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Fractal - v1.0 - 22 Jul 2012

    Generate fractals 

Syntax
        fractal [, options]

Options
    HShape(numlist) The horizontal (x-axis) proportions of the shape
      (equivalent to x coordinates of the shape).  Values should be between 0
      and 1.0 and be in ascending order. The first value should be 0 and the
      last value 1.

    VShape(numlist) The vertical (y-axis) proportions of the shape
      (equivalent to y coordinates of the shape). The first value should be
      zero.  The last value cannot be zero.  There are no restrictions to
      other values.  If the last value is not one, all values will be divided
      by the last value.

    HRange(numlist) Horizontal Range (of x-axis) (defines the horizontal size
      of the shape).  Two numbers, default is 0, 100.

    VRange(numlist) Vertical Range (of y-axis) (defines the vertical size of
      the shape).  Two numbers, default is 0, 100.

    ITer(#) The number of iterations of self-similarity (default is 1)

    KEEPVars Keep values for all iterations (default is to save only the last
      iteration).

    SAVEGraph Saves and displays graphs of all iterations

Options for more complex fractals
    HSHAPE2(numlist) Horizontal coordinates for a second shape that can be
      substituted for the base shape probalistically. Same rules as for
      HSHAPE option.

    VSHAPE2(numlist) Vertical coordinates for a second shape that can be
      substituted for the base shape probalistically.  Same rules as for
      VSHAPE option.

    PROB2(#) Probability of occurance for this shape (must be a number
      between 0 and 1).

    HSHAPE3(numlist) Horizontal coordinates for a third shape that can be
      substituted for the base shape probalistically. Same rules as for
      HSHAPE option.

    VSHAPE3(numlist) Vertical coordinates for a third shape that can be
      substituted for the base shape probalistically. Same rules as for
      VSHAPE option.

    PROB3(#) Probability of occurance for this shape (must be a number
      between 0 and 1).


    The sum of PROB2 and PROB3 must be less than 1.0. The probability of
      using the first (base) shape is 1.0 - (PROB2 + PROB3).

    This set of commands allows the use of several shapes when generating the
      fractal.  Use the command -set seed # - to consistently create the same
      fractal.

Description
    Fractal generates points that correspond to the fractal given by the
      shape specified. Results are saved in the variables _frctlx and
      _frctly.

    Mandelbrot argues that the modelling of natural phenomena, including that
      of the stock market, is problematic.  The distribution most commonly
      used, the normal or Gaussian distribution, does not adequately account
      for natural variation.  Neither are natural phenomena independent,
      another common, but erroneous, assumption when modelling.  Mandelbrot
      argues that fractals can more accurately model the variation observed
      in nature.

    This routine allows the generation of data for use in modelling.  The
      examples presented do not begin with a dataset, since the purpose of
      the routine is to generate data, not to analyse it.

Example - a Fractal Cartoon
. clear
. fractal  ,hshape(0,.33,.67,1.0) vshape(0,.67,.33,1.0) hr(0 100) vr(0 100) ite
> r(4) savegraph
. graph combine _frctl1.gph _frctl2.gph _frctl3.gph _frctl4.gph

Another simple example - an Elliott wave-like shape
. clear
. fractal  ,hs(0,.2,.3,.40,.5,.618,.75,.85,1.0) vs(0,.25,.2,.35,.25,.516,.366,.
> 42,.2) hr(0 100) vr(0 100) iter(3) savegraph keepvars
. line _frctly1 _frctlx1, lw(vthick) || line _frctly2 _frctlx2, lw(thick) || li
> ne _frctly3 _frctlx3, lw(medium)

An example creating a fractal containing two shapes
. clear
. set seed 1234356789
. fractal  ,hs(0,.33,.67,1.0) vs(0,.67,.33,1.0) hr(0 100) vr(0 100) iter(4) sav
> egraph hshape2(0,.25,.5,.75,1.0) vshape2(0,-.25,1.25,.5,1.0) prob2(.4)
. graph combine _frctl1.gph _frctl2.gph _frctl3.gph _frctl4.gph

Author
Paul Millar
www.paulmillar.ca
paulmi@nipissingu.ca 

References
Mandelbrot and Hudson (2004).  The (Mis)Behavior of Markets.  New York: Basic B
> ooks.