Koch Curve or Koch Snowflake

What is Koch Curve?

The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a mathematical curve and one of the earliest fractal curves to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled “On a continuous curve without tangents, constructible from elementary geometry” by the Swedish mathematician Helge von Koch.

The progression for the area of the snowflake converges to 8/5 times the area of the original triangle, while the progression for the snowflake’s perimeter diverges to infinity. Consequently, the snowflake has a finite area bounded by an infinitely long line.




Construction

Step1:

Draw an equilateral triangle. You can draw it with a compass or protractor, or just eyeball it if you don’t want to spend too much time drawing the snowflake.

  • It’s best if the length of the sides are divisible by 3, because of the nature of this fractal. This will become clear in the next few steps.
  • Step2:

    Divide each side in three equal parts. This is why it is handy to have the sides divisible by three.

    Step3:

    Draw an equilateral triangle on each middle part. Measure the length of the middle third to know the length of the sides of these new triangles.

    Step4:

    Divide each outer side into thirds. You can see the 2nd generation of triangles covers a bit of the first. These three line segments shouldn’t be parted in three.

    Step5:


    Draw an equilateral triangle on each middle part.

  • Note how you draw each next generation of parts that are one 3rd of the mast one.

  • Representation as Lindenmayer system


    The Koch curve can be expressed by the following rewrite system (Lindenmayer system):

    Alphabet : F
    Constants : +, ?
    Axiom : F
    Production rules: F ? F+F–F+F

    Here, F means “draw forward”, – means “turn right 60°”, and + means “turn left 60°”.
    To create the Koch snowflake, one would use F++F++F (an equilateral triangle) as the axiom.

    To create a Koch Curve :

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    # Python program to print partial Koch Curve.
    # importing the libraries : turtle standard 
    # graphics library for python
    from turtle import *
      
    #function to create koch snowflake or koch curve
    def snowflake(lengthSide, levels):
        if levels == 0:
            forward(lengthSide)
            return
        lengthSide /= 3.0
        snowflake(lengthSide, levels-1)
        left(60)
        snowflake(lengthSide, levels-1)
        right(120)
        snowflake(lengthSide, levels-1)
        left(60)
        snowflake(lengthSide, levels-1)
      
    # main function
    if __name__ == "__main__":
      
        # defining the speed of the turtle
        speed(0)                   
        length = 300.0              
      
        # Pull the pen up – no drawing when moving.
        penup()                     
          
        # Move the turtle backward by distance, 
        # opposite to the direction the turtle 
        # is headed.
        # Do not change the turtle’s heading.
        backward(length/2.0)        
      
        # Pull the pen down – drawing when moving.
        pendown()         
      
        snowflake(length, 4)
      
        # To control the closing windows of the turtle
        mainloop() 

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    Output:

    To create a full snowflake with Koch curve, we need to repeat the same pattern three times. So lets try that out.

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    # Python program to print complete Koch Curve.
    from turtle import *
      
    # function to create koch snowflake or koch curve
    def snowflake(lengthSide, levels):
        if levels == 0:
            forward(lengthSide)
            return
        lengthSide /= 3.0
        snowflake(lengthSide, levels-1)
        left(60)
        snowflake(lengthSide, levels-1)
        right(120)
        snowflake(lengthSide, levels-1)
        left(60)
        snowflake(lengthSide, levels-1)
      
    # main function
    if __name__ == "__main__":
        # defining the speed of the turtle
        speed(0)                   
        length = 300.0   
      
        # Pull the pen up – no drawing when moving.
        # Move the turtle backward by distance, opposite
        # to the direction the turtle is headed.
        # Do not change the turtle’s heading.           
        penup()                     
      
        backward(length/2.0)
      
        # Pull the pen down – drawing when moving.        
        pendown()           
        for i in range(3):    
            snowflake(length, 4)
            right(120)
      
         # To control the closing windows of the turtle
        mainloop()       

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    Output:

    
    

    This article is contributed by Subhajit Saha. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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