Dragon Curve
A self-similar curve produced by repeatedly folding a strip of paper in half, then unfolding it at right angles.
Folding Paper
Take a strip of paper and fold it in half, always folding in the same direction. Unfold it so every crease is a right angle. Draw the resulting shape.
The L-system encoding:
Axiom: FX
X → X+YF+
Y → −FX−Y
F draws a segment. + turns left 90°. − turns right 90°. X and Y are non-drawing symbols that control the recursive unfolding.
Iteration 0: a single line segment. One fold. The axiom FX draws one step forward.
Unfolding
Each iteration doubles the number of segments. The key property: no two segments ever cross. The curve is non-self-intersecting at every level of iteration.
Iterations 1–5: the curve begins to curl. The right-angle structure is clear; there is a local symmetry around each fold point.
The Limit
At infinite iterations the curve fills a region of the plane. The boundary of that region has fractal dimension approximately 1.5236. The curve tiles the plane by rotation: four copies of the dragon curve, rotated 0°, 90°, 180°, and 270°, fit together without overlap or gap.
Iteration 12: the dragon is recognisable. The self-similar spiral motif repeats at every scale. The shape has a Hausdorff dimension strictly between 1 and 2.