In geometry, a hyperrectangle (also called a box, hyperbox, 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. If all of the edges are equal length, it is a hypercube.A hyperrectangle is a special case of a parallelotope.
| Hyperrectangle Orthotope | |
|---|---|
 A rectangular cuboid is a 3-orthotope | |
| Type | Prism |
| Faces | 2n |
| Edges | n × 2n−1 |
| Vertices | 2n |
| Schläfli symbol | {}×{}×···×{} = {}n[1] |
| Coxeter diagram |   ··· |
| Symmetry group | [2n−1], order 2n |
| Dual polyhedron | Rectangular n-fusil |
| Properties | convex, zonohedron, isogonal |
Types
A four-dimensional orthotope is likely a hypercuboid.[3]
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.[4]
Dual polytope
| n-fusil | |
|---|---|
 Example: 3-fusil | |
| Type | Prism |
| Faces | 2n |
| Vertices | 2n |
| Schläfli symbol | {}+{}+···+{} = n{}[1] |
| Coxeter diagram |  ... |
| Symmetry group | [2n−1], order 2n |
| Dual polyhedron | n-orthotope |
| Properties | convex, isotopal |
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.
| n | Example image |
|---|---|
| 1 |  Line segment { } |
| 2 | .png) Rhombus { } + { } = 2{ } |
| 3 |  Rhombic 3-orthoplex inside 3-orthotope { } + { } + { } = 3{ } |
See also
Notes
References
- Coxeter, Harold Scott MacDonald (1973). Regular Polytopes (3rd ed.). New York: Dover. pp. 122–123. ISBN 0-486-61480-8.