Hyperrectangle

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

Projections of K-cells onto the plane (from ${\displaystyle k=1}$ to ${\displaystyle 6}$). Only the edges of the higher-dimensional cells are shown.

In geometry, a hyperrectangle (also called a box, hyperbox, ${\displaystyle k}$-cell 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. ^[3] This means that a ${\displaystyle k}$-dimensional rectangular solid has each of its edges equal to one of the closed intervals used in the definition. Every ${\displaystyle k}$-cell is compact. ^{[4] [5]}

If all of the edges are equal length, it is a hypercube . A hyperrectangle is a special case of a parallelotope.

Formal definition

For every integer ${\displaystyle i}$ from ${\displaystyle 1}$ to ${\displaystyle k}$, let ${\displaystyle a_{i}}$ and ${\displaystyle b_{i}}$ be real numbers such that ${\displaystyle a_{i}<b_{i}}$. The set of all points ${\displaystyle x=(x_{1},\dots ,x_{k})}$ in ${\displaystyle \mathbb {R} ^{k}}$ whose coordinates satisfy the inequalities ${\displaystyle a_{i}\leq x_{i}\leq b_{i}}$ is a ${\displaystyle k}$-cell. ^[6]

Intuition

A ${\displaystyle k}$-cell of dimension ${\displaystyle k\leq 3}$ is especially simple. For example, a 1-cell is simply the interval ${\displaystyle [a,b]}$ with ${\displaystyle a<b}$. A 2-cell is the rectangle formed by the Cartesian product of two closed intervals, and a 3-cell is a rectangular solid.

The sides and edges of a ${\displaystyle k}$-cell need not be equal in (Euclidean) length; although the unit cube (which has boundaries of equal Euclidean length) is a 3-cell, the set of all 3-cells with equal-length edges is a strict subset of the set of all 3-cells.

Types

A four-dimensional orthotope is likely a hypercuboid. ^[7]

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. ^[8]

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	Rhombus { } + { } = 2{ }
3	Rhombic 3-orthoplex inside 3-orthotope { } + { } + { } = 3{ }

See also

• Minimum bounding rectangle
• Cuboid
• Hilbert cube

Notes

1. 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. Coxeter, 1973
3. Foran (1991)
4. Rudin (1976 :39)
5. Foran (1991 :24)
6. Rudin (1976 :31)
7. Hirotsu, Takashi (2022). "Normal-sized hypercuboids in a given hypercube". arXiv: 2211.15342 .
8. 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, doi:10.14778/3402707.3402743 .

References

• Coxeter, Harold Scott MacDonald (1973). Regular Polytopes (3rd ed.). New York: Dover. pp.  122–123. ISBN   0-486-61480-8.

External links

• Weisstein, Eric W. "Orthotope". MathWorld .
• Foran, James (1991-01-07). Fundamentals of Real Analysis. CRC Press. ISBN   9780824784539 . Retrieved 23 May 2014.
• Rudin, Walter (1976). Principles of Mathematical Analysis . McGraw-Hill.