 Larry Riddle, Agnes Scott College ## Lévy Dragon

#### Description Paul Lévy

Construction
Animation

#### Construction via triangles

Start with an isosceles right triangle L0. 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. Place each new triangle so that it points out from the original triangle to get L1. For the next step, repeat the process on each of the triangles that make up L1, as illustrated in the figure below. For each new iteration, replace each triangle in Lk by two isosceles right triangles. The Lévy dragon is the limiting set of this iterative construction. Notice that at each iteration, the sides of each triangle in Lk are scaled by a factor $${\bf{r}} = \frac{1}{{\sqrt 2 }}$$ [see IFS discussion below], so that the area is scaled by 1/2. But each triangle in Lk produces two new triangles, so the total area remains unchanged. Thus the area of the Lévy dragon is the same as the area of the original isosceles right triangle L0.

Construction
Animation

#### Construction via line segments

Begin with a horizontal line segment. Specify the left endpoint as the "initial" point. As the first iteration, replace this segment with two segments at right angles, each scaled by a ratio $${\bf{r}} = \frac{1}{{\sqrt 2 }}$$. Following the orientation along the original segment starting at the initial point, 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 again placed to the left moving along the segments of the first iteration from the initial point. Continue this construction, always placing the new segments to the left along the segments of the previous iteration. The line segments correspond to the hypotenuses of the isosceles right triangles generated in the construction described above. This generates the "dragon curve". The following figure shows the first three iterations for this construction. Each iteration can be represented symbolically by a sequence consisting of the letters R L, S, and B. One can imagine moving along the individual segments making up the curve starting at the left endpoint. A corner is labeled R if the curve makes a right turn there of 90°, is labeled L if the curve makes a left turn of 90° at that corner, is labeled S if the curve continues straight at that corner (0° turn), and is labeled B if the curve make a 180° turn at that corner. So the three iterations shown above are represented by the sequences R, RSR, and RSRLRSR, respectively. [Details]

#### IteratedFunctionSystem

Suppose that the original isosceles right triangle L0 is placed so that its hypotenuse is the unit interval on the x-axis and the opposite corner is at (1/2, 1/2). Then each leg is equal to $$\frac{1}{{\sqrt 2 }}$$. This means that we must scale the triangle L0 by the factor $${\bf{r}} = \frac{1}{{\sqrt 2 }}$$ to get each of the new triangles for L1. One triangle must then be rotated by 45°, while the other triangle must be rotated by -45° (i.e. in the clockwise direction) and translated by 1/2 in both the x and y directions. This yields the following IFS

 $${f_1}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {0.5} & { - 0.5} \\ {0.5} & {0.5} \\ \end{array}} \right]{\bf{x}}$$ scale by r, rotate by 45° $${f_2}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} { 0.5} & { 0.5} \\ {- 0.5} & { 0.5} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} 0.5 \\ 0.5 \\ \end{array}} \right]$$ scale by r, rotate by −45°

where $${\bf{r}} = \frac{1}{{\sqrt 2 }}$$ . The attractor of this IFS will be the Lévy dragon. The Lévy dragon consists of two self-similar pieces corresponding to the two functions in the iterated function system.

#### L-System

L-system
Animation
Angle 45
Axiom F
F —> +F−−F+ First four iterations of the L-system

The line segments generated by the L-system correspond to the hypotenuses of the isosceles right triangles generated in the construction described above.

#### SimilarityDimension

The Levy 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

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

#### SpecialProperties

Paul Lévy studied the curve now known as Lévy's Dragon as part of a general study of curves consisting of parts similar to the whole [see the original 1938 article by Lévy or the translation in Edgar]. In this he was motivated by the earlier work of Helge von Koch and the Koch curve.

Among several of the properties that Lévy observed was that the plane can be tiled by copies of the Lévy dragon. This means that there is a sequence of sets congruent to the Lévy dragon that are non-overlapping and whose unions is the entire plane. Click for more details.

Another property shown by Lévy is that the dragon has non-empty interior [see Edgar]. The paper by Duvall and Keesling investigates some properties of this interior. In addition, the authors develop a theoretical approach to computing the Hausdorff dimension of the topological boundary of attractors of iterated function systems, and apply this theory to estimate that the boundary of the Lévy dragon has Hausdorff dimension of approximately 1.934007183. Bailey, Kim, and Strichartz show that the interior of the Lévy dragon consists of a countable number of components, the largest of which is a hexagon that is only a speck on the dragon. Moreover, they found 16 different shapes for these components and conjectured that there were no others. Alster proved that the number of shapes is finite.

The copy of the Lévy dragon below lies on top of a grid of size 1/4 by 1/4 with the initial line segment going from the origin to the point (1,0). It demonstrates that the Lévy dragon lies within a rectangle with –1/2 ≤ x ≤ 3/2 and –1/4 ≤ y ≤ 1, i.e. a rectangle of width 2 and height 1.25 [Details]. Birth of a
Lévy Dragon
Animation
The Lévy dragon can be 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 Lévy 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 Lévy dragon into the Heighway dragon through a continuous transformation. Indeed, for each t in the interval [0,1] define an iterated function system by

Lévy
to
Heighway
Animation
$\begin{array}{l} {f_1}(t)({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {1/\sqrt 2 } & 0 \\ 0 & {1/\sqrt 2 } \\ \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\cos ({{45}^ \circ })} & { - \sin ({{45}^ \circ })} \\ {\sin ({{45}^ \circ })} & {\cos ({{45}^ \circ })} \\ \end{array}} \right]{\bf{x}} \\ {f_2}(t)({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {1/\sqrt 2 } & 0 \\ 0 & {1/\sqrt 2 } \\ \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\cos ({{-45}^ \circ } + {{180}^ \circ }t)} & { - \sin ({{-45}^ \circ } + {{180}^ \circ }t)} \\ {\sin ({{-45}^ \circ } + {{180}^ \circ }t)} & {\cos ({{-45}^ \circ } + {{180}^ \circ }t)} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {0.5 + 0.5t} \\ {0.5 - 0.5t} \\ \end{array}} \right] \\ \end{array}$ 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 Lévy dragon and A(1) equal to the Heighway dragon. The animation shows A(t) as t goes from 0 to 1.

#### Lévy Tapestry

Rather than start with just one horizontal line segment, one can start the construction of the Lévy Dragon with a square. The iteration steps are repeated for each of the four sides of the square. There are two choices as one can orient the initial triangles to point inside the square or to point outside the square. The resulting images are sometimes called a Lévy Tapestry. Click on the images for more details.

#### Lévy Diamonds

Instead of starting the construction with a isosceles right triangle, we start with a trapezoid C0 with a base of length 1 and the other three sides of length 1/2. Replace this trapezoid with three scaled copies whose bases lie on the equal sides of the old trapezoid. Place each new trapezoid so that it points out from the original trapezoid to get C1. For the next step, repeat the process on each of the trapezoids that make up C1 to get C2, as illustrated in the figure below. Repeating this construction ad infinitum yields the following figure that looks like it is formed of many diamond shapes of smaller and smaller scale. #### Embroidery

The Lévy Dragon in cross-stitch Larry Riddle, January 2009
10" x 8" (16 point canvas)
Click on picture for a larger view
[Cross-stitch design (pdf) created with IFS Construction Kit]

The Lévy Dragon Curve (12 iterations) in back stitch Larry Riddle, August 2020
11" x 14" (18 point canvas)
Click on picture for a larger view

The Lévy Tapestry Inside (12 iterations) in back stitch Larry Riddle, January 2013
12.5" x 12.5" (10 point canvas)
Click on picture for a larger view
[Back stitch design (pdf) created with IFS Construction Kit]

The Lévy Tapestry Outside (12 iterations) in back stitch Larry Riddle, April 2013
12.5" x 12.5" (18 point canvas)
Click on picture for a larger view
[Back stitch design (pdf) created with IFS Construction Kit]

#### References

1. P. Lévy, "Les courbes planes ou gauches et les surfaces composée de parties semblales au tout," Journal de l'École Polytechnique (1938), 227-247, 249-291.
2. P. Lévy, "Plane or space curves and surfaces consisting of parts similar to the whole," in Classics on Fractals, Gerald A. Edgar, Editor, Addison-Wesley, 181-239.
3. P. Duvall and J. Keesling. "The Hausdorff dimensions of the boundary of the Lévy dragon," Geometry and topology in dynamics (Winston-Salem, NC, 1998/San Antonio, TX, 1999), 87–97, Contemp. Math., Vol. 246, Amer. Math. Soc., 1999. .
4. P. Duvall and J. Keesling, "The dimension of the boundary of the Lévy Dragon," International Journal of Mathematics and Mathematical Sciences, vol. 20, no. 4, pp. 627-632, 1997.
5. Scott Bailey, Theodore Kim and Robert Strichartz. "Inside the Lévy Dragon," American Mathematical Monthly, Vol 109, no. 8 (October 2002), 689-703 [available at JSTOR (subscription required)]. A companion website is available at https://web.archive.org/web/20180613013153/http://www.mathlab.cornell.edu/~twk6/. You can also listen to a talk by Scott Bailey on "How to Tame a Dragon" given at Clayton State College in 2011 (Flash required).
6. E. Alster, "The finite number of interior component shapes of the Lévy dragon," Discrete and Computational Geometry, Vol. 43, No. 4 (2010), 855-875.