Dragon curve
From Free net encyclopedia
A dragon curve is the generic name for a member of a family of self similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems.
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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 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.
It can be written as a Lindenmayer system with
- angle 90°
- initial string FX
- string rewriting rules
- X <math>\mapsto</math> X+YF+
- Y <math>\mapsto</math> -FX-Y
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>.
| Image:Fractal dragon curve.jpg | Image:Pavement with dragons.png | Image:DragonFractal4.png |
[Un]Folding the Dragon
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:
- 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
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.
This pattern in turn suggests the following method of creating models of iterations of the Heighway dragon curve by 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).
Image:Dragon curve paper strip.png
This pattern also gives a method for determining the direction of the nth turn in the turn sequence of a Heighway dragon iteration. First, express n in the form k2m where k is an odd number. The direction of the nth 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 nth turn is R; if k mod 4 is 3 then the nth turn is L.
For example, to determine the direction of turn 76376:
- 76376 = 9547 x 8.
- 9547 = 2386x4 + 3
- so 9547 mod 4 = 3
- so turn 76376 is L
Twindragon
The twindragon (also known as the Davis-Knuth dragon) can be constructed by placing two Heighway dragon curves back-to-back. It is the limit set of the following iterated function system:
- <math>f_1(z)=\frac{(1+i)z}{2}</math>
- <math>f_2(z)=\frac{(1+i)z+1-i}{2}</math>.
| Image:Lévy's dragon curve (IFS).jpg | Image:Twindragon.png |
Terdragon
The terdragon can be written as a Lidenmayer system:
- angle 120°
- initial string F
- string rewriting rules
- F <math>\mapsto</math> 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}}
\mbox{ and }\lambda^*=\frac{1}{2}+\frac{i}{2\sqrt{3}}</math>.
Lévy dragon
The Lévy C curve is sometimes known as the Lévy dragon.
| Image:Terdragon.png | Image:Lévy's C-curve (IFS).jpg |
External links
- Dragon Curve—from MathWorld
- Paperfolding and the Dragon curve
- The Mystery of the Paper Dragon from the Metaphysics of Chaosde:Drachenkurve