Difference between revisions of "Fractal Dimension"

From Soft-Matter
Jump to: navigation, search
Line 6: Line 6:
  
 
[[image:Fractaldimensions1.png]]
 
[[image:Fractaldimensions1.png]]
 +
Figure 1. Koch snowflake
  
There are many examples of fractals in nature. A koch snowflake is an idealization (and a good one) of an actual snowflake.  
+
There are many examples of fractals in nature. A koch snowflake is an idealization (and a good one) of an actual snowflake. Figure 1 shows the first four iterations of a Koch snowflake. If we continue with the iterations infinitely, we will have a Koch snowflake and the length of curve between any two points is infinite. The fractal dimension of a fractal line can be understood intuitively to describe an object that is too big to be a one-dimensional object, but too thin to be a two-dimensional object.
 +
 
 +
[[image:Fractaldimensions2.png]]
  
 
A non-exhaustive list of fractals and their fractal dimension can be found at http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension
 
A non-exhaustive list of fractals and their fractal dimension can be found at http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension

Revision as of 17:17, 4 December 2011

Introduction

A fractal dimension is a statistical quantity that describes how a fractal appears to fill space. A fractal is an object that displays a property known as self-similarity, i.e. a geometric shape that can be reduced to smaller parts, with each smaller part being a reduced copy of the whole. There are several specific definitions of fractal dimensions, but the most important ones include Renyi dimensions and Haussdorf dimensions.

Examples of fractals

Fractaldimensions1.png Figure 1. Koch snowflake

There are many examples of fractals in nature. A koch snowflake is an idealization (and a good one) of an actual snowflake. Figure 1 shows the first four iterations of a Koch snowflake. If we continue with the iterations infinitely, we will have a Koch snowflake and the length of curve between any two points is infinite. The fractal dimension of a fractal line can be understood intuitively to describe an object that is too big to be a one-dimensional object, but too thin to be a two-dimensional object.

File:Fractaldimensions2.png

A non-exhaustive list of fractals and their fractal dimension can be found at http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension

Fractals and polymers

Keyword in references:

G. Lois, J. Blawzdziewicz, and C. S. O'Hern, "Protein folding on rugged energy landscapes: Conformational diffusion on fractal networks", Phys. Rev. E 81 (2010) 051907