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===Self-similarity===
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).
* Quasi-self-similar. (Typically where micro-structures are distorted versions of the macro-structure).
* Statistical. (Where an empirical measure remains constant across scales).

{|
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|{{SingleImage|imageWidthPlusTen=410|imageURL=http://philipgalanter.com/generative_art/graphics/fractalANI.gif|caption=Zoom-in on Koch fractal}}
|{{SingleImage|imageWidthPlusTen=410|imageURL=http://philipgalanter.com/generative_art/graphics/nonfractalANI.gif|caption=Zoom-in on non-fractal curve}}
|}

===Occurrences of fractals=== 
Taken loosely, fractals can occur in a number of ways. For example:

* Simple iterative recursive geometric constructions can create fractals.
* Some natural phenomina exhibit a fractal nature. e.g. Clouds, coast lines, some plants, some forms of shattered material, lightning, mountains, patterns in frost on windows and other forms of crystal growth, etc.
* 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).

===Fractals as recursive functions===
Fractals can be defined and generated by recursive functions. For example, take the factorial function:
:<var>5! = 5 * 4 * 3 * 2 * 1</var>
:<var>N! = N * (N - 1) * (N - 2) * ... * 1</var>

=====Non-recursive factorial function=====
  '''function''' FACTORIAL(m)
      total = 1
      '''for''' n = 1 to m
          total = total * m
      '''end'''
      '''return''' total

=====Recursive factorial function=====
  '''function''' FACTORIAL(m)
      '''if''' m > 1
          '''return''' m = m * FACTORIAL(m-1)
      '''else'''
          '''return''' 1
      '''end'''

=====Recursive Koch curve=====
{{SingleImage|imageWidthPlusTen=410|imageURL=http://philipgalanter.com/generative_art/graphics/kochcurveprogression.gif|caption=The recursion of a Koch curve}}
  //   Given:
  //   res                      :length of smallest line to be drawn
  //   DRAW(x1, y1, x2, y2)     :draws a point from (x1, y1) to (x2, y2)
  //   DIST(x1, y1, x2, y2)     :distance between points (x1, y1) and (x2, y2)
  //   BEND(x1, y1, x2, y2)     :takes one line segment and bends it into 4 returning tx1, ty,2, tx2, ..., ty5
  <br>
  '''function''' KOCH(x1, y1, x2, y2, res)
      '''if''' DIST(x1, y1, x2, y2) < res
          DRAW(x1, y1, x2, y2)
      '''else'''
          BEND(x1,y1,x2,y2)
          KOCH(tx1, ty1, tx2, ty2, res)
          KOCH(tx2, ty2, tx3, ty3, res)
          KOCH(tx3, ty3, tx4, ty4, res)
          KOCH(tx4, ty4, tx5, ty5, res)
      '''endif'''
  <br>

=====Stochastic fractal river=====
Both fractals and L-systems can have stochastic (random/chance) forms.
  //   Given:
  //   res                      :length of smallest line to be drawn
  //   DIST(x1, y1, x2, y2)     :distance between points (x1, y1) and (x2, y2)
  //   BEND(x1, y1, x2, y2)     :returns x3,y3 randomly between and offset
  <br>
 '''function''' RIVERBEND(x1, y1, x2, y2, res)
      '''if''' DIST(x1, y1, x2, y2) > res
          BEND(x1, y1, x2, y2)
          RIVERBEND(x1, y1, x3, y3)
          RIVERBEND(x3, y3, x2, y2)
      '''else'''
          DRAW(x1, y1, x2, y2)
      '''endif'''
  <br>

===Fractal or Box Counting Dimension=== 
<br />
In traditional units a Koch curve will have infinite length but 0 area.  This is not a useful measurement.<br />
A measurement of dimension filling. For example, D = 1.4 means it fills more than a line, but less than a plane.<br />
<br />
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 />
Using this formula for increasingly smaller divisions, if the object is a fractal the dimension will be about the same.<br /> 
If the dimension changes with scale the object is not a fractal.<br />
<br />
{{SingleImage|imageWidthPlusTen=510|imageURL=http://philipgalanter.com/generative_art/graphics/box_counting_formula.png|caption=}}<br />
{{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 />
{{SingleImage|imageWidthPlusTen=510|imageURL=http://www.viz.tamu.edu/courses/viza626/10Fall/fractImg2.gif|caption=Koch curve breaks the segment into 4 segments}}<br />
{{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 />
{{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 />
{{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 />