An additional aspect of surfaces that is generalized above is that of dimension. We are intuitively familiar with objects of dimension up to three: points, curves, surfaces, and volumes. As we build structure in geometry, we try to keep all our definitions applicable to any number of dimensions.

The Cartesian charts of a manifold determine an unambiguous dimensionality, since they are maps to \({\mathbb{R}^{n}}\) for a specific integer \({n}\). In contrast, short of a manifold structure there are several definitions of dimension, some of which result in non-integer values (called fractal dimensions). It is also important to note that properties that hold in easily visualized lower dimensional manifolds do not always remain valid in higher dimensions.