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
Topological 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, and is conjectured to be equal to the Lyapunov Dimension.

**References**

Nayfeh, A. H. and Balachandran, B. *Applied Nonlinear Dynamics: Analytical, Computational, and Experimental
Methods.* New York: Wiley, pp. 538-541, 1995.

Peitgen, H.-O. and Richter, D. H. *The Beauty of Fractals: Images of Complex Dynamical Systems.* New York:
Springer-Verlag, 1986.

Wheeden, R. L. and Zygmund, A. *Measure and Integral: An Introduction to Real Analysis.*
New York: M. Dekker, 1977.

© 1996-9

1999-05-26