Dragon Curve
Take a strip of paper, fold it in half repeatedly, then unfold every crease to 90°. The resulting shape is the Heighway dragon — a space-filling curve built from nothing but right-angle folds. At each iteration the number of segments doubles, and the curve never crosses itself.
Sₙ₊₁ = Sₙ + R + reverse(flip(Sₙ)) dimH = 2 segments = 2n
Paper folding and the dragon curve
The Heighway dragon was first described by NASA physicists John Heighway, Bruce Banks, and William Harter in the 1960s. The construction is beautifully simple: take a strip of paper, fold it in half (always in the same direction), repeat, then unfold every crease to a 90° angle. After n folds, the paper traces a path with 2n segments that never crosses itself.
The construction rule
Algorithmically, the dragon curve is built by maintaining a sequence of turns (left = L, right = R). At each iteration, you take the current sequence, append R, then append the reverse of the current sequence with every turn flipped (L↔R). Starting from empty: iteration 1 gives R, iteration 2 gives R-R-L, iteration 3 gives R-R-L-R-R-L-L, and so on. Each turn becomes a 90° left or right turn when drawing the curve.
The twindragon
The twindragon (or Davis-Knuth dragon) is formed by placing two Heighway dragons back-to-back, sharing their starting segment but facing opposite directions. The result tiles the plane — copies of the twindragon can fill space without gaps or overlaps. It has a more symmetric appearance than the single dragon.
Space-filling properties
The boundary of the Heighway dragon has Hausdorff dimension approximately 1.5236, but the curve itself is space-filling — in the limit, it covers a region of the plane with Hausdorff dimension 2. The dragon is also a rep-tile: four copies of itself can be arranged to form a larger copy. This self-similarity is the heart of its fractal nature.
Coloring by iteration depth
When segments are colored by the iteration at which they were added, the fractal’s recursive structure becomes visible. The earliest segments (from the first few folds) appear in one color, while later refinements layer on top in different hues — revealing the self-similar structure at every scale.