ABSTRACT
Our discussion from here onward deals with exponent dimensions that estimate the dimension of irregular objects, such as fractals. For such irregular or natural objects (as opposed to mathematically determined objects), it's probably impossible to get an exact measure of dimension. We can, however, get closer and closer to an exact measure by using smaller and smaller measuring tools. To indicate that possibility, the relation D = logN/log(llr) (Eq. 21.3) needs a way to show that the equation gives a closer and closer answer as the ruler length gets smaller and smaller and approaches zero. Chaologists do that with a limit term. With such a limit term added and with ruler length e written in place of scaling ratio r, Equation 21.3 becomes
. logN Dc=hml (1/)'
I'll call the dimension De as defined in that fashion the capacity, following Farmer et al. ( 1983). (Some people call itthe capacity dimension or the limit capacity.) Technically, capacity is only definable in the range of extremely small ruler lengths E, specifically in the limit as e goes to zero.
