube Cross-polytope Simplex
Dimensions by number	Zero One Two Three Four Five Six (degrees of freedom) Seven Eight Nine Ten n-dimensions Negative dimensions
Category