nLab 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. The collection of topological cubes forms a topological cocubical set?.

related entries

• cubical set

• cubical type theory

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