In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, an orthotope
[Coxeter, 1973] (also called a hyperrectangle or a box) is the generalization of a
rectangle 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 ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ ...
of
intervals. 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 (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions ...
.
A hyperrectangle is a special case of a
parallelotope.
Types
A three-dimensional orthotope is also called a right rectangular
prism, rectangular
cuboid, or rectangular
parallelepiped.
The special case of an ''n''-dimensional orthotope where all edges have equal length is the ''n''-
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the on ...
.
By analogy, the term "hyperrectangle" or "box" can refer to Cartesian products of
orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
intervals of other kinds, such as ranges of keys in
database theory or ranges of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s, rather than
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s.
[See e.g. .]
Dual polytope
The
dual polytope of an ''n''-orthotope has been variously called a rectangular n-
orthoplex, rhombic ''n''-fusil, or ''n''-
lozenge
Lozenge or losange may refer to:
*Lozenge (shape), a type of rhombus
*Throat lozenge, a tablet intended to be dissolved slowly in the mouth to suppress throat ailments
*Lozenge (heraldry), a diamond-shaped object that can be placed on the field of ...
. It is constructed by 2''n'' 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
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
. A 2-fusil is a
rhombus
In plane Euclidean geometry, a rhombus (plural 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. Th ...
. Its plane cross selections in all pairs of axes are
rhombi.
See also
*
Minimum bounding box
*
Cuboid
Notes
References
*
External links
*
*
{{Dimension topics
Polytopes
Prismatoid polyhedra
Multi-dimensional geometry