Koch Snowflake (also known as the Koch Curve, Koch Star or Koch Island) — it is a mathematical curve and one of the earliest fractal curves described. It is based on the Koch curve, published by the Swedish mathematician Helge von Koch in a 1904 article entitled “On a continuous curve without tangents, constructed from elementary geometry.”

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

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construction

Step 1:

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

It is best if the length of the sides is divisible by 3 due to the nature of this fractal. This will become clear in the next few steps.

Step 2:

Divide each side into three equal parts. This is why it is convenient to divide the sides into three. 

Step 3:

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

Step 4:

Separate each outside by a third. You can see the 2nd generation of triangles cover a bit of the 1st. These three segments should not be divided into three. 

Step 5:

Draw an equilateral triangle on each middle section.

Notice how you draw each successive generation of parts that are third in relation to the mast.

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Lindenmayer system view

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The Koch Curve can be expressed by the following rewriting system ( Lindenmayer system ):

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

Here F means "pull forward", — means turn right 60 °, and + means turn left 60 °. 
To create a Koch snowflake, you can use F ++ F ++ F (equilateral triangle) as an axiom.

To create a Koch curve:

# Python program to print a partial Koch curve. # import libraries: turtle standard # graphical library for python from turtle import *    # function to create 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__" :   # determine the speed of the turtle speed ( 0 )  length = 300.0      # Pull the handle up - no drawing while moving. penup ()     # Move the turtle back a distance, # opposite the direction of the turtle # headed by.   # Do not change the turtle’s course. backward ( length / 2.0 )    # Pull pen down - draw while moving . pendown ()    snowflake (length, 4 )   # To control the closing of turtle windows mainloop ()

Output: < / p>

To create a complete snowflake with a Koch curve, we need to repeat the same drawing three times. So let’s try this.

# Python program for printing the full Koch curve. from turtle import *   # function to create Koch snowflake or Koch curve def snowflake (lengthSide, levels): if levels = = 0 : forward (lengthSide) return < p> 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 __" : # determine the speed of the turtle speed ( 0 )  length = 300.0       # Pull handle up - no graphic while moving. # Move the turtle back the opposite distance # in the direction the turtle is heading # Do not change the turtle’s course. penup ()    backward (length / 2.0 )    # Pull the handle down - draw while moving. pendown ()  for i in range ( 3 ) :  snowflake (length, 4 ) right ( 120 )   # To control the closing of turtle windows   mainloop ()

Output:

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