Explanation Given: First three Steps of generation of quadratic Koch Curve as below- Formula used: Fractal dimension D of the object is found from N = R D Where, N is the factor by which number of element increases when ruler is shortened by a factor R. Taking logarithm on both the sides- log(N) = log(R D ) log(N) = D log(R) [applying the rule log(a x ) = x log(a) to right hand side] D = log ( N ) log ( R ) An object is defined as fractal if it has a non-integer dimension. Calculation: a. It can be observed from the steps of quadratic Koch curve generation- For R = 1, N = 1 = 1 3 2 For R = 4,N = 8 = 4 3 2 For R = 16, N = 8 × 8 = 64 = 16 3 2 Thus, in general, N = R 3 2 b. Fractal dimension = log ( N ) log ( R ) = 3 2 = 1.5 In every step of the quadratic Koch curve generation ruler size is reduced by a factor of 4 and the number of elements increases by factor of 8...

Using & Understanding Mathematics, Books a la Carte edition (7th Edition)

7th Edition

ISBN: 9780134716015

Author: Jeffrey O. Bennett, William L. Briggs

Publisher: PEARSON

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Question

Chapter 10.C, Problem 27E

To determine

a. The relation between N and R for quadratic Koch curve.

b. The fractal dimension of quadratic coach curve and conclusion about the length of quadratic Koch curve with conclusions

c. Why the total area of quadratic Koch Island is the same as the area of the original square and the length of the Koch Island.

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Chapter 10.C, Problem 26E

Chapter 10.C, Problem 28E

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