I’m not sure this should be modeled as “fractional dimension”, though. It seems like standard “volume” in 3 dimensions is the thing you’re talking about, and your measurement granularity is determining how you include or exclude “pockets” in the crumple as part of the volume.
See also https://en.wikipedia.org/wiki/Coastline_paradox . At different scales, you’ll be measuring different things and different features or granularity of variance will matter.
I’m not sure this should be modeled as “fractional dimension”, though. It seems like standard “volume” in 3 dimensions is the thing you’re talking about, and your measurement granularity is determining how you include or exclude “pockets” in the crumple as part of the volume.
The metric I had in mind that the function would report is the log of the ratio of the measure at slightly different scales, not the measure itself.