# Hyperrectangle

Hyperrectangle
Orthotope
A rectangular cuboid is a 3-orthotope
TypePrism
Faces2n
Edgesn × 2n−1
Vertices2n
Schläfli symbol{}×{}×···×{} = {}n[1]
Coxeter diagram···
Symmetry group[2n−1], order 2n
Dual polyhedronRectangular n-fusil
Propertiesconvex, zonohedron, isogonal

In geometry, a hyperrectangle (also called a box, hyperbox, or orthotope[2]), is the generalization of a rectangle (a plane figure) and the rectangular cuboid (a solid figure) to higher dimensions. A necessary and sufficient condition is that it is congruent to the Cartesian product of finite intervals. If all of the edges are equal length, it is a hypercube. A hyperrectangle is a special case of a parallelotope.

## 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-cube or hypercube.[2]

By analogy, the term "hyperrectangle" can refer 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.[4]

## Dual polytope

n-fusil
Example: 3-fusil
TypePrism
Faces2n
Vertices2n
Schläfli symbol{}+{}+···+{} = n{}[1]
Coxeter diagram ...
Symmetry group[2n−1], order 2n
Dual polyhedronn-orthotope
Propertiesconvex, 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: { } + { } + ... + { } or n{ }.

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
Line segment
{ }
2
Rhombus
{ } + { } = 2{ }
3
Rhombic 3-orthoplex inside 3-orthotope
{ } + { } + { } = 3{ }