Koch Curves: Rewriting System, Geometry and Application

: Problem statement: Recently, new Koch curves have been generated by dividing the initiator into three unequal parts. There is no formal rewriting system to generate such kind of curves. Approach: It is required to measure the new changed geometrical properties. Generalized rewriting systems for the new Koch curves have been developed. Results: New formulas have been given to measure their geometrical properties. Conclusion/Recommendations: The geometrical properties of new Koch curves make them more suitable as antennas in wireless communication than the conventional Koch curve.


INTRODUCTION
The classical Koch curve is, mathematically, continuous everywhere but differentiable nowhere. It is an example of bounded curve of infinite length (Schroeder, 2009;Falconer, 2003). Koch snowflake curve is used as a fractal antenna. Nathan (2005) described the importance of fractal antennas explicitly for wireless technologies with emphasis to military services. For a detailed study on Koch fractal antenna and related various new findings, one may refer to (Elkamchouchi and Nasr, 2007;Ghatak et al., 2009;Kordzadeh and Kashani, 2009;Krishna et al., 2009;Mirzapour and Hassani, 2009;Song et al., 2008;Werner and Suman, 2003;Zhang and Kishk, 2006) and several cross references thereof. McClure (2008) has discussed the vibration modes of a drum shaped like a Koch snowflake. Epstien and Adeeb (2008) derived stiffness of the Koch curve. Jibrael et al. (2008) simulated quadratic Koch antenna and explored its antenna properties. Further, Vinoy et al. (2002) presented a new way of generation of variants of a Koch curve by varying indentation angle and gave a formula to calculate their fractal dimension and studied the impact of fractal dimension in the design of multi-resonant fractal antennas (Vinoy et al., 2004). Generated new Koch curves as new examples of superior fractals by dividing the initiator into three unequal parts. A comprehensive review of literature on superior fractals, which are constructed using superior iterates, is given by Singh et al. (2011).
The purpose of this study is to develop rewriting rules for superior Koch curves and suggest formulas for calculation of their fundamental mathematical properties.
Rewriting system for koch curves: In this section, we develop general production rules to draw superior Koch curves. We divide the production rules for Koch curves at different scaling factors into two parts. All the symbols used in the following rewriting systems carry similar meanings as in that of Koch curve. We use the same initiator, as shown in Fig. 1a, in all the generations.
Case 1: Koch curves at scaling factor 1 S n = , where n is an odd number.
Let 1 S 5 = , then one of the possible Koch curves is shown in Fig. 2. The following is the rewriting system for such a Koch curve: Figure 2 Koch middle one-fifth curve for (r1, r2, r3) = (2/5, 1/5, 2/5) with its initiator: Case 2: Koch curves at scaling factor S=1/n, where n is an even number. Let the scaling factor be ¼, then rewriting system for the two possible Koch middle one-fourth curves (see the generators in Fig. 3) is given below.
Where m is the total number of iterations.
Considering: Theoretically, a Koch loop can accommodate wire of infinite length. However, with the reducing scaling factor, growth rate of the perimeter decreases.
The general formula to calculate the dimension of a Koch curve is given by: Kochcurve log(n 1) Dimension , n N and n 3 log n From (10), it can be calculated that the dimension of Koch curves at n = 4, 5 and 6 is approximately 1.16, 1.113 and 1.086 respectively. According to (Vinoy et al., 2004;Song et al., 2008) (8)) and can accommodate long length of wire (cf. (9)) and therefore will have better multiband characteristics due to smaller fractal dimension (cf. (10)).

CONCLUSION
Following formulas have been derived in this study: Rewriting system for superior Koch curves at.
Scaling factor: s = 1/n, where n is an odd natural number and n ≥ 3: = , as they are more compact (Result 2(i)), can accommodate long length of wire (Result 2(ii)) and will have better multiband characteristic due to smaller fractal dimension (Result 2(iii)).