Infinite complexity from folding paper
R = Right turn, L = Left turn
The dragon curve was discovered in 1966 by NASA physicist John Heighway while folding paper strips. Fold a strip in half repeatedly (always in the same direction), then unfold with 90° turns at each crease. The resulting path is the dragon curve—infinite complexity from the simplest possible operation!
The curve can be generated using a Lindenmayer system (L-system). F and G both mean "draw forward", + means turn right 90°, − means turn left 90°. After n iterations, the string describes exactly how to draw the n-th order dragon curve.
Despite being a curve (1D), the dragon curve has fractal dimension exactly 2—it's a space-filling curve! As iterations increase, it fills a bounded region completely. Yet it never crosses itself, always maintaining a single connected path. How can a line fill an area?
The dragon curve contains two copies of itself at each scale: one rotated 45° clockwise and scaled by 1/√2, one rotated 135° counterclockwise. This √2 ratio appears everywhere—in the paper folding, the area calculations, and even the curve's bounding box proportions.