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1 Fun with Fractals Dr. Bori Mazzag Redwood Empire Mathematics Tournament March 25, 2006

2 Images of some fractals

3 What are fractals, anyway? Important aspects of fractals: Self-similarity

4 What are fractals, anyway? Important aspects of fractals: Self-similarity Recursive rule to create figures

5 What are fractals, anyway? Important aspects of fractals: Self-similarity Recursive rule to create figures Fractional dimensions

6 In fractals... Self-similarity

7 Self-similarity In fractals... in nature...

8 and in art. (M.C. Escher)

9 Recursive rules and the Mandelbrot sequence General recursive rule: s N+1 = f(s N )

10 Recursive rules and the Mandelbrot sequence General recursive rule: s N+1 = f(s N ) Mandelbrot sequence: s 0 = s s N+1 = s 2 N + s

11 Recursive rules and the Mandelbrot sequence General recursive rule: s N+1 = f(s N ) Mandelbrot sequence: s 0 = s s N+1 = s 2 N + s Let s try it on a few starting points: s 0 = 1 s 1 = s s = = 2 s 2 = s s = = 5 (escaping sequence)

12 Recursive rules and the Mandelbrot sequence General recursive rule: s N+1 = f(s N ) Mandelbrot sequence: s 0 = s s N+1 = s 2 N + s Let s try it on a few starting points: s 0 = 1 s 1 = s s = = 2 s 2 = s s = = 5 (escaping sequence) s 0 = 1 s 1 = ( 1) 2 1 = 0 s 2 = = 1 (periodic sequence)

13 s 0 = 0.75 s 1 = ( 0.75) = 0.19 s 2 = ( 0.19) = 0.72 as N gets large, s N 0.5, so -0.5 is an attractor

14 s 0 = 0.75 s 1 = ( 0.75) = 0.19 s 2 = ( 0.19) = 0.72 as N gets large, s N 0.5, so -0.5 is an attractor b a+bi a

15 s 0 = 0.75 s 1 = ( 0.75) = 0.19 s 2 = ( 0.19) = 0.72 as N gets large, s N 0.5, so -0.5 is an attractor b a+bi a (i) seeds of periodic sequences and sequences that approach attractors are colored black (ii) seeds if escaping sequences are colored

16 Creating a fractal Recursion with pictures Generating the Koch snowflake

17 Creating a fractal Recursion with pictures Generating the Koch snowflake Generating Sierpinski s gasket

18 Calculating the perimenter of the the Koch snowflake

19 Calculating the perimenter of the the Koch snowflake Start Step 1 Step 2 Step 3 ( 3 4 ) ( ) 2 ( ) 3 3 3

20 Calculating the perimenter of the the Koch snowflake Start Step 1 Step 2 Step 3 ( 3 4 ) ( ) 2 ( ) As the number of steps, N increases, so does ( 4 N 3) and the perimenter becomes infinite.

21 The area of the Koch snowflake - activity Start Step 1 Step 2 Step 3 Step N Number of edges # new triangles - ( 3 area of each new triangle - 1 ) 9 A total new area A total area A A A

22 The area of the Koch snowflake Start Step 1 Step 2 Step N Number of edges N # new triangles N 1 ( A of each new triangle - 1 ) ( 9 A 1 ) 2 ( 9 A 1 ) N 9 A total new A A ( ( 4 9) 1 ( 3) A 4 N 1 ( 9) 1 3) A Total new area: A + ( ( 1 3) A + 4 ( 1 ( 9) 3) A ( 1 ( 9) 3) A N 1 ( 1 9) 3) A Using: a + ar + ar ar N 1 = a(rn 1) r 1 Area of the Koch snowflake is 8 5 A = 1.6A. we get:

23 The area of the Koch snowflake Start Step 1 Step 2 Step N Number of edges N # new triangles N 1 ( A of each new triangle - 1 ) ( 9 A 1 ) 2 ( 9 A 1 ) N 9 A total new A A ( ( 4 9) 1 ( 3) A 4 N 1 ( 9) 1 3) A Total new area: A + ( ( 1 3) A + 4 ( 1 ( 9) 3) A ( 1 ( 9) 3) A N 1 ( 1 9) 3) A Using: a + ar + ar ar N 1 = a(rn 1) r 1 Area of the Koch snowflake is 8 5 A = 1.6A. we get: The Koch snowflake has finite area and infinite perimeter!

24 Computing dimensions Figure Dimension # of copies Line 1 3 Square 2 9 Cube 3 27 Shape d n = 3 d

25 Dimension of the Koch snowflake We get 4 new line segments or 4 self-similar copies. So, in our formula Compute d, the dimension: d = ln 4 ln 3 = = 3 d. The Koch snowflake has fractional dimensions!

26 Applications of fractals Ben-Jacob: bacterial growth in stressed environments Biological applications (i) Bacterial growth under stress (ii) Classifying home-range searches (iii) Branching systems : structure of blood vessels

27 Applications of fractals II. Aerial view of rivers in Norway Mathematical and physical science connections (i) Connections with other areas of mathematics (dynamical systems and chaos) (ii) material science (iii) modeling terrains (iv) modeling watersheds (rivers and creeks) (v) measuring coastlines

28 Some references Stunning fractal images: Fractal-like images in nature: Mathematical details on dimensions, self-similarity and connections to chaos on Bob Devaney s page Chaos in the Classroom : More mathematics:

29 Thank you! Bori Mazzag Humboldt State University

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