Rectangular cuboid

A rectangular cuboid is a 3-orthotope
Type Prism
Facets 2n
Edges n×2n-1
Vertices 2n
Schläfli symbol {}×{}×···×{} = {}n[1]
Coxeter-Dynkin diagram CDel node 1.pngCDel 2.pngCDel node 1.png···CDel node 1.png
Symmetry group [2n−1], order 2n
Dual Rectangular n-fusil
Properties convex, zonohedron, isogonal

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.

A hyperrectangle is a special case of a parallelotope.


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

Rectangular fusil

Example: 3-fusil
Facets 2n
Vertices 2n
Schläfli symbol {}+{}+···+{} = n{}[1]
Coxeter-Dynkin diagram CDel node 1.pngCDel sum.pngCDel node 1.pngCDel sum.png ... CDel sum.pngCDel node 1.png
Symmetry group [2n−1], order 2n
Dual n-orthotope
Properties convex, 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
Cross graph 1.svg

{ }
CDel node 1.png
Rhombus (polygon).png

{ } + { } = 2{ }
CDel node 1.pngCDel sum.pngCDel node 1.png
Dual orthotope-orthoplex.svg

Rhombic 3-orthoplex inside 3-orthotope
{ } + { } + { } = 3{ }
CDel node 1.pngCDel sum.pngCDel node 1.pngCDel sum.pngCDel node 1.png

See also


  1. ^ a b 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. ^ a b 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.