Image:3fractals2.jpg

From Wikipedia, the free encyclopedia

Image
File history
File links

$Image:3fractals2.jpg$
Size of this preview: 800 × 294 pixels
Full resolution‎ (1,990 × 732 pixels, file size: 493 KB, MIME type: image/jpeg)

These are examples of Multiplicative Cascade fractals. To create these distributions there are a few steps to take. Firstly, we must create a lattice of points which will be our underlying probability density field. Then we will populate this lattice with randomly placed points, insisting that the probability that the points be placed are proportional to the cell probability.

The fractal is constructed as follows: The space is split into four equal parts, each part is then assigned a probability from the set ${p 1, p 2, p 3, p 4}$ without replacement. Where $p_i \in [0,1]$ . Each subspace is then divided again and assigned probabilities randomly from the same set and this is continued to the $N t h$ level.

At the $N t h$ level the probability of a cell being occupied is the product of the cell's $p i$ and its parents and ancestors up to level 1 i.e. all the cells above it. In constructing this model down to level 8 we produce a $48$ array of cells each with its own probability. To then place particle in the space we invoke a Monte Carlo rejection scheme. Choosing x and y coordinates randomly we simply test if a random number between 0 and 1 is less or greater than the cell probability. We typically dust the probability density field with 5,000 points in a space of 256 x 256.

Model I (left plot): ${p 1, p 2, p 3, p 4} = {1,1,1,0}$ Model II (middle plot): ${p 1, p 2, p 3, p 4} = {1,0.75,0.75,0.5}$ Model III (right plot): ${p 1, p 2, p 3, p 4} = {1,0.5,0.5,0.25}$

I, the copyright holder of this work, hereby grant the permission to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.

File history

Click on a date/time to view the file as it appeared at that time.

	Date/Time	Dimensions	User	Comment
current	14:10, 9 August 2007	1,990×732 (493 KB)	Csabiu (Talk \| contribs)
	13:30, 9 August 2007	1,990×732 (493 KB)	Csabiu (Talk \| contribs)	(These are examples of multiplicative cascade fractal.)