Fractals (#125)

I lied to Morton G…

I told him I thought a turtle graphics solution would be a unique
approach not
many people would try. Boy was I wrong! That turns out to be the
easiest way
to attack this problem and the solvers knew that a lot better than I

The solutions are very diverse and I had a hard time picking what to
show. Some
solutions allowed you to customize the fractal drawn. Others offered
forms of output, including some pretty pictures. Jesse M.'s
solution did
it all and I do recommend having a peek at that code.

For this summary though, I’m going to show Donald B.'s solution. I
think it
captured the technique most people used well and it also included a
unique form
of output. Let’s take a look:

def instructions(level=1, list=[:forward])
if level == 1
instructions(level-1, do |item|
case item
when :forward
[:forward, :left, :forward, :right,
:forward, :right, :forward, :left, :forward]
else item

This method sums up the shortcuts pretty much everyone used. First, we
turtle graphics style drawing. If we assume a turtle starts facing
east, the
Array in the middle of this method contains the instructions to draw
level one
of the fractal. It’s much easier to work with a format like this than
it is to
edit a large String or Array of paths.

The other shortcut was a simplification of the drawing pattern. It
turns out
that just replacing all occurrences of the level zero pattern with the
level one
pattern is all it takes to transform any level of the pattern into the

Knowing that, this method is a recursive implementation of those
shortcuts. It
begins with the level zero pattern and for each level it performs the
replacement I mentioned earlier. The end result will be a long path for
the entire target level.

The next step is to convert those directions into and collection of
points that
could be plotted:

DIRECTIONS = [:east, :south, :west, :north]

def plot(instructions)
p = [0, 0]
points = [p]
direction = :east
instructions.each do |item|
case item
when :forward
p = p.dup
case direction
when :east: p[0] += 1
when :south: p[1] -= 1
when :west: p[0] -= 1
when :north: p[1] += 1
points << p
when :left
direction = DIRECTIONS[(DIRECTIONS.index(direction) - 1) % 4]
when :right
direction = DIRECTIONS[(DIRECTIONS.index(direction) + 1) % 4]

This method generates a collection of points from the path of :forward,
and :right instructions. It works by tracking the direction the turtle
currently facing and moving by incrementing the points as indicated for
direction. Turns simply change the direction.

Now, it’s not so easy to flatten the coordinates into an ASCII art
string and we
really want to see these fractals as an image anyway, so Donald opted
for an
unusual means of output. Donald’s program builds a web page that can
draw the
fractal (as long as it is displayed in Safari or Firefox). Here’s the

def to_html(points, width=600, height=400)
length = [width/ {|p| p[0]}.max,
height/ {|p| p[1]}.max].min! {|p| {|x| x*length}}
s = ‘<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" ' s <’
s << “\n”
s << “”
s << “”
s << “”
s << “”

The work here is pretty simple. First the code finds a good line width
for the
fractal by measuring how much space the canvas has for width and height.
the points are then scaled by that length. This is done with simple
multiplication since the scale was previously one.

The output is some HTML preamble followed by a JavaScript section that
does the
real work. The prepares a drawing context and then literally just plays
the dots with the points. There’s a little more HTML to close things up
invoke the drawing routine and, presto, we have a self drawing web page.

As written, this code draws fractals upside down (in comparison to the
examples). If you want to flip them over, just change to two #{p[1]}
interpolations in the JavaScript to #{height - p[1]}.

We’re just short the code to kick-off the process:

if $0 == FILE
puts to_html(plot(instructions(ARGV[0].to_i))) if ARGV.length == 1

As usual, this code is trivial. Generate instructions() for the
lever, hand those off to plot() to get the coordinates, and filter those
to_html() to get a printable web page.

My thanks to all who showed me just how powerful turtle graphics can be.
should have never doubted.

Tomorrow we will give a little time to a problem that has been making
the rounds
in the hope of making ourselves more employable…