Fractals

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(→Fractal or Box Counting Dimension)
 
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[[Image:http://www-viz.tamu.edu/students/ablev/genart/fractalANI.gif]]
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===Self-similarity===
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Fractals are structures that exhibit self-similarity at all scales. Such self-similarity may be:
Fractals are structures that exhibit self-similarity at all scales. Such self-similarity may be:
* Exact. (Typically systems which are created by an iterative recursive function).
* Exact. (Typically systems which are created by an iterative recursive function).
* Quasi-self-similar. (Typically where micro-structures are distorted versions of the macro-structure).
* Quasi-self-similar. (Typically where micro-structures are distorted versions of the macro-structure).
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* Statisical. (Where an empirical measure remains constant across scales).
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* Statistical. (Where an empirical measure remains constant across scales).
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<html><div style="clear:none"><div style="border:1px solid black; background:#dddddd; padding: 5px; font-size:smaller; width:400px; float:left; margin:5px"><img src="http://www-viz.tamu.edu/students/ablev/genart/fractalANI.gif" style="float:none">Fractal example - Koch Curve</div></html>
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<html><div style="border:1px solid black; background:#dddddd; padding: 5px; font-size:smaller; width:400px; float:left; margin:5px"><img src="http://www-viz.tamu.edu/students/ablev/genart/nonfractalANI.gif" style="float:none">Non-fractal function example</div></div></html>
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|{{SingleImage|imageWidthPlusTen=410|imageURL=http://philipgalanter.com/generative_art/graphics/fractalANI.gif|caption=Zoom-in on Koch fractal}}
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|{{SingleImage|imageWidthPlusTen=410|imageURL=http://philipgalanter.com/generative_art/graphics/nonfractalANI.gif|caption=Zoom-in on non-fractal curve}}
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<html><div style="clear:none; width:100%"> hullo</div></html>
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|}
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===Occurrences of fractals===
Taken loosely, fractals can occur in a number of ways. For example:
Taken loosely, fractals can occur in a number of ways. For example:
* Some mathematical functions define sets of points where the boundry between in-set and out-of-set points exhibit self-similarity across scales. e.g. The Julia set, the Mandelbrot set, and others.
* Some mathematical functions define sets of points where the boundry between in-set and out-of-set points exhibit self-similarity across scales. e.g. The Julia set, the Mandelbrot set, and others.
* In chaotic systems a phase diagram will exhibit fractal behavior about a strange attractor. (To be covered later).
* In chaotic systems a phase diagram will exhibit fractal behavior about a strange attractor. (To be covered later).
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===Fractals as recursive functions===
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Fractals can be defined and generated by recursive functions. For example, take the factorial function:
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:<var>5! = 5 * 4 * 3 * 2 * 1</var>
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:<var>N! = N * (N - 1) * (N - 2) * ... * 1</var>
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=====Non-recursive factorial function=====
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  '''function''' FACTORIAL(m)
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      total = 1
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      '''for''' n = 1 to m
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          total = total * m
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      '''end'''
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      '''return''' total
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=====Recursive factorial function=====
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  '''function''' FACTORIAL(m)
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      '''if''' m > 1
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          '''return''' m = m * FACTORIAL(m-1)
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      '''else'''
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          '''return''' 1
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      '''end'''
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=====Recursive Koch curve=====
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{{SingleImage|imageWidthPlusTen=410|imageURL=http://philipgalanter.com/generative_art/graphics/kochcurveprogression.gif|caption=The recursion of a Koch curve}}
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  //  Given:
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  //  res                      :length of smallest line to be drawn
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  //  DRAW(x1, y1, x2, y2)    :draws a point from (x1, y1) to (x2, y2)
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  //  DIST(x1, y1, x2, y2)    :distance between points (x1, y1) and (x2, y2)
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  //  BEND(x1, y1, x2, y2)    :takes one line segment and bends it into 4 returning tx1, ty,2, tx2, ..., ty5
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  <br>
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  '''function''' KOCH(x1, y1, x2, y2, res)
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      '''if''' DIST(x1, y1, x2, y2) < res
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          DRAW(x1, y1, x2, y2)
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      '''else'''
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          BEND(x1,y1,x2,y2)
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          KOCH(tx1, ty1, tx2, ty2, res)
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          KOCH(tx2, ty2, tx3, ty3, res)
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          KOCH(tx3, ty3, tx4, ty4, res)
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          KOCH(tx4, ty4, tx5, ty5, res)
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      '''endif'''
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  <br>
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=====Stochastic fractal river=====
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Both fractals and L-systems can have stochastic (random/chance) forms.
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  //  Given:
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  //  res                      :length of smallest line to be drawn
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  //  DIST(x1, y1, x2, y2)    :distance between points (x1, y1) and (x2, y2)
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  //  BEND(x1, y1, x2, y2)    :returns x3,y3 randomly between and offset
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  <br>
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'''function''' RIVERBEND(x1, y1, x2, y2, res)
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      '''if''' DIST(x1, y1, x2, y2) > res
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          BEND(x1, y1, x2, y2)
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          RIVERBEND(x1, y1, x3, y3)
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          RIVERBEND(x3, y3, x2, y2)
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      '''else'''
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          DRAW(x1, y1, x2, y2)
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      '''endif'''
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  <br>
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===Fractal or Box Counting Dimension===
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<br />
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In traditional units a Koch curve will have infinite length but 0 area.  This is not a useful measurement.<br />
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A measurement of dimension filling. For example, D = 1.4 means it fills more than a line, but less than a plane.<br />
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<br />
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Where epsilon is the relative size of the division (1, 1/2, 1/3, etc.), and N is the number of copies or boxes.<br />
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Using this formula for increasingly smaller divisions, if the object is a fractal the dimension will be about the same.<br />
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If the dimension changes with scale the object is not a fractal.<br />
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<br />
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{{SingleImage|imageWidthPlusTen=510|imageURL=http://philipgalanter.com/generative_art/graphics/box_counting_formula.png|caption=}}<br />
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{{SingleImage|imageWidthPlusTen=510|imageURL=http://www.viz.tamu.edu/courses/viza626/10Fall/fractImg1.gif|caption=Cantor dust breaks the segment in thirds and discards the middle one}}<br />
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{{SingleImage|imageWidthPlusTen=510|imageURL=http://www.viz.tamu.edu/courses/viza626/10Fall/fractImg2.gif|caption=Koch curve breaks the segment into 4 segments}}<br />
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{{SingleImage|imageWidthPlusTen=510|imageURL=http://www.viz.tamu.edu/courses/viza626/10Fall/fractImg3.gif|caption=Note that a line segment has a fractal dimension of 1}}<br />
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{{SingleImage|imageWidthPlusTen=510|imageURL=http://www.viz.tamu.edu/courses/viza626/10Fall/fractImg4.gif|caption=Note that a square has a fractal dimension of 2}}<br />
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{{SingleImage|imageWidthPlusTen=600|imageURL=http://philipgalanter.com/generative_art/graphics/sierp-det.GIF|caption=The Sierpinski triangle is constructed yielding 3 copies at 1/2 size so the fractal dimension = log(3)/log(2) = 1.58}}<br />{{SingleImage|imageWidthPlusTen=1000|imageURL=http://philipgalanter.com/generative_art/graphics/box_counting_dimension.png|caption=Here is an illustration of the box counting method to measure the fractal dimension of actual objects, in this case the coastline of England}}<br />

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