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.

Quick Facts Hyperrectangle Orthotope, Type ...

Hyperrectangle Orthotope
A rectangular cuboid is a 3-orthotope
Type	Prism
Faces	2n
Edges	n × 2n−1
Vertices	2n
Schläfli symbol	{}×{}×···×{} = {}n[1]
Coxeter diagram	···
Symmetry group	[2n−1], order 2n
Dual polyhedron	Rectangular n-fusil
Properties	convex, zonohedron, isogonal

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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]

Quick Facts n-fusil, Type ...

n-fusil
Example: 3-fusil
Type	Prism
Faces	2n
Vertices	2n
Schläfli symbol	{}+{}+···+{} = n{}[1]
Coxeter diagram	...
Symmetry group	[2n−1], order 2n
Dual polyhedron	n-orthotope
Properties	convex, isotopal

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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.

More information n, Example image ...

n	Example image
1	Line segment { }
2	Rhombus { } + { } = 2{ }
3	Rhombic 3-orthoplex inside 3-orthotope { } + { } + { } = 3{ }

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• Minimum bounding rectangle
• Cuboid