Hyperrectangle
Hyperrectangle northotope 


A rectangular cuboid is a 3orthotope 

Type  Prism 
Facets  2n 
Vertices  2^{n} 
Schläfli symbol  {} × {} ... × {}^{[1]} 
CoxeterDynkin diagram  ... 
Symmetry group  [2^{n1}], order 2^{n} 
Dual  Rectangular nfusil 
Properties  convex, zonohedron, isogonal 
In geometry, an northotope^{[2]} (also called a hyperrectangle or a box) is the generalization of a rectangle for higher dimensions, formally defined as the Cartesian product of intervals.
Types
A threedimensional orthotope is also called a right rectangular prism, rectangular cuboid, or rectangular parallelepiped.
A special case of an northotope, where all edges are equal length, is the ncube.^{[2]}
By analogy, the term "hyperrectangle" or "box" refers 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
nfusil  

Example: 3fusil 

Facets  2^{n} 
Vertices  2n 
Schläfli symbol  {} + {} + ... + {} 
CoxeterDynkin diagram  ... 
Symmetry group  [2^{n1}], order 2^{n} 
Dual  northotope 
Properties  convex, isotopal 
The dual polytope of an northotope has been variously called a rectangular northoplex, rhombic nfusil, or nlozenge. It is constructed by 2n points located in the center of the orthotope rectangular faces.
An nfusil's Schläfli symbol can be represented by a sum of n orthogonal line segments: { } + { } + ... + { }.
A 1fusil is a line segment. A 2fusil is a rhombus. Its plane cross selections in all pairs of axes are rhombi.
n  Example image 

1 
{ } 
2 
{ } + { } 
3 
Rhombic 3orthoplex inside 3orthotope { } + { } + { } 
See also
Notes
 ^ N.W. Johnson: Geometries and Transformations, (2018) ISBN 9781107103405 Chapter 11: Finite symmetry groups, 11.5 Spherical Coxeter groups, p.251
 ^ ^{a} ^{b} Coxeter, 1973
 ^ 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
 Coxeter, Harold Scott MacDonald (1973). Regular Polytopes (3rd ed.). New York: Dover. pp. 122–123. ISBN 0486614808.
External links
 Weisstein, Eric W. "Rectangular parallelepiped". MathWorld.
 Weisstein, Eric W. "Orthotope". MathWorld.