Hyperrectangle

In geometry, an orthotope[2] (also called a hyperrectangle or a box) is the generalization of a rectangle to higher dimensions. A necessary and sufficient condition is that it is congruent to the Cartesian product of intervals. If all of the edges are equal length, it is a hypercube.

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

A hyperrectangle is a special case of a parallelotope.

Types

A three-dimensional orthotope is also called a right rectangular prism, rectangular cuboid, or rectangular parallelepiped.

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.[2]

By analogy, the term "hyperrectangle" or "box" 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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