Geometric

Dragon curve

Discovered by John Heighway, Bruce Banks, and William Harter at NASA in 1966 (and popularized by Mandelbrot, then by appearing on the chapter dividers of Crichton’s Jurassic Park). Fold a strip of paper in half repeatedly, always in the same direction, then unfold so each crease is a right angle. The trace is the dragon curve.

At a glance

Constructions

• Paper-folding sequence: fold a strip of paper in half n times always in the same direction; unfold so each crease is 90°. The trace is the level-n dragon curve.
• L-system: axiom F X, rules X → X + Y F +, Y → − F X − Y; + means turn left 90°, − means turn right.
• IFS: two affine contractions, each a rotation by ±45° followed by scaling by 1/√2.

Surprising properties

• Plane-filling: the curve is space-filling (dimension 2), it covers a region of positive area in the limit.
• Self-similar boundary: the boundary of the filled region has Hausdorff dimension ≈ 1.5236, exactly the positive real root of 2x4 − 2x2 − 1 = 0.
• Tessellates the plane: four dragon curves rotated by 90° each tile a region around the starting point.
• No self-intersection: at every finite step, the curve never crosses itself, despite filling space.

References

• Davis, C. & Knuth, D., “Number representations and dragon curves,” J. Recreational Mathematics, 1970.
• Mandelbrot, B., The Fractal Geometry of Nature, W. H. Freeman, 1982.
• Edgar, G., Classics on Fractals, Westview, 1993.
• Koch snowflake · Sierpinski triangle

Try it

Run an interactive playground at /tools/dragon.