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 | |
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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 |
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]
n-fusil | |
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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 |
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1 | Line segment { } |
2 | Rhombus { } + { } = 2{ } |
3 | Rhombic 3-orthoplex inside 3-orthotope { } + { } + { } = 3{ } |