Perhaps the best known Space Filling Curve is the Hilbert Curve.

to hilbert :count :size :angle
if :count > 0
[
  rt 90*:angle
  hilbert (:count - 1) :size (:angle * (0-1) )
  fd :size 
  rt 90*:angle * (0-1)
  hilbert (:count - 1) :size :angle
  fd :size
  hilbert (:count - 1) :size :angle
  rt 90*:angle * (0-1)
  fd :size
  hilbert (:count - 1) :size (:angle * (0-1) )
  rt 90*:angle
]
end
hilbert 1 (2*63/1) (1)
fd (63/1)
hilbert 2 (2*63/3) (1)
fd (63/3)
hilbert 3 (2*63/7) (1)
fd (63/7)
setxy 10 150
hilbert 4 (2*63/15) (1)
fd (63/15)
hilbert 5 (2*63/31) (1)
fd (63/31)
hilbert 6 (2*63/63) (1)

Let's see how it looks with a bevel on each corner. I'm going to replace all calls to 'rt' (right turn) with a new function, rti, which "cuts" the corner.

to rti :angle :size
 rt :angle/2
 fd :size/(4*sqrt(2))
 rt :angle/2
end
to hilbert :count :size :angle
if :count > 0
[
  rti 90*:angle :size
  hilbert (:count - 1) :size (:angle * (0-1) )
  fd :size*(2/4)
  rti 90*:angle * (0-1) :size
  hilbert (:count - 1) :size :angle
  fd :size*(2/4)
  hilbert (:count - 1) :size :angle
  rti 90*:angle * (0-1) :size
  fd :size*(2/4)
  hilbert (:count - 1) :size (:angle * (0-1) )
  rti 90*:angle :size
]
end
hilbert 1 (2*63/1) (1)
fd (63/1)
hilbert 2 (2*63/3) (1)
fd (63/3)
hilbert 3 (2*63/7) (1)
setxy 10 150
fd (63/7)
hilbert 4 (2*63/15) (1)
fd (63/15)
hilbert 5 (2*63/31) (1)
fd (63/31)
hilbert 6 (2*63/63) (1)

External Links

• Portrait of the Hilbert Curve
• Wikipedia: Hilbert Curve

See Also

• Space Filling Curves (includes the Peano curve)
• Flowsnake (the Peano-Gosper curve)
• Dragon curve