# Difference between revisions of "Fractal Dimension"

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[[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. |

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+ | [[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

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.

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