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9.3 A fractal example: Van Koch snowflake

Using recursion, it’s very easy to generate in LOGO some special curves called fractals in mathematics.

Here are the first steps to create the Van Koch broken line:

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Between two steps:

  1. each segment is divided into three equal part.
  2. an equilateral triangle is drawn on the middle segment.
  3. finally, this middle segment is erased.

What is important: Let’s have a look at step 2, we can see that the broken lines contains four identical motifs corresponding to precedent step with a 3 lesser size. Here we have found the recursive structure of the fractal.

Let’s call Ln,ℓ the motif of size ℓ, corresponding to step n.
To draw this motif:

  1. We draw Ln-1,ℓ∕3
  2. We turn left 60 degrees
  3. We draw Ln-1,ℓ∕3
  4. We trun right 120 degrees
  5. We draw Ln-1,ℓ∕3
  6. We trun left 60 degrees
  7. We draw Ln-1,ℓ∕3

With LOGO, it’s very easy to write:

# :l motif size  
# :p step  
to line :l :p  
if :p=0 [fd :l] [  
  line :l/3 :p-1 lt 60 line :l/3 :p-1 rt 120 line :l/3 :p-1 lt 60 line :l/3 :p-1  
]  
end

If we draw an equilateral triangle with three Van Koch lines, we obtain a beautiful Van Koch snowflake.

# :l side length  
to snowflake :l :p  
repeat 3[line :l :p rt 120]  
end

Then run: snowflake 200 6

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