Take a line segment one inch long At stage 1 remove the midd

Take a line segment one inch long. At stage 1, remove the middle third and replace it with the top two legs of an equilateral triangle. At stage 2, remove the middle thirds of each of the four line segments and replace them with the top two legs of an equilateral triangle. Rinse, lather and repeat. See diagram. Write an expression for the length of the figure after the nth stage. After 10 iterations of this process, what is the length of the curve. If you stretch the curve flat, what is this length comparable to? The height of a human? The length of a school bus? Something bigger? What about after 100 iterations? What is this comparable to? What is the minimum number of iterations would it take to stretch from here to my high school in Florida (about 3,200 miles)? If we iterate this process infinitely many times, we get what is called the Koch Curve. This is a fractal since it exhibits self-similarity (one small piece looks like a scaled-down version of the bigger curve) and as such it is infinitely complex. You may notice that it is somehow caught in between being a 1-dimensional line and a 2-dimensional shape, between a line and a plane. In fact, the dimension, D is given by -D = log_1/3 4 is fractional (hence the term fractal). Find the dimension of the Koch Curve.

Solution

(a) from the above given diagram we can find out

In stage 0 , number of side = 1  

In stage 1 , number of side = 4

In stage 2 , number of side = 16

In stage 3 , number of side = 64

given sequence is in G.P with a =1 , r = 4/1 or, 16/4 or, 64/4 = 4

So, In stage n , number of side = a r^n-1 = 1 x 4^n-1 = 4^n-1

(b)

After each iteration, the number of sides of the Koch snowflake increases by a factor of 4, so the number of sides after n iterations is given by:

N_n = N_n-1 . 4 = 3. 4ⁿ

If the original equilateral triangle has sides of length s, the length of each side of the snowflake after n iterations is:

S_n = S_n-1 / 3 = s / 3ⁿ

So, S₁₀ = 1 / 3¹⁰ , now number of sides after 10 iterations = 4^10-1 = 4⁹                                                                           so length of the curve = 1 / 3¹⁰ x 4⁹ = 4.439 inch

stretching the curve flat means perimeter, the perimeter of the snowflake after n iterations is:

P_n = N_n . S_n = 3 . s . (4/3)ⁿ = 3. 1 . (4/3)¹⁰ = 53.2731 inch

for 100 iterations, P_n =3. 1 . (4/3)¹⁰⁰ = 9.3539 x 10¹² inch

(c) Now, 3 . s . (4/3)ⁿ = 3200 miles converts this to inches = 2.0275 x 10⁸ inches

So, (4/3)ⁿ = 2.0275 x 10⁸ / 3 = 67583333.33

taking log both sides we get, n log (4/3) = log 67583333.33

So, n = 63

(d) -D = log_1/34 = log 4 / log (1/3) = -1.2618 so, D = 1.2618