View source for Dragon curve

{{short description|Fractal constructible with L-systems}}
{{redirects|Dragon fractal|the filled Julia sets|Douady rabbit}}
{{redirect|Twindragon|other uses|Twin Dragon (disambiguation)}}
{{Use dmy dates|date=May 2019|cs1-dates=y}}
[[File:Fractal dragon curve.jpg|thumb|Heighway dragon curve]]
A '''dragon curve''' is any member of a family of [[Self-similarity|self-similar]] [[fractal curves]], which can be approximated by [[recursion|recursive]] methods such as [[Lindenmayer system]]s. The dragon curve is probably most commonly thought of as the shape that is generated from repeatedly folding a strip of paper in half, although there are other curves that are called dragon curves that are generated differently.

== Heighway dragon ==
The '''Heighway dragon''' (also known as the '''Harter–Heighway dragon''' or the '''Jurassic Park dragon''') was first investigated by [[NASA]] physicists John Heighway, Bruce Banks, and William Harter. It was described by [[Martin Gardner]] in his [[Scientific American]] column ''[[Mathematical Games (column)|Mathematical Games]]'' in 1967. Many of its properties were first published by [[Chandler Davis]] and [[Donald Knuth]].  It appeared on the section title pages of the [[Michael Crichton]] novel ''[[Jurassic Park (novel)|Jurassic Park]]''.<ref name=tabachnikov>{{citation
 | last = Tabachnikov | first = Sergei
 | doi = 10.1007/s00283-013-9428-y
 | issue = 1
 | journal = The Mathematical Intelligencer
 | mr = 3166985
 | pages = 13–17
 | title = Dragon curves revisited
 | volume = 36
 | year = 2014| s2cid = 14420269
 }}</ref>

=== Construction ===
[[File:Dragon Curve unfolding zoom numbered.gif|300px|Recursive construction of the curve|alt=|thumb]]
[[File:Dragon Curve adding corners trails rectangular numbered R.gif|300px|Recursive construction of the curve|alt=|thumb]]
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:<ref>{{citation
 | last = Edgar | first = Gerald
 | editor1-first = Gerald
 | editor1-last = Edgar
 | contribution = Heighway's Dragon
 | doi = 10.1007/978-0-387-74749-1
 | edition = 2nd
 | isbn = 978-0-387-74748-4
 | mr = 2356043
 | pages = 20–22
 | publisher = Springer | location = New York
 | series = Undergraduate Texts in Mathematics
 | title = Measure, Topology, and Fractal Geometry
 | year = 2008}}</ref>
[[File:Dragon curve iterations (2).svg|none|700px|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:

:<math>f_1(z)=\frac{(1+i)z}{2}</math>
:<math>f_2(z)=1-\frac{(1-i)z}{2}</math>

with the initial set of points <math>S_0=\{0,1\}</math>.

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

:<math>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}</math>

:<math>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}</math>

=== Folding the dragon ===
The Heighway dragon curve can be constructed by [[regular paperfolding sequence|folding a strip of paper]], which is how it was originally discovered.<ref name=tabachnikov/> 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).

[[File:Dragon curve paper strip.png|center|800px]]

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.<ref name=tabachnikov/>

=== Properties ===
* 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 <math>\textstyle{\sqrt{2}}</math>. Based on these self-similarities, many of its lengths are simple rational numbers.
{{multiple image|total_width=800|align=center|image1=Dimensions fractale dragon.png|caption1=Lengths|image2=Auto-similarity dragon curve.png|caption2=Self-similarities}}
[[File:Full tiling dragon.svg|thumb|upright=1.35|Tiling of the plane by dragon curves]]
* The dragon curve can [[Tessellation|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.<ref>{{harvtxt|Edgar|2008}}, "Heighway’s Dragon Tiles the Plane", pp. 74–75.</ref>
* As a [[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.<ref name=tabachnikov/>
* The boundary of the set covered by the dragon curve has infinite length, with fractal dimension <math display=block>2\log_2\lambda\approx 1.523627086202492,</math> where <math display=block>\lambda=\frac{1+(28-3\sqrt{87})^{1/3}+(28+3\sqrt{87})^{1/3}}{3}\approx 1.69562076956</math> is the real solution of the equation <math>\lambda^3-\lambda^2-2=0.</math><ref>{{harvtxt|Edgar|2008}}, "Heighway Dragon Boundary", pp. 194–195.</ref>

== Twindragon ==
[[File:Twindragon.png|thumb|Twindragon curve constructed from two Heighway dragons]]{{See also|Complex-base system#Base −1 ± i}}
The '''twindragon''' (also known as the '''Davis–Knuth dragon''') can be constructed by placing two Heighway dragon curves back to back (after flipping the original dragon curve vertically and horizontally). It is also the limit set of the following iterated function system:

:<math>f_1(z)=\frac{(1+i)z}{2}</math>
:<math>f_2(z)=1-\frac{(1+i)z}{2}</math>

where the initial shape is defined by the following set <math>S_0 = \{0,1,1-i\}</math>.

It can be also written as a [[Lindenmayer system]] – it only needs adding another section in the initial string:
* angle 90°
* initial string ''FX+FX+''
* string rewriting rules
** ''X'' {{mapsto}} ''X''+''YF''
** ''Y'' {{mapsto}} ''FX''−''Y''.
It is also the locus of points in the complex plane with the same integer part when written in [[Complex-base system#Base −1 ± i|base <math>(-1 \pm i)</math>]].<ref name="Knuth2">{{cite book |last=Knuth |first=Donald |title=The art of computer programming |publisher=Addison-Wesley |year=1998 |isbn=0-201-89684-2 |edition=3rd |volume=2 |location=Boston |pages=206 |chapter=Positional Number Systems |oclc=48246681 |authorlink=Donald Knuth}}</ref>

== Terdragon ==
[[File:Terdragon.png|thumb|left|200px|Terdragon curve.]]
[[File:Developing Terdragon Curve.jpg|thumb|right| text-bottom |300px| A sculpture depicting multiple iterations of the Lindenmayer system that generates the terdragon curve.<br>by [[Henry Segerman]]]]
The '''terdragon''' can be written as a [[Lindenmayer system]]:
* angle 120°
* initial string ''F''
* string rewriting rules
** ''F'' {{mapsto}} ''F+F−F''.

It is the limit set of the following iterated function system:

:<math>f_1(z)=\lambda z</math>
:<math>f_2(z)=\frac{i}{\sqrt{3}}z + \lambda</math>
:<math>f_3(z)=\lambda z + \lambda^*</math>
:<math>\mbox{where }\lambda=\frac{1}{2}-\frac{i}{2\sqrt{3}}
\text{ and }\lambda^*=\frac{1}{2}+\frac{i}{2\sqrt{3}}.</math>

== Lévy dragon ==
The [[Lévy C curve]] is sometimes known as the '''Lévy dragon'''.<ref>{{citation
 | last1 = Bailey | first1 = Scott
 | last2 = Kim | first2 = Theodore
 | last3 = Strichartz | first3 = Robert S.
 | doi = 10.2307/3072395
 | issue = 8
 | journal = [[The American Mathematical Monthly]]
 | mr = 1927621
 | pages = 689–703
 | title = Inside the Lévy dragon
 | volume = 109
 | year = 2002| jstor = 3072395
 }}.</ref>

{|
|[[File:Lévy's C-curve (IFS).jpg|thumb|200px|Lévy C curve.]]
|}

== Occurrences of the dragon curve in solution sets ==
Having obtained the set of solutions to a linear differential equation, any linear combination of the solutions will, because of the [[superposition principle]], also obey the original equation. In other words, new solutions are obtained by applying a function to the set of existing solutions. This is similar to how an iterated function system produces new points in a set, though not all IFS are linear functions.
In a conceptually similar vein, a set of [[Littlewood polynomial]]s can be arrived at by such iterated applications of a set of functions.

A Littlewood polynomial is a polynomial: <math> p(x) = \sum_{i=0}^n a_i x^i \, </math> where all <math>a_i = \pm 1</math>.

For some <math>|w| < 1</math> we define the following functions:
:<math> f_+(z) = 1 + wz</math>
:<math> f_-(z) = 1 - wz</math>

Starting at z=0 we can generate all Littlewood polynomials of degree d using these functions iteratively d+1 times.<ref name='ncafe'>{{Cite web|url=http://golem.ph.utexas.edu/category/2009/12/this_weeks_finds_in_mathematic_46.html|title = The n-Category Café}}</ref> For instance: <math>f_+(f_-(f_-(0))) = 1 + (1-w)w = 1 + 1w - 1w^2</math>

It can be seen that for <math>w=(1+i)/2</math>, the above pair of functions is equivalent to the IFS formulation of the Heighway dragon. That is, the Heighway dragon, iterated to a certain iteration, describe the set of all Littlewood polynomials up to a certain degree, evaluated at the point <math>w=(1+i)/2</math>.
Indeed, when plotting a sufficiently high number of roots of the Littlewood polynomials, structures similar to the dragon curve appear at points close to these coordinates.<ref name='ncafe' /><ref>{{Cite web|url=http://math.ucr.edu/home/baez/week285.html|title=Week285}}</ref><ref>{{Cite web|url=http://johncarlosbaez.wordpress.com/2011/12/11/the-beauty-of-roots/|title = The Beauty of Roots|date = 11 December 2011}}</ref>

== See also ==
* [[List of fractals by Hausdorff dimension]]
* [[Complex-base system]]

== References ==
{{reflist}}

== External links ==
{{Commons}}
* {{mathworld|urlname=DragonCurve|title=Dragon Curve|mode=cs2}}
* [https://www.youtube.com/watch?v=v678Em6qyzk Knuth on the Dragon Curve]

{{Fractals}}
{{Mathematics of paper folding}}

[[Category:Fractal curves]]
[[Category:Paper folding]]
[[Category:L-systems]]