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| [[Image:Dragon curve animation.gif|right|300 px]]
| | I'm a 46 years old and working at the college (Industrial and Labor Relations).<br>In my spare time I'm trying to learn Arabic. I have been there and look forward to go there anytime soon. I like to read, preferably on my ipad. I like to watch The Big Bang Theory and The Big Bang Theory as well as documentaries about nature. I enjoy Boxing.<br><br>Here is my web blog Oltramare, [http://archives.tdg.ch/geneve/actu/banquier-genevois-35-millions-chaire-religion-politique-2011-09-20 archives.tdg.ch], |
| 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.
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| ==Heighway dragon==
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| [[Image:Fractal dragon curve.jpg|thumb|Heighway dragon curve]]
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| 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'' 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]]''.
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| ===Construction===
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| [[Image:DragonCurve animation.gif|right|400px|Recursive construction of the curve]]
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| It can be written as a [[Lindenmayer system]] with
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| * angle 90°
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| * initial string ''FX''
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| * string rewriting rules
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| ** ''X'' {{mapsto}} ''X''+''YF''+
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| ** ''Y'' {{mapsto}} −''FX''−''Y''.
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| That can be described this way : Starting from a base segment, replace each segment by 2 segments with a right angle and with a rotation of 45° alternatively to the right and to the left:
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| [[Image:Dragon curve iterations (2).svg|none|700px|The first 5 iterations and the 9th]]
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| The Heighway dragon is also the limit set of the following [[iterated function system]] in the complex plane:
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| :<math>f_1(z)=\frac{(1+i)z}{2}</math>
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| :<math>f_2(z)=1-\frac{(1-i)z}{2}</math>
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| with the initial set of points <math>S_0=\{0,1\}</math>.
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| Using pairs of real numbers instead, this is the same as the two functions consisting of
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| :<math>f_1(x,y)= \frac{1}{\sqrt{2}}\begin{pmatrix} \cos 45^o & -\sin 45^o \\ \sin 45^0 & \cos 45^o \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}</math>
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| :<math>f_2(x,y)= \frac{1}{\sqrt{2}}\begin{pmatrix} \cos 135^o & -\sin 135^o \\ \sin 135^0 & \cos 135^o \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} 1 \\ 0 \end{pmatrix}</math>
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| This representation is more commonly used in software such as [[Apophysis (software)|Apophysis]].
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| ===[Un]Folding the Dragon===
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| Tracing an iteration of the Heighway dragon curve from one end to the other, one encounters a series of 90 degree turns, some to the right and some to the left. For the first few iterations the sequence of right (R) and left (L) turns is as follows:
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| :1st iteration: R
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| :2nd iteration: '''R''' R '''L'''
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| :3rd iteration: '''R''' R '''L''' R '''R''' L '''L'''
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| :4th iteration: '''R''' R '''L''' R '''R''' L '''L''' R '''R''' R '''L''' L '''R''' L '''L'''.
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| This suggests the following pattern: each iteration is formed by taking the previous iteration, adding an R at the end, and then taking the original iteration again, flipping it, switching each letter and adding the result after the R.
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| This pattern in turn suggests the following method of creating models of iterations of the Heighway dragon curve by [[paper folding|folding a strip of paper]]. 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).
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| [[Image:Dragon curve paper strip.png|center|800px]]
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| This pattern also gives a method for determining the direction of the ''n''th turn in the turn sequence of a Heighway dragon iteration. First, express ''n'' in the form ''k''2<sup>''m''</sup> where ''k'' is an odd number. The direction of the ''n''th turn is determined by ''k'' mod 4 i.e. the remainder left when ''k'' is divided by 4. If ''k'' mod 4 is 1 then the ''n''th turn is R; if ''k'' mod 4 is 3 then the ''n''th turn is L.
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| For example, to determine the direction of turn 76376:
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| :76376 = 9547 x 8.
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| :9547 = 2386x4 + 3
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| :so 9547 mod 4 = 3
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| :so turn 76376 is L
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| There is a simple one line non-recursive method of implementing the above ''k'' mod 4 method of finding the turn direction in code. Treating turn ''n'' as a binary number, calculate the following [[Boolean logic|boolean]] value:
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| :bool turn = (((n & −n) << 1) & n) != 0;
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| * "n & −n" leaves you with only one bit as a '1', the rightmost '1' in the binary expansion of ''n'';
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| * "<< 1" shifts the that bit one bit to the left;
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| * "& n" leaves you with either that single bit (if ''k'' mod 4 = 3) or a zero (if ''k'' mod 4 = 1).
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| * so "bool turn = (((n & −n) << 1) & n) != 0" is TRUE if the ''n''th turn is L; and is FALSE if the ''n''th turn is R.
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| ==== Gray code method ====
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| Another way of handling this is a reduction for the above algorithm. Using [[Gray code]], starting from zero, determine the change to the next value. If the change is a 1 turn left, and if it is 0 turn right. Given a binary input, B, the corresponding gray code, G, is given by "G = B XOR (B>>1)". Using ''G''<sub>''i''</sub> and ''G''<sub>''i''−1</sub>, turn equals" (not ''G''<sub>''i''</sub>) AND ''G''<sub>''i''−1</sub>".
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| * G = B^(B >> 1); This gets gray code from binary.
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| * T = (~G0)&G1; If T is equal to 0 turn clockwise else turn counterclockwise.
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| ===Dimensions===
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| * In spite of its strange aspect, the Heighway dragon curve has simple dimensions. Note that the dimensions 1, and 1.5 are [[Limit (mathematics)|limits]] and not actual values.
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| [[Image:Dimensions fractale dragon.gif|none|500px]]
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| * Its '''surface''' is also quite simple : If the initial segment equals 1, then its surface equals <math>\textstyle{\frac{1}{2}}</math>. This result comes from its paving properties.
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| * The curve never crosses itself.
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| * 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>.
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| [[Image:Auto-similarity dragon curve.gif|none|350px]]
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| * Its [[fractal dimension]] can be calculated : <math>\textstyle{\frac {\ln 2} {\ln \sqrt{2}} = 2}</math>. That makes it a [[space-filling curve]].
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| * Its '''boundary''' has an infinite length, since it increases by a similar factor every iteration.
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| * The fractal dimension of its boundary has been approximated numerically by Chang & Zhang .<ref name="chang">[http://poignance.coiraweb.com/math/Fractals/Dragon/Bound.html Fractal dimension of the boundary of the Dragon curve]</ref>).
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| In fact it can be found analytically:<ref>"[http://demonstrations.wolfram.com/TheBoundaryOfPeriodicIteratedFunctionSystems/ The Boundary of Periodic Iterated Function Systems]" by Jarek Duda, [[The Wolfram Demonstrations Project]]. Recurrent construction of the boundary of dragon curve.</ref>
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| <math>\log_2\left(\frac{1+\sqrt[3]{73-6\sqrt{87}}+\sqrt[3]{73+6\sqrt{87}}}{3}\right)\cong 1.523627086202492.</math> This is the root of the equation <math>\textstyle{4^x(2^x-1)=4(2^x+1).}</math>
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| ===Tiling===
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| The dragon curve can tile the plane in many ways.
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| <center>
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| <gallery>
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| Image:Dragon tiling1.svg|1st element with 4 curves
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| Image:Dragon tiling2.svg|2nd element with 4 curves
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| Image:Dragon tiling3.svg|3rd element with 4 curves
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| Image:Dragon tiling dragon.svg|The dragon curve can tile itself
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| Image:Dragon tiling4.svg|1st element with 2 curves
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| Image:Dragon tiling5.svg|2nd element with 2 curves (twindragon)
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| Image:Dragon tiling6.svg|3rd element with 2 curves
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| Image:Full tiling dragon.svg|Example of plane tiling
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| Image:Full tiling dragon2.svg|Example of plane tiling
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| Image:Full tiling dragon3.svg|Example of plane tiling
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| Image:Dragon spiral tiling.png|Dragon curves of increasing sizes (ratio sqrt(2)) form an infinite spiral. 4 of these spirals (with rotation 90°) tile the plane.
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| </gallery>
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| </center>
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| ==Twindragon==
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| The '''twindragon''' (also known as the '''Davis-Knuth dragon''') can be constructed by placing two Heighway dragon curves back-to-back. It is also the limit set of the following iterated function system:
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| :<math>f_1(z)=\frac{(1+i)z}{2}</math>
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| :<math>f_2(z)=1-\frac{(1+i)z}{2}</math>
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| where the initial shape is defined by the following set <math>S_0 = \{0,1,1-i\}</math>.
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| It can be also written as a [[Lindenmayer system]] – it only needs adding another section in initial string:
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| * angle 90°
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| * initial string ''FX+FX+''
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| * string rewriting rules
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| ** ''X'' {{mapsto}} ''X''+''YF''
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| ** ''Y'' {{mapsto}} ''FX''−''Y''.
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| {|
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| |[[Image:Lévy's dragon curve (IFS).jpg|thumb|Twindragon curve.]]
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| |[[Image:Twindragon.png|thumb|Twindragon curve constructed from two Heighway dragons.]]
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| |}
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| ==Terdragon==
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| [[Image:Terdragon.png|thumb|right|250px|Terdragon curve.]]
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| The '''terdragon''' can be written as a [[Lindenmayer system]]:
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| * angle 120°
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| * initial string ''F''
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| * string rewriting rules
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| ** ''F'' {{mapsto}} ''F+F−F''.
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| It is the limit set of the following iterated function system:
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| :<math>f_1(z)=\lambda z</math>
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| :<math>f_2(z)=\frac{i}{\sqrt{3}}z + \lambda</math>
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| :<math>f_3(z)=\lambda z + \lambda^*</math>
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| :<math>\mbox{where }\lambda=\frac{1}{2}-\frac{i}{2\sqrt{3}}
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| \text{ and }\lambda^*=\frac{1}{2}+\frac{i}{2\sqrt{3}}.</math>
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| ==Lévy dragon==
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| The [[Lévy C curve]] is sometimes known as the '''Lévy dragon'''{{citation needed|date=November 2012}}.
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| {|
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| |[[Image:Lévy's C-curve (IFS).jpg|thumb|200px|Lévy C curve.]]
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| |}
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| == Occurrences of the dragon curve in solution sets ==
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| Having obtained the set of solutions to a 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 produce new points in a set, though not all IFS are linear functions.
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| In a conceptually similar vein, a set of [[Littlewood polynomial]]s can be arrived at by such iterated applications of a set of functions.
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| 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>.
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| For some |w| < 1 we define the following functions:
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| :<math> f_+(z) = 1 + wz</math>
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| :<math> f_-(z) = 1 - wz</math>
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| Starting at z=0 we can generate all Littlewood polynomials of degree d using these functions iteratively d+1 times.<ref name='ncafe'>http://golem.ph.utexas.edu/category/2009/12/this_weeks_finds_in_mathematic_46.html</ref> For instance: <math>f_+(f_-(f_-(0))) = 1 + (1-w)w = 1 + 1w - 1w^2</math>
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| It can be seen that for '''w = (1+i)/2''', 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 '''w = (1+i)/2'''.
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| 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>http://math.ucr.edu/home/baez/week285.html</ref><ref>http://johncarlosbaez.wordpress.com/2011/12/11/the-beauty-of-roots/</ref>
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| == See also ==
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| * [[List of fractals by Hausdorff dimension]]
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| * [[Complex base systems]]
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| * [[Regular paperfolding sequence]]
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| ==Notes==
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| <references />
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| ==External links==
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| {{Commons|Dragon curve}}
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| * [http://mathworld.wolfram.com/DragonCurve.html Dragon Curve]—from [[MathWorld]]
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| * [http://www.eschertile.com/pic/dragon3.gif Tile made by David Chow]
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| * [http://www.eschertile.com/java/Dragon/dragon.htm Twin dragon tile with JAVA]
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| * {{cite web|last=Eastaway|first=Rob|title=Dragon Curves|url=http://www.numberphile.com/videos/dragon_curve.html|work=Numberphile|publisher=[[Brady Haran]]}}
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| * {{cite web|last=Bernard|first=Pierre (animation)|title=Dragon Curve to Music|url=http://www.numberphile.com/videos/dragon_curve_music.html|work=Numberphile|publisher=[[Brady Haran]]|coauthors=Stewart, Alan (music)}}
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| {{Fractals}}
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| [[Category:Fractal curves]]
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| [[Category:Paper folding]]
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