For simple bodies, such as cube or sphere, the surface scales with the volume as $$ S=\alpha V^\beta,$$ Where $$ \beta=\frac{2}{3}, $$ And $\alpha$ is of the order of unity.
While this is intuitively obvious for "simple" non-regular shapes, it clearly breaks for fractal surfaces. I am therefore looking for clear mathematical statements:
- When the scaling above is applicable?
- What is the range of variation of factor $\alpha$?
Remark: this question is inspired by my answer to the infamous "chicken surface problem", which provoked such heated debates.
Update To generalize a bit: in $n$ dimensions we expect for a closed surface/volume: $$ V\propto R^n, \\ S\propto R^{n-1}, $$ where $R$ is the length scale. When is this not true?