Ideally irregular surfaces, of dimension greater than two, in theory and practice

We apply Mandelbrot's [1] concept of fractal (noninteger) dimension D to surfaces: We describe how surfaces can provide environments (e.g. for adsorbates) with 2≤D<3. Conventional D=2 characterizes smooth surfaces; carriers of D>2 are "ideally irregular" (self-similar) surfaces; and D→3 obtains in the limit where a surface visits every point of a volume. We present three major methods to get D from experiment. Case studies give ample instances for well-defined D>2. Some eminent consequences are implied.