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.
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.
, 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.
