Hyperrectangle: Difference between revisions - Wikipedia

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{{short description|Generalization of a rectangle for higher dimensions}}

{{Merge|K-cell (mathematics)|discuss=Talk:K-cell (mathematics)#Proposed merge of Hyperrectangle with K-cell (mathematics)|date=July 2024}}

{| cellpadding="5" style="background:#fff; float:right; margin-left:10px; width:250px;" class="wikitable"

{{Infobox polyhedron

|-

!| name style="background:#e7dcc3;" colspan="2"| Hyperrectangle<br>Orthotope

| image = Cuboid no label.svg

|-

|colspan=2|[[Image:Cuboid nocaption label.svg|240px|Rectangular cuboid]]<br> = A rectangular [[cuboid]] is a 3-orthotope

| styletype ="background:#aeae74;"|Type|| [[Prism (geometry)|Prism]]▼

|-

| faces = {{math|2''n''}}

▲| style="background:#aeae74;"|Type||[[Prism (geometry)|Prism]]

| edges = {{math|''n'' × 2^{''n''&minus;1}}}

|-

| vertices = {{math|2^''n''}}

| style="background:#e7dcc3;"|[[Facet (geometry)|Facets]]||2''n''

| vertex_config = <!--list faces around a vertex-->

|-

| styleschläfli ="background:#e7dcc3;"|[[Schläfli symbol]]|{{math|1={}×{}×···×{} = {}^''n''}}<ref name=johnson>[[Norman Johnson (mathematician)|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>▼

| style="background:#e7dcc3;"|[[Edge (geometry)|Edges]]||''n''×2^''n''-1

| wythoff = <!--Wythoff symbol-->

|-

| conway = <!--Conway polyhedron notation-->

| style="background:#e7dcc3;"|[[Vertex (geometry)|Vertices]]||2^''n''

| stylecoxeter ="background:#e7dcc3;"|[[Coxeter-Dynkin diagram]]||{{CDD|node_1|2|node_1}}···{{CDD|node_1}}▼

|-

| symmetry = {{math|[2^''n''−1]}}, order {{math|2^''n''}}

▲| style="background:#e7dcc3;"|[[Schläfli symbol]]||{}×{}×···×{} = {}^''n''<ref name=johnson>[[Norman Johnson (mathematician)|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>

| rotation_group = <!--rotation group-->

|-

| surface_area = <!--some simple formula(e)-->

▲| style="background:#e7dcc3;"|[[Coxeter-Dynkin diagram]]||{{CDD|node_1|2|node_1}}···{{CDD|node_1}}

| volume = <!--some simple formula(e)-->

|-

| angle = <!--dihedral angle-->

| style="background:#e7dcc3;"|[[Coxeter notation|Symmetry group]]||[2^''n''−1], order 2^''n''

| dual = [[#Dual polytope|Rectangular {{mvar|n}}-fusil]]

|-

| styleproperties ="background:#e7dcc3;"|Properties|| [[Convex polyhedron|convex]], [[zonohedron]], [[isogonal figure|isogonal]]▼

| style="background:#e7dcc3;"|[[Dual polytope|Dual]]||[[#Dual polytope|Rectangular ''n''-fusil]]

|}}▼

|-

▲| style="background:#e7dcc3;"|Properties||[[Convex polyhedron|convex]], [[zonohedron]], [[isogonal figure|isogonal]]

▲|}

In [[geometry]], ana '''hyperrectangle''' or(also called a '''orthotopebox'''<ref name=regpoly>Coxeter, 1973</ref>'''hyperbox''', sometimes called aor '''hyperboxorthotope'''<ref name="Kovalerchuk2018"regpoly>{{citeCoxeter, book | last=Kovalerchuk | first=B. | title=Visual Knowledge Discovery and Machine Learning | publisher=Springer International Publishing | series=Intelligent Systems Reference Library | year=2018 | isbn=978-3-319-73040-0 | url=https://books.google.com.br/books?id=l8FHDwAAQBAJ&pg=PA115 | access-date=2023-09-25 | page=115}}1973</ref> or simply a "box"), is the generalization of a [[rectangle]] (a [[plane figure]]) and the [[rectangular cuboid]] (a [[solid figure]]) to [[higher dimension]]s.

A [[necessary and sufficient condition]] is that it is [[Congruence (geometry)|congruent]] to the [[Cartesian product]] of finite [[interval (mathematics)|intervals]]. If all of the edges are equal length, it is a ''[[hypercube]]''.

A hyperrectangle is a special case of a [[parallelohedron#Related shapes|parallelotope]].

==Types==

A four-dimensional orthotope is likely a hypercuboid.<ref>{{cnCite arXiv| title=Normal-sized hypercuboids in a given hypercube | last1=Hirotsu | first1=Takashi | date=2022 | arxiv=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 [[integer]]s, rather than [[real number]]s.<ref>See e.g. {{citation

Line 44 ⟶ 42:

| url = http://www.vldb.org/pvldb/vol4/p1075-zhang.pdf

| volume = 4

| year = 2011}}| doi = 10.<14778/ref>3402707.3402743

}}.</ref>

{{-}}

==Dual polytope==

{{Infobox polyhedron

{| cellpadding="5" style="background:#fff; float:right; margin-left:10px; width:250px;" class="wikitable"

| name = {{mvar|n}}-fusil

|-

| image = File:Rhombic 3-orthoplex.svg

! style="background:#e7dcc3;" colspan="2"|''n''-fusil

| caption = Example: 3-fusil

|-

| type = [[Prism (geometry)|Prism]]

|colspan=2 align=center|[[File:Rhombic 3-orthoplex.svg|240px|Rectangular fusil]]<br>Example: 3-fusil

| faces = {{math|2''n''}}

|-

| edges =

| style="background:#e7dcc3;"|[[Facet (geometry)|Facets]]||2^''n''

| vertices = {{math|2^''n''}}

|-

| vertex_config = <!--list faces around a vertex-->

| style="background:#e7dcc3;"|[[Vertex (geometry)|Vertices]]||2''n''

| schläfli = {{math|1={}+{}+···+{} = {{mvar|n}}{} }}<ref name=johnson/>

|-

| wythoff = <!--Wythoff symbol-->

| style="background:#e7dcc3;"|[[Schläfli symbol]]||{}+{}+···+{} = ''n''{}<ref name=johnson/>

| conway = <!--Conway polyhedron notation-->

|-

| stylecoxeter ="background:#e7dcc3;"|[[Coxeter-Dynkin diagram]]||{{CDD|node_1|sum|node_1|sum}} ... {{CDD|sum|node_1}}

| symmetry = {{math|[2^''n''−1]}}, order {{math|2^''n''}}

|-

| rotation_group = <!--rotation group-->

| style="background:#e7dcc3;"|[[Coxeter notation|Symmetry group]]||[2^''n''−1], order 2^''n''

| surface_area = <!--some simple formula(e)-->

|-

| volume = <!--some simple formula(e)-->

| style="background:#e7dcc3;"|[[Dual polytope|Dual]]||''n''-orthotope

| angle = <!--dihedral angle-->

|-

| dual = {{mvar|n}}-orthotope

| styleproperties ="background:#e7dcc3;"|Properties|| [[Convex polyhedron|convex]], [[isotopalIsotopal figure|isotopal]]

|}}

The [[dual polytope]] of an ''n''-orthotope has been variously called a rectangular n-[[orthoplex]], rhombic ''n''-[[Fusil (geometry)|fusil]], or ''n''-[[Lozenge (shape)|lozenge]]. It is constructed by 2''n'' points located in the center of the orthotope rectangular faces.▼

▲The [[dual polytope]] of an ''{{mvar|n''}}-orthotope has been variously called a rectangular {{mvar|n}}-[[orthoplex]], rhombic ''{{mvar|n''}}-[[Fusil (geometry)|fusil]], or ''{{mvar|n''}}-[[Lozenge (shape)|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: {&nbsp;{math|{ } + {&nbsp; } + ... + {&nbsp; } }} or {{math|''n''{&nbsp; }.}}

A 1-fusil is a [[line segment]]. A 2-fusil is a [[rhombus]]. Its plane cross selections in all pairs of axes are [[rhombus|rhombi]].

{| class=wikitable

! {{mvar|n}}

!n

! Example image

|- align=center

!1

|[[File:Cross graph 1.svg|160px]]<br>[[Line segment]]<br>{{math|{&nbsp; } }}]]<br>{{CDD|node_1}}

|- align=center

!2

|[[File:Rhombus (polygon).png|160px]]<br>[[Rhombus]]<br>{{math|1={&nbsp; } + {&nbsp; }]] = 2{&nbsp; } }}<br>{{CDD|node_1|sum|node_1}}

|- align=center

!3

|[[File:Dual orthotope-orthoplex.svg|160px]]<br>Rhombic 3-orthoplex inside [[cuboid|3-orthotope]]<br>{&nbsp;{math|1={ } + {&nbsp; } + {&nbsp; } = 3{&nbsp; } }}<br>{{CDD|node_1|sum|node_1|sum|node_1}}

|}

==See also==

* [[Minimum bounding boxrectangle]]

* [[Cuboid]]