Peano Curve

- plane filling curve

Blue Ball

To draw this curve, the turtle moves through a patch of ground, which is divided into 9 squares.

Starting at the lower left, it first turns left, then right 3 times, left 3 times and a final right turn. The turtle emerges upper right having moved diagonally through each of the 9 squares.

By recursion, each square can be further divided into 9 smaller squares. The turtle ends up passing through every point on the patch of ground.

 Turtle Path

To New
  # set default screen, pen and turtle values
  ResetAll SetScreenSize [400 400] HideTurtle
  SetSC Black SetPC Green SetPS 1 PenUp
End

To Init :Level
  GlobalMake "Size (SqRt 2)*(64/(Power 3 :Level))
  PenUp SetPos [-192 -192] SetH 45 Forward :Size
  PenDown     # comment out for curved corners
End

To Peano :Level
  If :Level < 0 [Stop]
  Peano :Level-1
  Left90 :Size Peano :Level-1
  Repeat 3 [Rite90 :Size Peano :Level-1]
  Repeat 3 [Left90 :Size Peano :Level-1]
  Rite90 :Size Peano :Level-1
End

To Left90 :Radius
  Forward :Radius Left 90
  Forward :Radius
End

To Rite90 :Radius
  Forward :Radius Right 90
  Forward :Radius
End

To rArc :Angle :Radius
  # clockwise arc drawn relative to turtle heading
  Arc :Radius Heading :Angle+Heading Right :Angle
End

To Go :Level
  New Init :Level Peano :Level
End

To GoX
  New For [Level 0 3] [
    SetPC :Level+1 SetPW 7-(2*:Level)
    Init :Level Peano :Level Wait 30]
End

Enter go 0 for the basic path as in the diagram above.
Enter go 1, go 2 or go 3 for increasingly complex paths.
Around go 4 or go 5 the turtle fills the screen completely.

Add any pair of procedures for different styles of left and right 90 degree turns, which produce quite different looking curves.
Comment out the PenDown in the Init procedure for corners using rArcs.

To Left90 :Radius
  # curved left 90 corner
  Right 90 Back :Radius Left 90
  rArc 90 :Radius
  Left 90 Forward :Radius Left 90
End

To Rite90 :Radius
  # curved right 90 corner
  Left 90 Back :Radius
  rArc 90 :Radius
  Forward :Radius Right 90
End

To Left90 :Radius
  # octagon corner
  Forward 0.414*:Radius Left 45
  Forward 0.828*:Radius Left 45
  Forward 0.414*:Radius
End

To Rite90 :Radius
  # octagon corner
  Forward 0.414*:Radius Right 45
  Forward 0.828*:Radius Right 45
  Forward 0.414*:Radius
End

To Left90 :Radius
  # left 45 turn
  Left 45 Forward 1.414*:Radius Left 45
End

To Rite90 :Radius
  # right 45 turn
  Right 45 Forward 1.414*:Radius Right 45
End

To Left90 :Radius
  # anti-curve left 90 corner
  Forward :Radius Left 180
  rArc 90 :Radius
  Forward :Radius
End

To Rite90 :Radius
  # anti-curve right 90 corner
  Forward :Radius Right 90
  rArc 90 :Radius
  Left 90 Forward :Radius
End

Paste one pair of corners into the XLogo editor.

Peano Curve
Peano Curve

Procedures blue
Variables pink
Comments green
Library gray