Everything about Fractal Dimension totally explained
In
fractal geometry, the
fractal dimension,
D, is a statistical quantity that gives an indication of how completely a
fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension and none of them should be treated as the universal one. From the theoretical point of view the most important are the
Hausdorff dimension, the
packing dimension and, more generally, the
Rényi dimensions. On the other hand the
box-counting dimension and
correlation dimension are widely used in practice, partly due to their ease of implementation.
Although for some classical fractals all these dimensions do coincide, in general they're not equivalent. For example, what is the dimension of the
Koch snowflake? It has
topological dimension one, but it's by no means a curve-- the length of the curve between any two points on it's infinite. No small piece of it's line-like, but neither is it like a piece of the plane or any other. In some sense, we could say that it's too big to be thought of as a one-dimensional object, but too thin to be a two-dimensional object, leading to the question of whether its dimension might best be described in some sense by number between one and two. This is just one simple way of motivating the idea of fractal dimension.
The many definitions
There are two main approaches to generate a fractal structure. One is growing from a unit object, and the other is to construct the subsequent divisions of an original structure, like the
Sierpinski triangle (Fig.(2)). Here we follow the second approach to define the dimension of fractal structures.
If we take an object with linear size equal to 1 residing in Euclidean dimension, and reduce its linear size to be
in each spatial direction, it takes
number of self similar objects to cover the original object(Fig.(1)). However, the dimension defined by
» .
is still equal to its topological or Euclidean dimension, image analysis, acoustics, Riemann zeta zeros and even (electro)chemical processes .
Practical dimension estimates are very sensitive to numerical or experimental noise, and particularly sensitive to limitations on the amount of data. Claims based on fractal dimension estimates, particularly claims of low-dimensional dynamical behaviour, should always be taken with a handful of salt — there's an inevitable ceiling, unless
very large numbers of data points are presented.
Further Information
Get more info on 'Fractal Dimension'.
|
External Link Exchanges
Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:
<a href="http://fractal_dimension.totallyexplained.com">Fractal dimension Totally Explained</a>
Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned. |
