Hyperrectangle

In geometry, a hyperrectangle (also called a box, hyperbox, ${\displaystyle k}${\displaystyle k}-cell or orthotope^[2] ), 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.^[3] This means that a ${\displaystyle k}${\displaystyle k}-dimensional rectangular solid has each of its edges equal to one of the closed intervals used in the definition. Every ${\displaystyle k}${\displaystyle k}-cell is compact.^{[4] [5]}

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

For every integer ${\displaystyle i}${\displaystyle i} from ${\displaystyle 1}${\displaystyle 1} to ${\displaystyle k}${\displaystyle k}, let ${\displaystyle a_{i}}${\displaystyle a_{i}} and ${\displaystyle b_{i}}${\displaystyle b_{i}} be real numbers such that ${\displaystyle a_{i}<b_{i}}${\displaystyle a_{i}<b_{i}}. The set of all points ${\displaystyle x=(x_{1},\dots ,x_{k})}${\displaystyle x=(x_{1},\dots ,x_{k})} in ${\displaystyle \mathbb {R} ^{k}}${\displaystyle \mathbb {R} ^{k}} whose coordinates satisfy the inequalities ${\displaystyle a_{i}\leq x_{i}\leq b_{i}}${\displaystyle a_{i}\leq x_{i}\leq b_{i}} is a ${\displaystyle k}${\displaystyle k}-cell.^[6]

A ${\displaystyle k}${\displaystyle k}-cell of dimension ${\displaystyle k\leq 3}${\displaystyle k\leq 3} is especially simple. For example, a 1-cell is simply the interval ${\displaystyle [a,b]}${\displaystyle [a,b]} with ${\displaystyle a<b}${\displaystyle a<b}. 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 ${\displaystyle k}${\displaystyle 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.^[7]

The special case of an n-dimensional orthotope where all edges have equal length is the n-cube or hypercube.^[2]

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.^[8]

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.

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

• Coxeter, Harold Scott MacDonald (1973). Regular Polytopes (3rd ed.). New York: Dover. pp. 122–123. ISBN 0-486-61480-8.

• Weisstein, Eric W. "Orthotope". MathWorld .
• Foran, James (1991年01月07日). Fundamentals of Real Analysis. CRC Press. ISBN 9780824784539 . Retrieved 23 May 2014.
• Rudin, Walter (1976). Principles of Mathematical Analysis . McGraw-Hill.

• Polytopes
• Prismatoid polyhedra
• Multi-dimensional geometry

• Articles with short description
• Short description matches Wikidata