lated 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.
An alternative method to compute d is to partition the curve in Figure 3 into three similar pieces as shown in Figure 5. Then the equation for d is
which can be reduced to the same algebraic equation as (8) to yield the same solution for the fractal dimension d.
Since the boundary of dragon curve is composed of four copies of the curve we just examined, the fractal dimension of the entire boundary is also d = 1.523627085.
Algorithms to Generate the Boundary of the Dragon Curve
In accordance with the two different methods for determining the dimension of the boundary of the dragon curve, we present two different algorithms for generating the four sections making up the boundary. Like the first method for calculating d, Algorithm 1 uses the idea of separating the boundary into an infinite number of copies while Algorithm 2 corresponds with the second method by viewing the boundary as three similar pieces. From Figure 6 and Figure 7, we see that Algorithm 1 is quicker to converge and aesthetically more pleasing than Algorithm 2.
Algorithm 1 :
To produce the entire boundary of the dragon curve, four similar pieces of the border, generated by the above algorithms, have to be arranged as shown in Figure 8. With a suitable scaling of n and the orientation of the curve, the pieces should fit together the boundary dragon curve
The boundary of the dragon curve can also be generated using the chaos game. For the generation of the boundary of the dragon curve using the chaos game, we utilize the IFS structure that we had analyzed earlier in Figure 5. According to the theory of the chaos game , the boundary as shown in Figure 5 can be generated by a random iterated system which corresponds the these three linear affine transformations.
T1(x,y) = ((x-y)/4-1/2, (x+y)/4+1)
T2(x,y) = ((y-x)/4+1/2, -(x+y)/4+1)
T3(x,y) = ((x+y)/2+1, (y-x)/2)
Because the complete boundary is pieced together from four similar fractals, the computer is required to "play" four similar chaos games at the same time to generate the entire border. Figure 9 sho