 by .
Using (14), we get that the area enclosed by the boundary of the dragon curve is:
To reconcile this answer for the area with our first solution in equation (13), recall that the first solution uses the length a as shown in Figure 12. From the Pythagorean Theorem we know that a2=9b2+b2 so that b2=a2/10. From here it becomes apparent that the two solutions from equations (13) and (15) are one and the same since S=5b2=a2/2.
In addition we can also calculate the area of each small shape of which the dragon curve is composed. Since it is self-evident that these shapes are similar we can determine the relationship between their size as shown in Figure 13.
Using the relationship between their linear dimension, we can determine the relationship between their areas and we find that .
From Figure 13 it is apparent that
Solving for T0, we get T0=0.4S.
Although at this point in time the dragon curve and its border seem no more than quaint examples of fractals, self-sirnilar structures creating order from chaos, there may come a time when the results from this little mathematical investigation prove to be useful in other areas of science as well. Fractals and chaos are after all, the footprints of nature, appearing here and everywhere and what is science but the quest for knowledge, about nature, this universe that we live in.
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2. Bansley, M., Fractals Everywhere, Academic Press, San Diego, 1988.
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5.Hutchinson, J., Fractal and Self-Similarity, Indiana University Journal of Mathematics 30 , 1981.