Dragon curve
Fold a strip of paper in half n times and open every fold to a right angle: the edge draws the Heighway dragon.
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Formulas and key values
- Construction rule
Each fold appends a copy of the whole path, reversed and turned 90° about its end.
- Arc lengthIn the graph ≈ 32
- SegmentsIn the graph = 1024
- Boundary dimension
n is the number of folds; the ends are held 1 apart.
Explainer video
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Opens the Creator with a ready-made brief for this curve. Free-form generation is in beta.
Start the video01 · Definition
What is the Dragon curve?
Take a strip of paper, fold it in half, fold it again in the same direction, and keep going. Unfold it so every crease makes a right angle, and the strip traces the dragon curve. NASA physicists John Heighway, Bruce Banks and William Harter studied it in the 1960s; Martin Gardner introduced it to readers of Scientific American in 1967.
Algebraically, each fold appends a copy of the whole path, reversed and turned 90° about its end. The curve never crosses itself, although it touches itself at corners. Four copies around a point tile the plane, and the boundary of the limit shape has fractal dimension about 1.5236.
02 · Properties
Key properties
- After n folds the path has 2ⁿ segments, all meeting at right angles.
- With the endpoints held at distance 1, the length is (√2)ⁿ.
- The turn sequence is the regular paper-folding sequence.
- It never crosses itself; it only touches itself at corners.
- Copies of the limit shape tile the plane; its boundary has dimension ≈ 1.5236.
03 · Questions
Dragon curve: common questions
- How is the Dragon curve constructed?
- Construction rule: Each fold appends a copy of the whole path, reversed and turned 90° about its end.. n is the number of folds; the ends are held 1 apart.
- What is the arc length of the Dragon curve?
- Arc length: L_n = (√2)^n. n is the number of folds; the ends are held 1 apart.
04 · Keep exploring