Agnes Scott College
Larry Riddle, Agnes Scott College

Heighway Dragon

Description


 

Construction
Animation

Construction via line segments

Begin with a line segment. As the first iteration, replace this segment with two segments, each scaled by a ratio \({\bf{r}} = \frac{1}{{\sqrt 2 }}\) in such a way that the original segment would have been the hypotenuse of an isosceles right triangle. Following along the original segment, we place the two new segments to the left. For the second iteration, replace each of the segments with two new segments at right angles, each scaled by the ratio r. The new segments are placed to the left then to the right along the segments of the first iteration. Continue this construction, always alternating the new segments between left and right along the segments of the previous iteration. This generates the "dragon curve". The following figure shows the first three iterations for this construction.

construction

Construction via paper folds

Here is how Heighway originally constructed the dragon. Fold a long narrow sheet in half. Fold again in the same direction (Example). Continue folding, always folding in the same direction. After several folds, open the sheet so that every fold is a right angle and view the sheet from the edge. In general, n folds produces an order-n dragon. The following pictures show an order-3 dragon (left) and order-4 dragon (right) constructed from a narrow slice of a manila folder.

heighwayfold3a heighwayfold4a

 
 
 

Construction
Animation

Construction via triangles

For yet another way to construct the Heighway dragon, start with an isosceles right triangle H0. Replace this triangle with two isosceles right triangles so that the hypotenuses of the new triangles lie on the equal sides of the old triangle. Working around H0, starting from the left side, we place the first new triangle so that it points out from the old triangle and the other new triangle so that it points into the old triangle to get H1. For the next step, repeat the process on each of the triangles that make up H1, this time placing the new triangles so they point Out, In, In, Out, respectively, as illustrated in the figure below. This pattern of Out, In, In, Out is repeated for each subsequent iteration of the construction.

The Heighway dragon is the limiting set of this iterative construction. Notice that the line segments that make up the hypotenuses of the triangles form the same path as described in the first construction above.


Iterated
Function
System

To think of this construction as a result of affine transformations, consider the first iteration of the line segment from (0,0) to (1,0) as shown by the two red line segments in the following figure.


IFS
Animation
Because of the need to alternate left and right when constructing the dragon, we need to think of constructing the second line segment by a rotation through 135° followed by a translation of the origin to the point (1,0). This yields the following IFS
 
\({f_1}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {1/2} & { - 1/2} \\ {1/2} & {1/2} \\ \end{array}} \right]{\bf{x}}\)
 
   scale by r, rotate by 45°
 
\({f_2}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} { - 1/2} & { - 1/2} \\ {1/2} & { - 1/2} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} 1 \\ 0 \\ \end{array}} \right]\)
 
   scale by r, rotate by 135°
 

where \({\bf{r}} = \frac{1}{{\sqrt 2 }}\). The attractor of this IFS will be the same as the dragon curve. The Heighway dragon consists of two self-similar pieces corresponding to the two functions in the IFS [Demonstration].

IFSselfSimilar
[Enlarge]

L-System

 
 

L-system
Animation
Angle 45
Axiom FX
F —> Z
X —> +FX−−FY+
Y —> −FX++FY−


First four iterations of the L-system

Similarity
Dimension

The Heighway dragon is self-similar with 2 non-overlapping copies of itself, each scaled by the factor r < 1. Therefore the similarity dimension, d, of the attractor of the IFS is the solution to

\[\sum\limits_{k = 1}^2 {{r^d}} = 1 \quad \Rightarrow \quad d = \frac{{\log (1/2)}}{{\log (1/\sqrt 2 )}} = 2\]

Special
Properties

The Heighway dragon was first investigated by physicists John Heighway, Bruce Banks, and William Harter in 1966. It was described by Martin Gardner in his Mathematical Games column of Scientific American where he stated that Harter used the dragon as a cover design for a booklet prepared for a NASA seminar on group theory. Many of its properties were first published by Chandler Davis and Donald Knuth.

The Heighway dragon as described above starting with a line segment of length 1 has area 1/2 [Details]. If the initial line segment has length b, then the area of the Heighway dragon will be b2/2 [see the derivation of area based on the Heighway boundary].

The copy of the Heighway dragon below lies on top of a grid of size 1/6 by 1/6 with it's tail located at the origin. It demonstrates that the Heighway dragon lies within a rectangle with –2/6 ≤ x ≤ 7/6 and –2/6 ≤ y ≤ 4/6, i.e. a rectangle of width 1.5 and height 1 [Details].

heighwaySize

The Heighway dragon can be used to tile the plane [Details].

HeighwayTilingSmall
[See this design in back stitch]

The boundary of the Heighway dragon is also a fractal with dimension 1.523627 [Details].

boundary

The Heighway dragon is composed of an infinite union of geometrically similar components that sprial around two curves ending at (0,0) and (1,0). Each component consists of four copies of the dragon (two twindragons), one copy of the component scaled by 1/2, and two copies of the component scaled by 1/4 [Details].

components

Let H be the attractor for the IFS. There is a continuous curve f defined on [0,1] such that f([0,1]) = H and such that f is a space filling curve. This means that H has a non-empty interior. For a proof, see the book by Edgar.


Birth of a
Heighway Dragon
Animation
The Heighway dragon is constructed by replacing a line segment with two segments at 45°. If the angle between the line segments is less than 45° then a different dragon curve will be formed. If we let the angle grow from 0° to 45°, we can watch the Heighway dragon being born. See the animation.

The construction of the Lévy dragon and the Heighway dragon are very similar. In each case one can start with an isosceles right triangle and replace this triangle with two isosceles right triangles so that the hypotenuse of each new triangle lies on one of the equal sides of the old triangle. The difference is how those new triangles are placed relative to sides of the old triangle. For the Lévy dragon, both are placed towards the "outside"; for the Heighway dragon, one is placed pointing out while the other is placed pointing in. Because of this similarity, it is perhaps not surprising that one can transform the Heighway dragon into the Lévy dragon through a continuous transformation. Indeed, for each t in the interval [0,1] define an iterated function system by


Heighway
to
Lévy
Animation
\[\begin{array}{l} {f_1}(t)({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {1/\sqrt 2 } & 0 \\ 0 & {1/\sqrt 2 } \\ \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\cos ({{45}^ \circ })} & { - \sin ({{45}^ \circ })} \\ {\sin ({{45}^ \circ })} & {\cos ({{45}^ \circ })} \\ \end{array}} \right]{\bf{x}} \\ {f_2}(t)({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {1/\sqrt 2 } & 0 \\ 0 & {1/\sqrt 2 } \\ \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\cos ({{135}^ \circ } - {{180}^ \circ }t)} & { - \sin ({{135}^ \circ } - {{180}^ \circ }t)} \\ {\sin ({{135}^ \circ } - {{180}^ \circ }t)} & {\cos ({{135}^ \circ } - {{180}^ \circ }t)} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {1 - 0.5t} \\ {0.5t} \\ \end{array}} \right] \\ \end{array}\]

Let A(t) be the unique attractor of the iterated function system corresponding to the value t. Then A is a continuous function from the unit interval to the space of compact sets with the Hausdorff topology, with A(0) equal to the Heighway dragon and A(1) equal to the Lévy dragon. The animation shows A(t) as t goes from 0 to 1.

Variations

Twindragon

The twindragon is constructed by placing two Heighway dragons back-to-back.

twoHeighways2

Terdragon

The terdragon is a variation of the paper folding construction of the Heighway dragon in which the folds trisect the paper at each step instead of bisecting it. It was first introduced by Davis and Knuth in their 1970 paper on dragon curves.

terdragon

Golden Dragon

The golden dragon is constructed similarly to the Heighway dragon construction via line segments, except using different scaling factors and angles. It has this name because its similarity dimension is the golden ratio.

goldenDragon

Heighway Z2 Dragon

If the elements of the Z2 cyclic group are applied to the functions in the Heighway dragon IFS, a new IFS is created whose attractor exhibits rotational symmetry through 180°.

HeighwayZ2-sm

References

  1. Bannon, Thomas. "Fractals and Transformations," Mathematics Teacher, March 1991, 178-185.
  2. Crichton, Michael. Jurassic Park, Ballantine Books, New York 1990. If you read the book or saw the movie, you will remember that chaos theory, in particular the idea of sensitive dependence on initial conditions, was a major reason why the mathematician Ian Malcolm believed the idea of creating a prehistoric world of dinosaurs would never work. Each chapter of the book is illustrated with an iteration of the Heighway Dragon.
  3. Davis, Chandler and Donald J. Knuth. "Number representations and dragon curves" J. Recreational Math. 3 (1970) 66-81 (Part 1), 133-149 (Part 2). Reprinted with extra addendum in Selected Papers on Fun and Games, Donald Knuth, CSLI Publications, 2011.
  4. Edgar, Gerald A. Measure, Topology, and Fractal Geometry, Sringer-Verlag, 1990.
  5. Gardner, Martin. "Mathematical Games," Scientific American, March, April, July, 1967. [Reprinted in his Mathematical Magic Show, Knopf, 1977]
  6. Jones, Juw. "Fractals Before Mandelbrot-A Selective History," in Fractals and Chaos, Crilly, Earnshaw, and Jones, Editors, Springer-Verlag 1991.
  7. Helmberg, Gilbert. Getting Acquainted with Fractals, de Gruyter Publishing, 2007. [See Preview at Google Books]
  8. Lauwerier, Hans. Fractals, Princeton Science Library, Princeton, N.J. 1991 (Translated by Sophia Gill-Hoffstadt from the 1987 edition), p.50.
  9. Mandelbrot, Benoit. The Fractal Geometry of Nature, W.H. Freeman and Co. 1983, p66. [Preview available at Google Books]
  10. McWorter Jr., William A. and Jane Morrill Tazelaar. "Creating Fractals," Byte, August 1987, 123-132.
  11. Prusinkiewicz, Przemyslaw and James Hanan. Lindenmayer Systems, Fractals, and Plants, Lecture Notes in Biomathematics #79, Springer-Verlag 1989.
  12. Prusinkiewicz, Przemyslaw and Aristid Lindenmayer. The Algorithmic Beauty of Plants, Springer-Verlag, 1990. [Available from the Algorithmic Botany website]
  13. Angle Chang and Tianrong Zhang. "The Fractal Geometry of the Boundary of Dragon Curves," J. Recreational Mathematics, Vol. 30, No. 1 (1999-2000), 9-22. [Available at http://poignance.coiraweb.com/math/Fractals/Dragon/Bound.html]
  14. Ngai, Sze-Man and Nhu Nguyen. "The Heighway Dragon Revisited," Discrete and Computational Geometry, Vol. 29, no. 4 (2003), 603-623. [Available at http://math.georgiasouthern.edu/~smngai/research.html].
  15. Benedek, Agnes and Rafael Panzone. "On some notable plane sets. II. Dragons", Rev. Un. Mat. Argentina 39 (1994), no. 1-2, 76–89.