k-cell (mathematics)

Projections of K-cells onto the plane (from k=1 to 6. Only the edges of the higher-dimensional cells are shown.

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.^[1] This essentially means that is a k-dimensional rectangular solid, with each of its edges being 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 R² such that the sides of the rectangles are parallel to the coordinate axes.

Formal definition

Let a_i ∈ R and b_i ∈ R. If a_i < b_i for all i = 1,...,k, the set of all points x = (x₁,...,x_k) in R^k whose coordinates satisfy the inequalities a_i ≤ x_i ≤ b_i is a k-cell.^[2] Every k-cell is compact.^[3]

Intuition

A k-cell of dimension k ≤ 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.

Note that 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.

References