k-cell (mathematics)

Higher-dimensional version of a rectangle or rectangular solid

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

A ${\displaystyle k}$-cell is a higher-dimensional version of a rectangle or rectangular solid. It is the Cartesian product of ${\displaystyle k}$ closed intervals on the real line.^[1] 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. The ${\displaystyle k}$ intervals need not be identical. For example, a 2-cell is a rectangle in ${\displaystyle \mathbb {R} ^{2}}$ such that the sides of the rectangles are parallel to the coordinate axes. Every ${\displaystyle k}$-cell is compact.^[2][3]

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 for all ${\displaystyle a_{i}<b_{i}}$. 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.^[4]

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.

Notes

1. Foran (1991)
2. Rudin (1976:39)
3. Foran (1991:24)
4. Rudin (1976:31)

References

• 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.