nLab cube

The -cube

The nn-cube


The nn-dimensional cube, or simply nn-cube, is a generalisation of the ordinary cube (or 33-cube) to arbitrary dimensions. It comes in many guises.

As a cubical set

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

As a topological space

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

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

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


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.