Koch Snowflake

What is is?

Wolfram's Koch Snowflake.

The Recipe

1. Take a line.
   
2. Divide into three parts.
3. Take the middle part and create an equilateral triangle bump from it.
4. Repeat steps using each of the lines from the new shape.

Visually

To make the Koch snowflake, you start with three lines that make an equilateral triangle.

Ruby Version

# A vector implementation I made
require 'vector.rb'

class Koch
 # Creates a Koch line from point p1 to point p2
 def initialize(p1,p2)
  @points = [p1,p2]
 end


 # Creates the next step in the fractal
 def step
  p1 = @points.shift
  new_array = [p1]

  # For all points in the shape,
  # divide the line from next p1
  # to next p2.
  while(@points.length > 0)
   p2 = @points.shift

   third = Vector.between(p1,p2)
    third.multiply(0.333)
   two_third = Vector.between(p1,p2)
    two_third.multiply(0.666)

   # Adds p1 to the delta between p1 and p2 to
   # get the actual 1/3 and 2/3 way points.
   third.add_vector(p1)
   two_third.add_vector(p1)

   # Creates bump roughly equivalent to an equilateral triangle
   # This is the key line here: the midpoint is moved along the normal
   # (perpendicular line) of the line.  If you do Vector.between(two_third,third)
   # the normal points the opposite way and the bump will be backwards.
   top = Vector.midpoint(third,two_third)
    top.add_vector(Vector.between(third,two_third).normal)
   
   # Adds the subdivisions into the point array.
   new_array << third << top << two_third << p2

   # Moves to the next point
   p1 = p2
  end

  @points = new_array
 end


 def gl_paint
  GL.Begin(GL::LINE_STRIP)
  for p in @points 
   GL.Vertex2f(p.x,p.y)
  end
  GL.End
 end
end