Hyperrectangle

{{short description|Generalization of a rectangle for higher dimensions}} {{Infobox polyhedron | name = Hyperrectangle Orthotope | image = Cuboid no label.svg | caption = A rectangular cuboid is a 3-orthotope | type = Prism | faces = {{math|2''n''}} | edges = {{math|''n'' × 2''n''−1}} | vertices = {{math|2''n''}} | vertex_config =  | schläfli = {{math|1={}×{}×···×{} = {}''n''}}<ref name=johnson>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</ref> | wythoff =  | conway =  | coxeter = {{CDD|node_1|2|node_1}}···{{CDD|node_1}} | symmetry = {{math|[2''n''−1]}}, order {{math|2''n''}} | rotation_group =  | surface_area =  | volume =  | angle =  | dual = Rectangular {{mvar|n}}-fusil | properties = convex, zonohedron, isogonal }} thumb|Projections of <math>k</math>-cells onto the plane (from <math>k\in\{1,\dots{},6\}</math>). Only the edges of the higher-dimensional cells are shown. In geometry, a '''hyperrectangle''' (also called a '''box''', '''hyperbox''', '''<math>k</math>-cell''' or '''orthotope'''<ref name=regpoly>Coxeter, 1973</ref>), 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.<ref>{{harvcoltxt|Foran|1991}}</ref> This means that a <math>k</math>-dimensional rectangular solid has each of its edges equal to one of the closed intervals used in the definition. Every <math>k</math>-cell is compact.<ref>{{harvcoltxt|Rudin|1976|p=39}}</ref><ref>{{harvcoltxt|Foran|1991|p=24}}</ref>

If all of the edges are equal length, it is a ''hypercube''. A hyperrectangle is a special case of a parallelotope.

== Formal definition == For every integer <math>i</math> from <math>1</math> to <math>k</math>, let <math>a_i</math> and <math>b_i</math> be real numbers such that <math>a_i < b_i</math>. The set of all points <math>x=(x_1,\dots,x_k)</math> in <math>\mathbb{R}^k</math> whose coordinates satisfy the inequalities <math>a_i\leq x_i\leq b_i</math> is a '''<math>k</math>-cell'''.<ref>{{harvcoltxt|Rudin|1976|p=31}}</ref>

== Intuition == A <math>k</math>-cell of dimension <math>k\leq 3</math> is especially simple. For example, a 1-cell is simply the interval <math>[a,b]</math> with <math>a < b</math>. A 2-cell is the rectangle formed by the Cartesian product of two closed intervals, and a 3-cell is a rectangular solid.

The sides and edges of a <math>k</math>-cell need not be equal in (Euclidean) length; although the unit cube (which has boundaries of equal Euclidean length) is a 3-cell, the set of all 3-cells with equal-length edges is a strict subset of the set of all 3-cells.

==Types== A four-dimensional orthotope is likely a hypercuboid.<ref>{{Cite arXiv| title=Normal-sized hypercuboids in a given hypercube | last1=Hirotsu | first1=Takashi | date=2022 | class=math.CO | eprint=2211.15342 }}</ref>

The special case of an {{mvar|n}}-dimensional orthotope where all edges have equal length is the {{mvar|n}}-cube or hypercube.<ref name=regpoly />

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.<ref>See e.g. {{citation | last1 = Zhang | first1 = Yi | last2 = Munagala | first2 = Kamesh | last3 = Yang | first3 = Jun | issue = 11 | journal = Proc. VLDB | pages = 1075–1086 | title = Storing matrices on disk: Theory and practice revisited | url = http://www.vldb.org/pvldb/vol4/p1075-zhang.pdf | volume = 4 | year = 2011| doi = 10.14778/3402707.3402743 }}.</ref>

==Dual polytope== {{Infobox polyhedron | name = {{mvar|n}}-fusil | image = File:Rhombic 3-orthoplex.svg | caption = Example: 3-fusil | type = Prism | faces = {{math|2''n''}} | edges = | vertices = {{math|2''n''}} | vertex_config =  | schläfli = {{math|1={}+{}+···+{} = {{mvar|n}}{} }}<ref name=johnson/> | wythoff =  | conway =  | coxeter = {{CDD|node_1|sum|node_1|sum}} ... {{CDD|sum|node_1}} | symmetry = {{math|[2''n''−1]}}, order {{math|2''n''}} | rotation_group =  | surface_area =  | volume =  | angle =  | dual = {{mvar|n}}-orthotope | properties = convex, isotopal }}

The dual polytope of an {{mvar|n}}-orthotope has been variously called a rectangular {{mvar|n}}-orthoplex, rhombic {{mvar|n}}-fusil, or {{mvar|n}}-lozenge. It is constructed by {{math|2''n''}} points located in the center of the orthotope rectangular faces.

An {{mvar|n}}-fusil's Schläfli symbol can be represented by a sum of {{mvar|n}} orthogonal line segments: {{math|{ } + { } + ... + { } }} or {{math|''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.

{| class=wikitable ! {{mvar|n}} ! Example image |- align=center !1 |160px Line segment {{math|{ } }} {{CDD|node_1}} |- align=center !2 |160px Rhombus {{math|1={ } + { } = 2{ } }} {{CDD|node_1|sum|node_1}} |- align=center !3 |160px Rhombic 3-orthoplex inside 3-orthotope {{math|1={ } + { } + { } = 3{ } }} {{CDD|node_1|sum|node_1|sum|node_1}} |}

==See also== * Minimum bounding rectangle * Cuboid * Hilbert cube

==Notes== {{reflist}}

== References == * {{cite book |last = Coxeter |first = Harold Scott MacDonald |author-link=H. S. M. Coxeter |title = Regular Polytopes |edition = 3rd |location = New York |publisher = Dover |year = 1973 |pages = [https://archive.org/details/regularpolytopes0000coxe/page/122 122–123] |isbn = 0-486-61480-8 }}

== External links == * {{MathWorld|title=Orthotope|urlname=Orthotope}} * {{cite book|last=Foran|first=James|title=Fundamentals of Real Analysis|url=https://books.google.com/books?id=sDjz8x0hJ44C&pg=PA24|accessdate=23 May 2014|date=1991-01-07|publisher=CRC Press|isbn=9780824784539}} * {{cite book|first=Walter|last=Rudin|author-link1=Walter Rudin|title=Principles of Mathematical Analysis|year=1976|publisher=McGraw-Hill}}

Category:Polytopes Category:Prismatoid polyhedra Category:Multi-dimensional geometry