Hyperrectangle Explained

In geometry, a hyperrectangle (also called a box, hyperbox,

-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. This means that a

k

-dimensional rectangular solid has each of its edges equal to one of the closed intervals used in the definition. Every

k

-cell is compact.

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

from to , let and be real numbers such that . The set of all points in whose coordinates satisfy the inequalities is a

k

-cell.

Intuition

A

-cell of dimension is especially simple. For example, a 1-cell is simply the interval with . 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

-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.^[3]

The special case of an -dimensional orthotope where all edges have equal length is the -cube or hypercube.

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

The dual polytope of an -orthotope has been variously called a rectangular -orthoplex, rhombic -fusil, or -lozenge. It is constructed by points located in the center of the orthotope rectangular faces.

An -fusil's Schläfli symbol can be represented by a sum of orthogonal line segments: or

A 1-fusil is a line segment. A 2-fusil is a rhombus. Its plane cross selections in all pairs of axes are rhombi.

	Example image
1	Line segment
2	Rhombus
3	Rhombic 3-orthoplex inside 3-orthotope

See also

• Minimum bounding rectangle
• Cuboid
• Hilbert cube

References

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

External links

• • Book: Foran, James. Fundamentals of Real Analysis. 23 May 2014. 1991-01-07. CRC Press. 9780824784539.
• Book: Rudin, Walter. Walter Rudin. Principles of Mathematical Analysis. 1976. McGraw-Hill.

Notes and References

1. [Norman Johnson (mathematician)|N.W. Johnson]
2. Coxeter, 1973
3. Normal-sized hypercuboids in a given hypercube . Hirotsu . Takashi . 2022 . 2211.15342 .
4. See e.g. .