A dragon curve is a recursive nonintersecting curve whose name derives from its resemblance to a certain mythical creature.
The curve can be constructed by representing a left turn by 1 and a right turn by 0. The first-order curve is then denoted 1. For higher order curves, append a 1 to
the end, then append the string of preceding digits with its middle digit complemented.
For example, the second-order curve is generated as follows: , and the third as .
Continuing gives 110110011100100... (OEIS A014577), which is sometimes known as the regular paperfolding sequence and written with s instead of 0s (Allouche and Shallit
2003, p. 155). A recurrence plot of the limiting
value of this sequence is illustrated above.
The same turn sequence gives an ideal zero-thickness ribbon with parallel hinges and right
angle folds. The illustration uses the first 15 turns, with 1 and 0 representing
opposite fold directions. It encodes the paperfolding sequence
without imposing finite-thickness constraints.
Representing the sequence of binary digits 1, 110, 1101100, 110110011100100, ... in octal
gives 1, 6, 154, 66344, ...(OEIS A003460; Gardner
1978, p. 216).
This procedure is equivalent to drawing a right angle and subsequently replacing each right angle with another
smaller right angle (Gardner 1978). In fact, the dragon
curve can be written as a Lindenmayer system
with initial string "FX", string
rewriting rules "X" -> "X+YF+", "Y" ->
"-FX-Y", and angle . The dragon curves of orders 1 to 9 are illustrated
above, with corners rounded to emphasize the path taken by the curve.
The regular paperfolding sequence can also be obtained from the fixed point beginning with of the Lindenmayer
system,
, , and , followed by the coding and to 1 and and to 0. For a two-sided extension, place a unit point mass at
each integer position carrying a 1. The Fourier
transform of its volume-averaged autocorrelation
is a pure point measure, meaning it consists only of
point masses (Baake et al. 2003).
, and the third as 
.
s instead of 0s (Allouche and Shallit
2003, p. 155). A recurrence plot of the limiting
value of this sequence is illustrated above.
. The dragon curves of orders 1 to 9 are illustrated
above, with corners rounded to emphasize the path taken by the curve.
of the Lindenmayer
system 
,
, 
, and 
, followed by the coding 
and 
to 1 and 
and 
to 0. For a two-sided extension, place a unit point mass at
each integer position carrying a 1. The Fourier
transform of its volume-averaged autocorrelation
is a pure point measure, meaning it consists only of
point masses (Baake et al. 2003).
