Hyperrectangle

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Hyperrectangle
n-orthotope
Rectangular cuboid
A rectangular cuboid is a 3-orthotope
Type Prism
Facets 2n
Vertices 2n
Symmetry group [2n-1], order 2n
Schläfli symbol { }n
Coxeter-Dynkin diagram CDel node 1.pngCDel 2.pngCDel node 1.png ... CDel node 1.png
Dual Rectangular n-fusil
Properties convex, zonohedron, isogonal

In geometry, an orthotope[1] (also called a hyperrectangle or a box) is the generalization of a rectangle for higher dimensions, formally defined as the Cartesian product of intervals.

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

A special case of an n-orthotope, where all edges are equal length, is the n-hypercube.[1]

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.[citation needed]

Dual polytope[edit]

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. Its plane cross selections in all pairs of axes are rhombi. An n-fusil's Schläfli symbol can be represented by a sum of n orthogonal line segments: { } + { } + ... + { }

n Example image
1 1-simplex t0.svg
{ }
CDel node f1.png
2 Rhombus (polygon).png
{ } + { }
CDel node f1.pngCDel 2.pngCDel node f1.png
3 Rhombic 3-orthoplex.svgDual orthotope-orthoplex.svg
Example rhombic 3-orthoplex inside 3-orthotope
{ } + { } + { }
CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.png

See also[edit]

Notes[edit]

  1. ^ a b Coxeter, 1973

References[edit]

External links[edit]