Broadly speaking, these notions of dimension are invariants of topological or metric spaces which seek to quantify the relation between the diameter of a set in a metric space, and the volume or measure of that set. For example, a ball in $$\mathbb{R}^3$$ is three-dimensional, as the volume of the ball is proportional to the cube of the radius—that is, $$\operatorname{vol}(B_r) \propto r^3$$. The dimension "sees" the cubic scaling law. Other notions of dimension generalize this basic idea to spaces where the volume may not be a priori defined, or where the scaling law involves non-integer powers.