Hyperrectangle

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Hyperrectangle
Orthotope
Cuboid no label.svg
A rectangular cuboid is a 3-orthotope
Type Prism
Faces 2n
Edges n × 2n1
Vertices 2n
Schläfli symbol {}×{}×···×{} = {}n [1]
Coxeter diagram CDel node 1.pngCDel 2.pngCDel node 1.png···CDel node 1.png
Symmetry group [2n−1], order 2n
Dual polyhedron Rectangular n-fusil
Properties convex, zonohedron, isogonal

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.

Contents

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
Rhombic 3-orthoplex.svg
Example: 3-fusil
Type Prism
Faces 2n
Vertices 2n
Schläfli symbol {}+{}+···+{} = n{} [1]
Coxeter diagram CDel node 1.pngCDel sum.pngCDel node 1.pngCDel sum.png ... CDel sum.pngCDel node 1.png
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.

nExample image
1 Cross graph 1.svg
Line segment
{ }
CDel node 1.png
2 Rhombus (polygon).png
Rhombus
{ } + { } = 2{ }
CDel node 1.pngCDel sum.pngCDel node 1.png
3 Dual orthotope-orthoplex.svg
Rhombic 3-orthoplex inside 3-orthotope
{ } + { } + { } = 3{ }
CDel node 1.pngCDel sum.pngCDel node 1.pngCDel sum.pngCDel node 1.png

See also

Notes

  1. 1 2 N.W. Johnson: Geometries and Transformations, (2018) ISBN   978-1-107-10340-5 Chapter 11: Finite symmetry groups, 11.5 Spherical Coxeter groups, p.251
  2. 1 2 Coxeter, 1973
  3. http://ui.adsabs.harvard.edu/abs/2022arXiv221115342H/abstract
  4. See e.g. Zhang, Yi; Munagala, Kamesh; Yang, Jun (2011), "Storing matrices on disk: Theory and practice revisited" (PDF), Proc. VLDB, 4 (11): 1075–1086.

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<span class="mw-page-title-main">4-polytope</span> Four-dimensional geometric object with flat sides

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<span class="mw-page-title-main">Prism (geometry)</span> Solid with 2 parallel n-gonal bases connected by n parallelograms

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In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.

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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 octahedron, and a 4-dimensional cross-polytope is a 16-cell. Its facets are simplexes of the previous dimension, while the cross-polytope's vertex figure is another cross-polytope from the previous dimension.

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<span class="mw-page-title-main">Orthant</span>

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In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors. It describes a kaleidoscopic construction: each graph "node" represents a mirror and the label attached to a branch encodes the dihedral angle order between two mirrors, that is, the amount by which the angle between the reflective planes can be multiplied to get 180 degrees. An unlabeled branch implicitly represents order-3, and each pair of nodes that is not connected by a branch at all represents a pair of mirrors at order-2.

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In geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in n-dimensional spaces with the Schläfli symbols {4,3...3,4} and containing the symmetry of Coxeter group Rn for n ≥ 3.

In geometry, a Schläfli orthoscheme is a type of simplex. The orthoscheme is the generalization of the right triangle to simplex figures of any number of dimensions. Orthoschemes are defined by a sequence of edges that are mutually orthogonal. They were introduced by Ludwig Schläfli, who called them orthoschemes and studied their volume in Euclidean, hyperbolic, and spherical geometries. H. S. M. Coxeter later named them after Schläfli. As right triangles provide the basis for trigonometry, orthoschemes form the basis of a trigonometry of n dimensions, as developed by Schoute who called it polygonometry. J.-P. Sydler and Børge Jessen studied orthoschemes extensively in connection with Hilbert's third problem.

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<span class="mw-page-title-main">Cantic 7-cube</span>

In seven-dimensional geometry, a cantic 7-cube or truncated 7-demicube as a uniform 7-polytope, being a truncation of the 7-demicube.

In geometry of 4 dimensions or higher, a double pyramid, duopyramid, or fusil is a polytope constructed by 2 orthogonal polytopes with edges connecting all pairs of vertices between the two. The term fusil is used by Norman Johnson as a rhombic-shape. The term duopyramid was used by George Olshevsky, as the dual of a duoprism.

References