# Edit Edit an Image Page: Blue Wash

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}} ===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 [http://www.dcs.gla.ac.uk/%7Ewpc/Fractal_Art.htm here] to learn more. ==Basic Recursive Method== {{hide|1= The featured image at the top of this page was created using the basic recursive method that will be described in this section. The canvas begins with rectangles of various sizes and colors, each of which undergo this basic recursive method. [[Image:BasicRectangle.png|right]] *Each rectangle is split into sub-rectangles ($a\,$ and $b\,$) by a random horizontal or vertical line ($\overline{PQ}\,$). *The random offset value used to decrease or increase the brightness level of each rectangle is determined randomly, but the mean of the [[Probability Distributions#Probability Density Function|probability density function]] of the random offset value is proportional to the square root of the area of each sub-rectangle. Algebraically, ::If a sub-rectangle has sides length $l\,$ and width $w\,$, ::And the mean of the probability density function $f(x)\,$ of the random offset value (''X'') is $E(X) = \int_{-\infty}^{\infty} x f(x) dx$, ::Then $\int_{-\infty}^{\infty} x f(x) dx \propto \sqrt{l * w}$ *The random offset value is then added to the brightness level of sub-rectangle $a\,$ and subtracted from the brightness level of sub-rectangle $b\,$ brightness to produce the next progression of the recursion method. The random value is designated such that, on average, the smaller sub-rectangle will have a smaller brightness offset than larger sub-rectangle. This method is repeated continuously to create a random fractal. }} ==Inclined Recursive Method - a more painterly effect== {{hide|1= Image:fs_64_100.gif|Random Fractal created using the basic method Image:kBlueWash.jpg|Random Fractal with higher 'k Image:kYellowWash.jpg|Random Fractal with lower 'k' In this method, each rectangle is again split into two sub-rectangles by random dividing line $\overline{PQ}\,$. However, instead of using a random offset value to directly increase or decrease the brightness of the two sub-rectangles, there is an inclined change in brightness. The random variable used in this procedure has a probability density function with: ::*a mean of $E(x)= 0\,$ ::*a standard deviation of $\sigma\approx k|\overline{PQ}|\,$, where ''k'' is a constant. The brightness of the dividing line itself is increased or decreased by the random variable, and the sub-rectangles $a\,$ and $b\,$ then form inclined plans in brightness that gradually increase or decrease to match the brightness of the dividing line. Similar to the basic method, the random variable is picked such that, on average, the smaller rectangles will have a smaller brightness offset than rectangles. Since ''k'' reflects the degree of randomness of the brightness of the color in each sub-rectangle, a higher ''k'' produces a more random fractal. The image on the above has a higher ''k'' constant than the image on the below. }} Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Topology Other None Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Topology Other None Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Topology Other Paul Cockshott, [http://www.dcs.gla.ac.uk/~wpc/ Paul Cockshott] *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. Yes, it is.