Hyperrectangle

In geometry, a hyperrectangle (also called a box, hyperbox, k -cell or orthotope ), 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.

In geometry, a hyperrectangle (also called a box, hyperbox, k -cell or orthotope ), 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. For every integer i from 1 to k , let a_i and b_i be real numbers such that a_i . The set of all points x=(x_1,\dots,x_k) in \mathbb{R}^k whose coordinates satisfy the inequalities a_i\leq x_i\leq b_i is a k -cell. ==Types== A four-dimensional orthotope is likely a hypercuboid. 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. ==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. ==See also== Minimum bounding rectangle Cuboid Hilbert cube ==Notes==

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