Once the Euclidean plane has been described in this language, it is actually a simple matter to extend its concept to arbitrary dimensions.

This can be generalized to an arbitrary finite dimension:

This method appeals to the definition, and allows generalization to arbitrary dimensions.

While the cross product works in P, it is not well-defined in arbitrary dimensions.

Graphics can be of arbitrary dimensions, resolution and colour depth, even on the same screen.

In linear algebra, the notion of orientation makes sense in arbitrary dimensions.

As of 2006, there is no general theorem to circumvent this difficulty in arbitrary dimension, although certain special cases have been resolved.

A number of algorithms are known for the three-dimensional case, as well as for arbitrary dimensions.

The algorithms discussed can be generalized to arbitrary dimensions, albeit with increased time and space complexity.

This can be extended to arbitrary dimensions.