Dragon curve

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Heighway Dragon Curve

A dragon curve is any member of a family of self-similar fractal curves, which can be approximated by recursive methods such as L-systems. It is also known as the shape generated by repeatedly folding a strip of paper in half, although there are other curves also generally 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" in 1967.

Many of his properties were first published by Chandler Davis and Donald Knuth. The figure appeared in the chapter titles of Michael Crichton's novel Jurassic Park.

Construction

Recursive construction of the curve

Heighway's dragon can be built from a base segment, repeatedly replacing each segment with two segments at a right angle and rotated 45° alternately to the right and left:

Heighway's dragon is also the limit set of the following iterative system of functions 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\yend{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\yend{pmatrix}}+{begin{pmatrix}1\0end{pmatrix}}}

Step-by-step construction

The dragon curve is built using the following steps:

From a segment, a rectangle triangle and isosceles are built, as the first two figures are shown. Then the initial segment is deleted.
The process of replacing a segment with two others for each old segment of the curve is repeated a few times, always alternating the orientation of the triangles.

The following figure shows the first thirteen steps:

Formation of the dragon curve

[D]ragon unfolding

Heighway's dragon curve can be constructed by folding a strip of paper, which is how it was originally discovered. Take a strip of paper and fold it in half to the right. Fold in half again to the right. If the strip is now opened, unfolding each fold to make it a 90 degree turn, the turn sequence would be RightRightLeft, that is, the second iteration of the Heighway dragon. Fold the strip in half again to the right, and the unfolded strip twist sequence is now DDIDDII - the third iteration of the Heighway dragon. Continue folding the strip in half to the right to create more iterations of the Highway dragon (in practice, the strip becomes too thick to fold completely after four or five iterations).

The folding patterns of this sequence of paper strips, as right (D) and left (I) fold sequences, are:

1.a iteration: D
2.a iteration: D D I
3.a iteration: D D I D D I I
4.a iteration: D D I D D I I D D D I I D I I

Each iteration can be found by copying the previous iteration, then a D, then a second copy of the previous iteration in reverse order with the letters I and D swapped.

Properties

You can see many self-similitude in the corner of Heighway's dragon. The most obvious is the repetition of the same tilted pattern 45° and with a reduction ratio ${displaystyle textstyle {sqrt {2}}}$ . According to these self-similarities, many of their lengths are simple rational numbers.

Longitudes

Self-similitude

Thessing of the plane through dragon curves

The dragon's curve can weave the plane. A possible enlosing replaces each edge of a square tesseal with a dragon curve, using the recursive definition of the dragon from a straight segment. The initial direction to expand each segment can be determined from the color of the boxes of a chess board, having vertical segments in the black and out white boxes, and expanding horizontal segments in white and out of black boxes.
As a filling curve of space that does not cross itself, the dragon curve has a fractal dimension exactly 2. For a dragon curve with an initial segment length 1, its area is 1/2, as can be seen in the plane's teselados.
The limit of the set covered by the dragon curve has an infinite length, with fractal dimension ${displaystyle 2log _{2}lambda approx 1.523627086202492,}$ 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 ${displaystyle lambda ^{3}-lambda ^{2}-2=0.}$

Demonstration

Inner Homotetia of the Dragon

The dragon completely fills a part of the plane, so its fractal dimension must be 2. The calculation of its dimension is done as in the Koch snowflake, since the constructions of both curves are similar.

In the first step of the construction, it is observed that from the initial segment the other signs of the first triangle are obtained by two similarities (one is indirect) of reason ${displaystyle {tfrac {sqrt {2}}{2}}}$ of centers the ends of the segment, and of angles ${displaystyle {tfrac {pi }{4}}}$ and ${displaystyle -{tfrac {pi }{4}}}$ radian (i.e., 45°). Denomination $s_{1}$ and $s_{2}$ These two similarities. By the construction itself, the $n$ - that figure obtained in the process, ${displaystyle D_{n}}$ , it is the gathering of images by $s_{1}$ and $s_{2}$ of the previous figure ${displaystyle D_{n-1}}$ :

{displaystyle D_{n}=s_{1}(D_{n-1})cup s_{2}(D_{n-1})}

Taking the limit of this relationship (when n tend towards +∞), and calling ${displaystyle D=D_{infty }}$ to the dragon's curve, you get:

{displaystyle D_{n}=s_{1}(D)cup s_{2}(D)}

I mean, $D$ is the meeting of two copies of itself, on a scale ${displaystyle {tfrac {sqrt {2}}{2}}={tfrac {1}{sqrt {2}}}}$ as you can see in the figure on the right.

So if it gets bigger D through a homothecia of reason $sqrt 2$ We get twice. DSame scale.

Yeah. D is of dimension d, his "volumen" multiplied by ${displaystyle {sqrt {2}}^{ d}}$ through this homotecia. Here it is concluded that ${displaystyle {sqrt {2}}^{ d}=2}$ and, therefore, ${displaystyle d=2}$ .

Twin Dragon

Curve of the twin dragon built from two Heighway dragons

The Twin Dragon (also known as the Davis-Knuth Dragon) can be built by placing two Heighway Dragon Curves one behind the other. It is also the set of limits of the following system of iterated functions:

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

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

where the initial form is defined by the following set ${displaystyle S_{0}={0,1,1-i}}$ .

Can also be generated as an L-system; you just need to add another section in the initial string:

Angle 90°
Initial chain FX + FX +
Chain rewriting rules
- X X + YF
- And FX - And

Terdragon

Curva terdragón

Curva de Lévy C

The dragon curve, 60° variant. His self-similitude is clearly visible

The terdragon can be spawned as an L-system:

Angle 120
Initial chain F
Chain rewriting rules
- F F + F − F

The figure is generated by the set of limits of the following system of iterated functions:

{displaystyle f_{1}(z)=lambda z}

{displaystyle f_{2}(z)={frac {i}{sqrt {3}}}z+lambda }

{displaystyle f_{3}(z)=lambda z+lambda ^{*}}

{displaystyle {mbox{where }}lambda ={frac {1}{2}}-{frac {i}{2{sqrt {3}}}}{text{ and }}lambda ^{*}={frac {1}{2}}+{frac {i}{2{sqrt {3}}}}.}

Lévy's Dragon

The Lévy C curve is sometimes known as the Lévy dragon.

Variants

A polyominous dragon

It is possible to change the angle of rotation from 90° to other angles. Changing to 120° produces a triangle structure, while choosing a 60° twist can turn a discrete dragon curve into a polyomino dragon. Like discrete dragon curves, dragon polyominoes approach the fractal dragon curve as they tend to the limit.

Occurrences of the dragon curve in solution sets

Having obtained the set of solutions of a linear differential equation, any linear combination of the solutions, due to the superposition principle, will also obey the original equation. That is, new solutions are obtained by applying a function to the set of existing solutions. This is similar to how a system of iterated functions produces new points in a set, although not all systems of iterated functions are linear functions.

In a conceptually similar vein, a set of Littlewood polynomials can be arrived at by such iterated maps of a set of functions.

A polynomial of Littlewood is the one that: ${displaystyle p(x)=sum _{i=0}^{n}a_{i}x^{i},}$ where everything is ${displaystyle a_{i}=pm 1}$ .

For some $<math alttext="{displaystyle |w|日本語w日本語.1{displaystyle Șw implied} <img alt="{displaystyle |w|$ The following functions are defined:

{displaystyle f_{+}(z)=1+wz}

{displaystyle f_{-}(z)=1-wz}

Starting at z = 0 you can generate all graded Littlewood polynomials using these functions iteratively ${displaystyle d+1}$ Sometimes. For example: ${displaystyle f_{+}(f_{-}(f_{-}(0)))=1+(1-w)w=1+1w-1w^{2}}$

You can see that stop ${displaystyle w=(1+i)/2}$ , the previous pair of functions is equivalent to the formulation of the iterative system of Heighway dragon functions. That is, Heighway's dragon, iterated to a certain iteration, describes the set of all Littlewood's polynomials to a certain degree, evaluated at the point ${displaystyle w=(1+i)/2}$ .

In fact, by plotting a sufficiently high number of roots of Littlewood polynomials, structures similar to the dragon curve appear at points close to these coordinates.

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