From Math Images
- This image is a random fractal that is created by continually dividing a rectangle into two parts and adjusting the brightness of each resulting part.
Basic DescriptionThis image is considered a random fractal, because the behavior and coloring of this image is determined randomly. The basic rule governing a fractal of this kind:
- 1. Take a rectangle
- 2. Divide the rectangle into two sub-rectangles at a random location
- 3. Adjust the brightnesses of each sub-rectangles by a random factor
- 4. Repeat for all current sub-rectangles (now considered "rectangles") in the image
The artist actually experimented with at least two methods to create images similar to the random fractal above that differed on how the brightness was adjusted in each split rectangle.
- The first (Basic) method involves simply increasing the brightness of one sub-rectangle and decreasing the brightness of the other sub-rectangle of a split rectangle by a random variable. This method was used to create the fractal featured at the top of this page.
- The second (Inclined) method involves adjusting the brightness of the two sub-rectangles gradually to make an inclined plane in brightness within each sub-rectangle. The brightness of the line where the rectangle is split is determined randomly, and the two resulting sub-rectangles then each form an increasing or decreasing plane in brightness to meet the brightness of the dividing line. If you compare the featured image at the top of the page (created by the first method) to the image on the right (created by this second method), you can see that this method results in a more "painterly" effect.
An interesting observation about the creation of these fractals is that with each iteration, the change in brightness and spatial aspects (splitting the rectangles) of the fractals become smaller and smaller and their influence on the images gradually diminishes. Thus, after a number of iterations, the fractals barely change in appearance. Please take a look at the applets below to test this observation.
Origins of this Fractal
The artist who created this fractal, Paul Cockshott, is a computer scientist and a reader at the University of Glasgow. The various math images featured on this page emerged from his scientific research in stereo range finding, which is a method used to estimate distance and depth from two images (one from the right perspective and one from the left). Cockshott initially used Gaussian patterns for this project, but found that fractal patterns worked better...and were also more beautiful. Click here to learn more.
A More Mathematical Explanation
- Note: understanding of this explanation requires: *Statistics, Calculus
Basic Recursive Method
Inclined Recursive Method - a more painterly effect
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Paul Cockshott, Paul Cockshott
Future Directions for this Page
- An applet letting users pick a few starting colors to start the random fractal and watch it produce an image, or even let them change k.
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