Hyperrectangle

In geometry, an 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.

Hyperrectangle
Orthotope
Rectangular cuboid
A rectangular cuboid is a 3-orthotope
TypePrism
Facets2n
Edgesn×2n-1
Vertices2n
Schläfli symbol{}×{}×···×{} = {}n[1]
Coxeter-Dynkin diagram···
Symmetry group[2n−1], order 2n
DualRectangular n-fusil
Propertiesconvex, zonohedron, isogonal

Types

A four-dimensional orthotope is likely a hypercuboid.

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.[3]

Dual polytope
n-fusil
Rectangular fusil
Example: 3-fusil
Facets2n
Vertices2n
Schläfli symbol{}+{}+···+{} = n{}[1]
Coxeter-Dynkin diagram ...
Symmetry group[2n−1], order 2n
Dualn-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
{ }
2
{ } + { } = 2{ }
3
Rhombic 3-orthoplex inside 3-orthotope
{ } + { } + { } = 3{ }

See also

Notes
  1. N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite symmetry groups, 11.5 Spherical Coxeter groups, p.251
  2. Coxeter, 1973
  3. 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
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