K-cell (mathematics)



A $$k$$-cell is a higher-dimensional version of a rectangle or rectangular solid. It is the Cartesian product of $$k$$ closed intervals on the real line. This means that a $$k$$-dimensional rectangular solid has each of its edges equal to one of the closed intervals used in the definition. The $$k$$ intervals need not be identical. For example, a 2-cell is a rectangle in $$\mathbb{R}^2$$ such that the sides of the rectangles are parallel to the coordinate axes. Every $$k$$-cell is compact.

Formal definition
For every integer $$i$$ from $$1$$ to $$k$$, let $$a_i$$ and $$b_i$$ be real numbers such that for all $$a_i < b_i$$. 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.

Intuition
A $$k$$-cell of dimension $$k\leq 3$$ is especially simple. For example, a 1-cell is simply the interval $$[a,b]$$ with $$a < b$$. 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 $$k$$-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.