Nonlinear Dynamics and Chaos
Nonlinear Dynamics and Chaos
2nd Edition
ISBN: 9780429972195
Author: Steven H. Strogatz
Publisher: Taylor & Francis
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Chapter 11.3, Problem 7E
Interpretation Introduction

Interpretation:

To construct the fractals von Koch snowflake curve, taking an equilateral triangle as S0 and doing the von Koch procedure on each of its three sides.

  1. To show that S1 looks like a David star.

  2. To draw S2 and S3.

  3. To show that the curve S = S has an infinite arc length.

  4. The area of the region bounded by S is to be found.

  5. To find the similarity dimension of S.

Concept Introduction:

  • Fractals are the complex geometric shapes with a fine structure at arbitrarily small scales.

  • The similarity dimension d is defined as:

    d =ln mln r, where the values of m and r are determined by inspection.

  • The procedure of constructing the Koch snowflake is by deleting the middle third of S0 and replacing it with the other side of the equilateral triangle. This way, Sn is obtained by applying this procedure on Sn-1

Expert Solution & Answer
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Answer to Problem 7E

Solution:

  1. It is shown that S1 looks like a David star.

  2. S2 and S3 is drawn.

  3. It is shown that S = S have an infinite arc length.

  4. The area of the region surrounded by S is 235.

  5. The similarity dimension of the Koch snowflake is d =ln(4)ln(3).

Explanation of Solution

  1. The von Koch snowflake curve can be constructed by the computer:

    Nonlinear Dynamics and Chaos, Chapter 11.3, Problem 7E , additional homework tip  1

  2. Nonlinear Dynamics and Chaos, Chapter 11.3, Problem 7E , additional homework tip  2

  3. The arc length of S0= 3; each iteration arc length is multiplied by 43.

    Therefore, for S = S:

    The arc length is =3(43)=

  4. S0 has a side length = 1, and the area is 34. In every iteration, a new equilateral triangle is added to for each side having the side length as 13.

    Therefore:

    The side length of Sn=(13)n

    The number of sides of Sn=3(4)n

    The area added in each iteration: Sn1to Sn=(Number of sides of Sn1)34(13(Side length of Sn1))2

    =3(4)n134(13(13)n1)2=3(4)n134((13)n)2=3(4)n134(32n)=3316(49)n

    Add the area of S0 and the area added after each iteration:

    34+3316n=1(49)n=34+331649149=34+331645=34(1+35)=235

    Thus, the area of the region enclosed by S is 235

  5. The Koch snowflakes are constructed by using three Koch curves. To recreate the Koch curve, each Koch curve would require four copies of itself scaled by the factor 13.

    Thus, the Koch snowflake can be constructed by taking four complete copies of itself and scaling each of them by factor 13, after which each scaled Koch snowflake is cut into thirds and placed onto the full-size Koch snowflake.

    Therefore, the similarity dimension d of the Koch snowflake is given by:

    4=3d

    Rearrange the above expression:-

    d =ln(4)ln(3)

    Thus, the similarity dimension of the Koch snowflake is d =ln(4)ln(3).

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