How to make pictures using postscript programs (part 9)
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A way to make statements be conditionally executed is the if or ifelse 
statement. This statement looks like
condition {code block, in braces} if
or
condition {code block in braces, if-part} {code-block, else-part} ifelse
The simplest condition is an equality test, command eq, which compares
the previous two numbers. So 
------------------------Example 19-------------------------------------------
%!PS-Adobe-2.0
%%BoundingBox: 0 0 230 20
 1 1 10 { 3 eq 
  {newpath 10 10 5 0 360 arc stroke} 
  {newpath 10 10 5 0 360 arc fill} ifelse 
 20 0 translate } for
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will draw a  horizontal row of circles of radius 5, spaced 20 units,
only number three will be an empty circle (stroke) whereas the other 
are full circles (fill).
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We can combine that with recursion to make fractals. This is just one way
to create fractals, these are self-similar fractals. The idea is that the
entire picture is the union of several smaller copies of the picture, so
we want to create the picture at level 7 (e.g.), we write it as several
smaller copies of the picture at level 6, and so on, each time decreasing
the level, and only at level 0, when all the accumulated scale factors are
making the picture very small, we just draw a dot, or triangle, or outline of
the picture: scaled that much, it looks like a dot anyway. The key for
the resulting shape is the choice of the scale factor and translations and
rotations you use to place the smaller copies.
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Below is code for the Sierpinski triangle, invented by Waclaw Sierpinski,
a polish mathematician (mostly classical number theory) in the first half
of the 20th century.
------------------------Example 20-------------------------------------------
%!PS-Adobe-2.0
%%BoundingBox: 0 0 300 300

/level 7 def

/Triangle {level 0 eq
{newpath 0 0 moveto 300 0 lineto 150 150 3 sqrt mul lineto closepath fill}
{gsave 0.5 0.5 scale
/level level 1 sub def %Decrease level
Triangle           %Left Triangle
300 0 translate
Triangle           %Right Triangle
-150 150 3 sqrt mul translate
Triangle           %Top Triangle
/level level 1 add def %Increase level again
grestore}
ifelse
} def

Triangle

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