In the article on Koch Curves we derived the famous koch-curve, by starting with a line, then a bumpy-line, then a self-similar recursive bumpy-line. The final result looked like this:

to koch_curve :length :limit
 ifelse :limit = 0
[ fw :length ]
[
koch_curve :length/3 :limit - 1
lt 60 koch_curve :length/3 :limit - 1
rt 120 koch_curve :length/3 :limit -1
lt 60 koch_curve :length/3 :limit - 1
]
end

koch_curve 240 5

Now we can put three of them together to make Koch's snowflake.

to koch_curve :length :limit
 ifelse :limit = 0
[ fw :length ]
[
koch_curve :length/3 :limit - 1
lt 60 koch_curve :length/3 :limit - 1
rt 120 koch_curve :length/3 :limit -1
lt 60 koch_curve :length/3 :limit - 1
]
end

to koch_snowflake :length :limit
repeat 3
  [
    koch_curve :length :limit
    rt 120
  ]
end

koch_snowflake 240 5

Layered Koch Snowflake

Layering several of them on top of each other looks pretty cool.

to koch_curve :length :limit
 ifelse :limit = 0
[ fw :length ]
[
koch_curve :length/3 :limit - 1
lt 60 koch_curve :length/3 :limit - 1
rt 120 koch_curve :length/3 :limit -1
lt 60 koch_curve :length/3 :limit - 1
]
end

to koch_snowflake :length :limit
repeat 3 
[
    koch_curve :length :limit
    rt 120
]
end

koch_snowflake 240 1
koch_snowflake 240 2
koch_snowflake 240 3
koch_snowflake 240 4
koch_snowflake 240 5

Reverse Koch Snowflake

If you're thinking what I'm thinking, then what you are thinking is "If you're thinking what I'm thinking, then what you are thinking is 'Why turn right? Why not turn left?'" Which is very clever of you. If we turn our 3 Koch curves around (by turning those right turns to left turns), they will face inward, not outward, and we'll have a reverse Koch Snowflake.

to koch_curve :length :limit
 ifelse :limit = 0
[ fw :length ]
[
koch_curve :length/3 :limit - 1
lt 60 koch_curve :length/3 :limit - 1
rt 120 koch_curve :length/3 :limit -1
lt 60 koch_curve :length/3 :limit - 1
]
end

to reverse_koch_snowflake :length :limit
repeat 3 
[
  koch_curve :length :limit
  lt 120
]
end

reverse_koch_snowflake 240 5

There we have an inside-outed Koch Snowflake.

Layered Reverse Koch Curve

We can layer that as well.

to koch_curve :length :limit
 ifelse :limit = 0
[ fw :length ]
[
koch_curve :length/3 :limit - 1
lt 60 koch_curve :length/3 :limit - 1
rt 120 koch_curve :length/3 :limit -1
lt 60 koch_curve :length/3 :limit - 1
]
end

to reverse_koch_snowflake :length :limit
repeat 3 
[
    koch_curve :length :limit
    lt 120
]
end

reverse_koch_snowflake 240 1
reverse_koch_snowflake 240 2
reverse_koch_snowflake 240 3
reverse_koch_snowflake 240 4
reverse_koch_snowflake 240 5

Extra Long Reverse Koch Snowflake

Oh, one more trick.

What happens if we make an extra long reverse Koch Snowflake? Instead of putting 3 reverse snowflakes together, let's put 6 of them together. We'll need a smaller angle: instead of 120 degrees between one, we'll only need 60 degrees.

(And since it's going to be a larger structure, I'll make the sides a bit smaller, 120 instead of 240)

to koch_curve :length :limit
 ifelse :limit = 0
[ fw :length ]
[
koch_curve :length/3 :limit - 1
lt 60 koch_curve :length/3 :limit - 1
rt 120 koch_curve :length/3 :limit -1
lt 60 koch_curve :length/3 :limit - 1
]
end

to reverse_koch_snowflake :length :limit
repeat 6 
[
  koch_curve :length :limit
  lt 60
]
end

reverse_koch_snowflake 120 5

Ah, a double length reverse Koch Snowflake is just a Koch Snowflake.

Why? Because the spaces between Koch Snowflakes are themselves Koch Snowflakes.

These Things Pack Together With No Room To Spare!

to koch_curve :length :limit
 ifelse :limit = 0
[ fw :length ]
[
koch_curve :length/3 :limit - 1
lt 60 koch_curve :length/3 :limit - 1
rt 120 koch_curve :length/3 :limit -1
lt 60 koch_curve :length/3 :limit - 1
]
end

to koch_snowflake :length :limit
repeat 3 
[
    koch_curve :length :limit
    rt 120
]
end


repeat 3 [
  koch_snowflake 120 4
  pu
   fd 80 lt 60 fd 80 rt 60 
  pd
]
setxy  5 290
repeat 4 [
  koch_snowflake 120 4
  pu
   fd 80 lt 60 fd 80 rt 60 
  pd
]
setxy  5 430
repeat 4 [
  koch_snowflake 120 4
  pu
   fd 80 lt 60 fd 80 rt 60 
  pd
]