A dimension also called the fractal dimension, Hausdorff dimension, and Hausdorff-Besicovitch dimension in which nonintegral values are permitted. Objects whose capacity dimension is different from their Lebesgue covering dimension are called fractals. The capacity dimension of a compact metric space is a real number such that if denotes the minimum number of open sets of diameter less than or equal to , then is proportional to as . Explicitly,
(if the limit exists), where is the number of elements forming a finite cover of the relevant metric space and is a bound on the diameter of the sets involved (informally, is the size of each element used to cover the set, which is taken to approach 0). If each element of a fractal is equally likely to be visited, then , where is the information dimension.
The capacity dimension satisfies
where is the correlation dimension (correcting the typo in Baker and Gollub 1996).
is a real number 
such that if 
denotes the minimum number of open sets of diameter less than or equal to 
, then 
is proportional to 
as 
. Explicitly,
is the number of elements forming a finite cover of the relevant metric space and 
is a bound on the diameter of the sets involved (informally, 
is the size of each element used to cover the set, which is taken to approach 0). If each element of a fractal is equally likely to be visited, then 
, where 
is the information dimension.
is the correlation dimension (correcting the typo in Baker and Gollub 1996).