Hyperrectangle

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

In geometry, an 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.

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-hypercube.^[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.^{[citation needed]}

Dual polytope

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. Its plane cross selections in all pairs of axes are rhombi. An n-fusil's Schläfli symbol can be represented by a sum of n orthogonal line segments: { } + { } + ... + { }

n	Example image
1	{ }
2	{ } + { }
3	Example 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.

v t e Dimension Dimensional spaces Vector space Euclidean space Affine space Projective space Free module Manifold Algebraic variety Spacetime Other dimensions Degrees of freedom Krull dimension Homological dimension (disambiguation) Lebesgue covering dimension Inductive dimension Hausdorff dimension Minkowski dimension Fractal dimension Polytopes and Shapes Hyperplane Hypersurface Hypercube Hypersphere Hyperrectangle Demihypercube Cross-polytope Simplex Value of the dimension Zero One Two Three Four Five Six Six degrees of freedom Seven Eight n-dimensions Category