The -dimensional cube, or simply -cube, is a generalisation of the ordinary cube (or -cube) to arbitrary dimensions. It comes in many guises.
The standard cubical -cube is the cubical set represented (as a presheaf) by the object in the cube category.
The standard topological -cube is the space , where is the unit interval. In general, any closed -cube is the topological product of closed intervals. The collection of topological cubes forms a topological cocubical set?.
There are also open -cubes, which contain all points of the closed -cube which are apart from the boundary. Open -cubes are the topological product of open intervals. The standard topological open -cube is the space , where is the open unit interval.
The open -cubes are the balls in -dimensional Cartesian space with respect to the metric derived from the supremum norm on .
Discussion of the 3-cube as a Platonic solid:
Last revised on December 10, 2022 at 11:23:33. See the history of this page for a list of all contributions to it.