Capacity Dimension
A Dimension also called the Fractal Dimension, Hausdorff Dimension, and Hausdorff-BesicovitchDimension in which nonintegral values are permitted. Objects whose capacity dimension is different from theirTopological Dimension are called Fractals. The capacity dimension of a compact MetricSpace 
is a Real Number 
such that if 
denotes the minimum number of open sets of diameterless 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 MetricSpace and is a bound on the diameter of the sets involved (informally, is the size of eachelement used to cover the set, which is taken to approach 0). If each element of a Fractal is equallylikely to be visited, then 
, where 
is the Information Dimension. Thecapacity dimension satisfies
where
is the Correlation Dimension, and is conjectured to be equal to the Lyapunov Dimension.See also Correlation Exponent, Dimension, Hausdorff Dimension, Kaplan-Yorke 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.
- Tetrahedral Surface
- Tetrahedroid
- Tetrahedron
- Tetrahedron 10-Compound
- Tetrahedron 5-Compound
- Tetrahedron Inscribing
- Tetrahemihexacron
- Tetrahemihexahedron
- Tetrakaidecagon
- Tetrakaidecahedron
- Tetrakis Hexahedron
- Tetranacci Number
- Tetrix
- Tetromino
- Thales' Theorem
- Theorem
- Theorema Egregium
- Theta Function
- Theta Operator
- Theta Subgroup
- Thiele's Interpolation Formula
- Thiessen Polytope
- Third Curvature
- Thirteenth
- Thomae's Theorem