Fractal Dimension

A quantitative exponent that characterizes how the size (measure, covering number, or measure scaling) of a set scales with resolution; in the Hausdorff sense, it is the critical exponent s at which the s-dimensional Hausdorff measure jumps from infinity to zero.

A real number (possibly non-integer) that quantifies how a set scales under magnification, capturing geometric complexity beyond the integer-valued topological dimension; multiple formalizations (Hausdorff, box-counting/Minkowski, packing) exist that measure scaling of measure, coverings, or counts.