On the Fractal Structure of the Boundary of Dragon Curve [pdf]
Angel Chang <email@example.com>
Tianrong Zhang <firstname.lastname@example.org>
Abstract: This paper discusses the fractal geometry of the boundary of the dragon curves by analyzing the boundary’s structure, calculating its Hausdorff dimension, and computing the area enclosed within the boundary. The algorithms for generating computer graphics for the boundary curves have also been included.
Fractals are everywhere in this world. From clouds, coastlines, leaves, to biological organs, many scientists can use fractal geometry to describe complex natural objects and processes [1-3]. Opening several new research areas in mathematics, geometry, and computer graphics, fractal geometry seems to describe natural shapes and forms more gracefully and succinctly than does Euclidean geometry,
The concept of scale invariance or self-similarity is an underlying theme in all fractal structures with many examples going back to the classical mathematicians. What we know so well as the Cantor set, the Koch curve, the Hilbert curve, are among these these classical irregular curves along with the dragon curve.
The dragon curve is a space filling curve with dimension 2 which originally came from the repeated folding of a long stripe of paper  in the same direction. After pleating the paper, it is then unfolded with each adjacent segments of paper formed into a right angle. The dragon curve emerges slowly with striking complexity and beauty. As shown in Figure 1, it is a construction that fills a finite area in a plane but has a fractal boundary. This same curve can also be constructed by a computer algorithm using IFS (Iterated Function Systems) or the chaos game .
Although the dragon curve has already been investigated in several papers, its boundary has remained unexplored. To alleviate this neglect, it is the purpose of this paper is to study the boundary of dragon curve. To do so, we partitioned the boundary into several pieces to find its IFS structure and then calculated the Hausdorff dimension of the boundary. Included in this paper are several algorithms to generate the boundary using a computer. Finally, we also computed the area enclosed by the border of the dragon curve.
IFS structure of the boundary
While it is evident that the boundary of the dragon curve is a fractal with recurring self-similarity, its exact structure remains elusive. To discover its hidden pattern, we partition the complete boundary of the dragon curve into four similar pieces as shown in Figure 2. In order to investigate the geometry of the boundary, we examine one section of the border as shown in Figure 3. Upon observation, it is apparent that this piece can be constructed with an infinite number of self-similar copies of itself with different sizes and rotated at various angles.
To analyze the curve in Figure 3, let its size be represented by the length of line segment as shown in Figure 4. Placed in opposite directions, the first two copies and their sizes can be represented by the line segments A1B1 and B1A2 with
The rest of the infinite number of copies can be grouped into pairs with each pair containing two exact duplicates placed in the opposite directions. All pairs form an infinite sequence such that each element in the sequence is reduced by a factor of 21/2 , and rotated by an angle of -45 degrees with respect to the previous one, i.e.
Hausdorff dimension of the boundary
Using the formula showed by Huchinson , the fractal dimension d of a self-similar curve constructed by N similar pieces with reduction factors can be calculated by solving the following equation:
For the curve as shown in Figure 3, it is easy to show that
By substituting (4) and (5) into (3), the equation for d can be rewritten as
When N ® ¥ , formula (6) results in
Solving equation (8) for x yields a real solution and two conjugated complex solutions. From the real solution, the fractal dimension d of the curve in Figure 3 can be expressed as
the numerical value of d is calculated to be approximately 1.523627085.
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