The Heighway Dragon Curve

Fold a strip of paper in half, again and again, then unfold every crease to a right angle — that's the whole construction.

History

Discovered by paper-folding — NASA physicists John Heighway, Bruce Banks, and William Harter noticed the shape while folding strips of paper in half repeatedly. It reached a wide audience through Martin Gardner’s 1967 Scientific American column, and later appeared on the section-break pages of Michael Crichton’s Jurassic Park.

Anecdote

Fold a long strip of paper in half repeatedly, always the same direction, then unfold every crease to a right angle. The silhouette is a finite approximation of the dragon curve — one of the rare fractals here you can generate with nothing but paper and your hands.

Five panels: a strip folded three times, then unfolded to right angles — The five steps at a glance: fold three times, then unfold every crease to a right angle.

Animation: a folded strip of paper unfolding, crease by crease, into the dragon curve — The same unfolding step, slowed down: watch each of the 7 creases open to a right angle, one at a time — the flat folded strip unfurls straight into an 8-segment dragon curve.

Construction

Encode a right fold as $+1$ . The turn sequence after $n$ folds is built from the previous one:

T_0 = (), \qquad T_{n+1} = T_n \frown (+1) \frown \mathrm{reverse}(-T_n)

where $\frown$ joins two sequences end to end, and $-T_n$ flips every sign in $T_n$ . It reads more easily worked out by hand than stated abstractly. Start from the empty sequence and build up:

$T_0 = ()$ — no folds yet, nothing to turn.
$T_1 = T_0 \frown (+1) \frown \mathrm{reverse}(-T_0) = (+1)$ — one fold, one turn.
$T_2 = T_1 \frown (+1) \frown \mathrm{reverse}(-T_1) = (+1, +1, -1)$ — take $T_1$ , tack on a $+1$ , then tack on $T_1$ ‘s signs flipped and reversed (here $T_1$ is a single element, so reversing does nothing and only the flip shows).
$T_3 = T_2 \frown (+1) \frown \mathrm{reverse}(-T_2) = (+1, +1, -1, +1, +1, -1, -1)$ — seven turns for three folds, matching $2^3-1$ .

Walk unit steps, turning by each entry before stepping, and this sequence traces exactly the silhouette a strip of paper makes after being folded $n$ times and unfolded to right angles (Cool Fact box, this chapter). The boundary has Hausdorff dimension exactly $D=2$ , and four copies tile the plane.

★Cool Fact

The dragon curve can tile the entire plane with copies of itself: four suitably rotated dragon curves fit together edge-to-edge with no gaps or overlaps.

Python implementation

Pseudocode first

Algorithm in one glance Folding

Mirror, reverse, turn

folds(n) {
    turns = {LEFT}
    repeat n times
        turns = turns + {LEFT} + reverse(flip(turns))
    return turns
}

The executable version keeps the same shape; shared plotting/export details live in fractalfair_helpers.py.

def folds(n):
    turns = [1]
    for _ in range(n):
        turns = turns + [1] + [-t for t in reversed(turns)]
    return turns

λThe Same Idea in Haskell

dragonPoints :: Int -> [P2 Double]
dragonPoints n =
  scanl (\pt h -> pt .+^ rotateBy (h/4) unitX) (p2 (0,0)) headings
  where
    turns    = map fromIntegral (foldSeq n)
    headings = 0 : scanl1 (+) turns

Heighway dragon curve — **Figure:** 14 simulated folds (16,383 segments), coloured from deep violet to pale gold. Every inward spiral is a smaller, later fold nested inside a larger, earlier one.