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 ...
, 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
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
to the
Cartesian product of
intervals. If all of the edges are equal length, it is a
hypercube.
A hyperrectangle is a special case of a
parallelotope.
Types
A three-dimensional orthotope is also called a right rectangular
prism
Prism usually refers to:
* Prism (optics), a transparent optical component with flat surfaces that refract light
* Prism (geometry), a kind of polyhedron
Prism may also refer to:
Science and mathematics
* Prism (geology), a type of sedimentary ...
, rectangular
cuboid, or rectangular
parallelepiped.
The special case of an ''n''-dimensional orthotope where all edges have equal length is the ''n''-
cube.
By analogy, the term "hyperrectangle" or "box" can refer to Cartesian products of
orthogonal 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 (), 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 languag ...
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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s.
[See e.g. .]
Dual polytope
The
dual polytope
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other ...
of an ''n''-orthotope has been variously called a rectangular n-
orthoplex
In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahed ...
, rhombic ''n''-fusil, or ''n''-
lozenge. 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. 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. The ...
. Its plane cross selections in all pairs of axes are
rhombi
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. The ...
.
See also
*
Minimum bounding box
*
Cuboid
Notes
References
*
External links
*
*
{{Dimension topics
Polytopes
Prismatoid polyhedra
Multi-dimensional geometry