Generalizing dimension

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.

An Illustrated Handbook