geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a hyperrectangle (also called a box, hyperbox,

k

-cell or orthotopeCoxeter, 1973), is the generalization of a

rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...

(a plane figure) and the

rectangular cuboid A rectangular cuboid is a special case of a cuboid with rectangular faces in which all of its dihedral angles are right angles. This shape is also called rectangular parallelepiped or orthogonal parallelepiped. Many writers just call these ...

solid figure Solid geometry or stereometry is the geometry of three-dimensional Euclidean space (3D space). A solid figure is the region of 3D space bounded by a two-dimensional closed surface; for example, a solid ball consists of a sphere and its inte ...

) to higher dimensions. A necessary and sufficient condition is that it is congruent to the

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

of finite intervals. This means that a

k

-dimensional rectangular solid has each of its edges equal to one of the closed intervals used in the definition. Every

k

-cell is

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

. If all of the edges are equal length, it is a ''

hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...

''. A hyperrectangle is a special case of a parallelotope.

Formal definition

For every integer

i

from

1

k

, let

a_i

and

b_i

real numbers In mathematics, a real number is a number that can be used to measurement, measure a continuous variable, continuous one-dimensional quantity such as a time, duration or temperature. Here, ''continuous'' means that pairs of values can have arbi ...

such that

a_i < b_i

. The set of all points

x=(x_1,\dots,x_k)

\mathbb^k

whose coordinates satisfy the inequalities

a_i\leq x_i\leq b_i

is a

k

-cell.

Intuition

k

-cell of dimension

k\leq 3

is especially simple. For example, a 1-cell is simply the interval

,b /math> with 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 k -cell need not be equal in (Euclidean) length; although the

unit cube A unit cube, more formally a cube of side 1, is a cube whose sides are 1 unit long.. See in particulap. 671. The volume of a 3-dimensional unit cube is 1 cubic unit, and its total surface area is 6 square units.. Unit hypercube The term '' ...

(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. The special case of an -dimensional orthotope where all edges have equal length is the -

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

or hypercube. By analogy, the term "hyperrectangle" can refer to Cartesian products of

orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...

intervals of other kinds, such as ranges of keys in

database theory Database theory encapsulates a broad range of topics related to the study and research of the theoretical realm of databases and database management systems. Theoretical aspects of data management include, among other areas, the foundations of q ...

or ranges of

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

s, rather than

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s.See e.g. .

Dual polytope

The dual polytope of an -orthotope has been variously called a rectangular -

orthoplex In geometry, a cross-polytope, hyperoctahedron, orthoplex, staurotope, or cocube is a regular polytope, regular, convex polytope that exists in ''n''-dimensions, dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensi ...

, rhombic - fusil, or - lozenge. It is constructed by points located in the center of the orthotope rectangular faces. An -fusil's

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...

can be represented by a sum of orthogonal line segments: or A 1-fusil is a

line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...

. A 2-fusil is a

rhombus In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhom ...

. Its plane cross selections in all pairs of axes are

rhombi In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhom ...

Notes

References

External links

* * * {{Dimension topics Polytopes Prismatoid polyhedra Multi-dimensional geometry