Larry Riddle, Agnes Scott College

Construction

Animation

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.

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.

Construction

Animation

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.

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].

L-system

Animation

Angle 45

Axiom FX

F —> Z

X —> +FX−−FY+

Y —> −FX++FY−

Axiom FX

F —> Z

X —> +FX−−FY+

Y —> −FX++FY−

First four iterations of the L-system

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

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

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 b^{2}/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].

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

[See this design in back stitch]

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

The Heighway dragon is composed of an infinite union of geometrically similar components that spiral 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].

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. [See Here for another example of a space filling curve generated by an iterated function system.]

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.

[See this design in back stitch]

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in
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