The -cube

# The $n$-cube

## Idea

The $n$-dimensional cube, or simply $n$-cube, is a generalisation of the ordinary cube (or $3$-cube) to arbitrary dimensions. It comes in many guises.

## As a cubical set

The standard cubical $n$-cube is the cubical set represented (as a presheaf) by the object $[n]$ in the cube category.

## As a topological space

The standard topological $n$-cube is the space $[0,1]^n$, where $[0,1]$ is the unit interval. In general, any closed $n$-cube is the topological product of closed intervals. The collection of topological cubes forms a topological cocubical set?.

There are also open $n$-cubes, which contain all points of the closed $n$-cube which are apart from the boundary. Open $n$-cubes are the topological product of open intervals. The standard topological open $n$-cube is the space $(0,1)^n$, where $(0,1)$ is the open unit interval.

The open $n$-cubes are the balls in $n$-dimensional Cartesian space $\mathrm{R}^n$ with respect to the metric derived from the supremum norm $\Vert x \Vert_\infty$ on $\mathrm{R}^n$.

## References

Discussion of the 3-cube as a Platonic solid:

• Felix Klein, chapter I.5 of Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade, 1884, translated as Lectures on the Icosahedron and the Resolution of Equations of Degree Five by George Morrice 1888, online version

Last revised on December 10, 2022 at 11:23:33. See the history of this page for a list of all contributions to it.