Hyperrectangle

For every integer $i$ from $1$ to $k$, let $a\_i$ and $b\_i$ be real numbers such that . The set of all points $x=(x\_1,\backslash dots,x\_k)$ in $\backslash mathbb\{R\}^k$ whose coordinates satisfy the inequalities $a\_i\backslash leq\; x\_i\backslash leq\; b\_i$ is a $k$-cell.{{harvcoltxt|Rudin|1976|p=31}}

A $k$-cell of dimension $k\backslash leq\; 3$ is especially simple. For example, a 1-cell is simply the interval $[a,b]$ with . 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 $k$-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.

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

The special case of an {{mvar|n}}-dimensional orthotope where all edges have equal length is the {{mvar|n}}-cube or hypercube.

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.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

}}.

{{-}}

{{Infobox polyhedron

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

| image = File:Rhombic 3-orthoplex.svg

| caption = Example: 3-fusil

| type = Prism

| faces = {{math|2n}}

| edges =

| vertices = {{math|2ⁿ}}

| vertex_config =

| schläfli = {{math|1={}+{}+···+{} = {{mvar|n}}{} }}

| wythoff =

| conway =

| coxeter = {{CDD|node_1|sum|node_1|sum}} ... {{CDD|sum|node_1}}

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

| 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|2n}} 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.

• Minimum bounding rectangle
• Cuboid
• Hilbert cube

{{reflist}}

• {{cite book

|last = Coxeter

|first = Harold Scott MacDonald

|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

}}

• {{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}}

{{Dimension topics}}

Category:Polytopes

Category:Prismatoid polyhedra

Category:Multi-dimensional geometry