Dragon curve

Recursive construction of the curve

Recursive construction of the curve

The Heighway dragon can be constructed from a base line segment by repeatedly replacing each segment by two segments with a right angle and with a rotation of 45° alternatively to the right and to the left:[2]

The first 5 iterations and the 9th

The Heighway dragon is also the limit set of the following iterated function system in the complex plane:

${\displaystyle f_{1}(z)={\frac {(1+i)z}{2}}}$

${\displaystyle f_{2}(z)=1-{\frac {(1-i)z}{2}}}$

with the initial set of points ${\displaystyle S_{0}=\{0,1\}}$.

Using pairs of real numbers instead, this is the same as the two functions consisting of

${\displaystyle f_{1}(x,y)={\frac {1}{\sqrt {2}}}{\begin{pmatrix}\cos 45^{\circ }&-\sin 45^{\circ }\\\sin 45^{\circ }&\cos 45^{\circ }\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}}$

${\displaystyle f_{2}(x,y)={\frac {1}{\sqrt {2}}}{\begin{pmatrix}\cos 135^{\circ }&-\sin 135^{\circ }\\\sin 135^{\circ }&\cos 135^{\circ }\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}+{\begin{pmatrix}1\\0\end{pmatrix}}}$

The Heighway dragon curve can be constructed by folding a strip of paper, which is how it was originally discovered.[1] Take a strip of paper and fold it in half to the right. Fold it in half again to the right. If the strip was opened out now, unbending each fold to become a 90-degree turn, the turn sequence would be RRL, i.e. the second iteration of the Heighway dragon. Fold the strip in half again to the right, and the turn sequence of the unfolded strip is now RRLRRLL – the third iteration of the Heighway dragon. Continuing folding the strip in half to the right to create further iterations of the Heighway dragon (in practice, the strip becomes too thick to fold sharply after four or five iterations).

The folding patterns of this sequence of paper strips, as sequences of right (R) and left (L) folds, are:

• 1st iteration: R
• 2nd iteration: R R L
• 3rd iteration: R R L R R L L
• 4th iteration: R R L R R L L R R R L L R L L.

Each iteration can be found by copying the previous iteration, then an R, then a second copy of the previous iteration in reverse order with the L and R letters swapped.[1]

• Many self-similarities can be seen in the Heighway dragon curve. The most obvious is the repetition of the same pattern tilted by 45° and with a reduction ratio of ${\displaystyle \textstyle {\sqrt {2}}}$. Based on these self-similarities, many of its lengths are simple rational numbers.

Lengths

Self-similarities

Tiling of the plane by dragon curves

• The dragon curve can tile the plane. One possible tiling replaces each edge of a square tiling with a dragon curve, using the recursive definition of the dragon starting from a line segment. The initial direction to expand each segment can be determined from a checkerboard coloring of a square tiling, expanding vertical segments into black tiles and out of white tiles, and expanding horizontal segments into white tiles and out of black ones.[3]
• As a non-self-crossing space-filling curve, the dragon curve has fractal dimension exactly 2. For a dragon curve with initial segment length 1, its area is 1/2, as can be seen from its tilings of the plane.[1]
• The boundary of the set covered by the dragon curve has infinite length, with fractal dimension
  $${\displaystyle 2\log _{2}\lambda \approx 1.523627086202492,}$$

  
  where
  $${\displaystyle \lambda ={\frac {1+(28-3{\sqrt {87}})^{1/3}+(28+3{\sqrt {87}})^{1/3}}{3}}\approx 1.69562076956}$$

  
  is the real solution of the equation ${\displaystyle \lambda ^{3}-\lambda ^{2}-2=0.}$[4]