One of the characteristics of definitions of dimension d of a figure is the scaling property that if you increase the one-dimensional lengths by a factor r, then the content of the figure (length, area, volume, fractal measure) increases by r^d. Here r and d are real numbers; r would have to be real because metrics have real ranges, but one could conceivably extend the domain of d to the set of complex numbers. Of course, the full definition of dimension as something like a Hausdorff or box-counting dimensions cannot yield anything but non-negative real numbers. Nonetheless, I am wondering if an extension of the scaling property to include the full set of complex numbers could not somehow be extended into something similar to dimension, including the cyclical nature of the new concept, since
r^(a+ib) = r^(a + 2b*pi/ln(r))
The technical details are not primarily what I am asking, but rather first whether this concept (which I hesitate to call "complex dimensions", since that term is already used for another concept) would be meaningful. If it is meaningful, then I presume it is used, so my next question would be how it is interpreted. Only if the concept has some meaning would my final question be where I could find the technical details of the concept. If the concept is not meaningful, then that would be the end of that. In any case, thanks.