Fractals Nicole MacFarlane December 1 st, 2014

What are Fractals? Fractals are never- ending patterns. Many objects in nature have what is called a ‘self- similar’ structure, in that a small portion of the object looks very much like the object as a whole, creating an infinite loop of the same shape.

History of Fractals One of the main developers of fractals was a Polish- born, French and American Mathematician Benoit Mandelbrot. He coined the term ‘fractal’ in 1975, after investigating a variety of self- similar structures in nature, and used geometry to help prove this theory of ‘self- similarity’.

Examples of Fractals For this presentation, I will demonstrate how to construct the following fractals, as well as how to determine their area and perimeter:  Koch Snowflake  Sierpinski’s Triangle

Koch Snowflake Start with an equilateral triangle, with sides of length 1. First iteration: Divide each line segment from the original triangle into 3 equal parts. Erase the middle part and substitute it with the top part of an equilateral triangle. Repeat this process infinitely many times to get the fractal. The video to the right shows the first seven iterations of the Koch Snowflake.

The Math Behind the Koch Snowflake Using the information from the previous slide on how to construct Koch’s Snowflake, we can use this formula to determine the number of sides, n: n = 3*4 a, in the a th iteration.  For iterations 0, 1, 2 and 3, the number of sides are 3, 12, 48 and 192, respectively. The determine the length of a side:  After every iteration, the length of a side is 1/3 the length of a side from the iteration preceding it. If we start with an equilateral triangle with side length 1, then the length of a side in iteration ‘a’ is: length = 1/3 a. For iterations 0 to 3, length = 1, 1/3, 1/9 and 1/27.

What is the perimeter of the Koch Snowflake (for the first three iterations)? Number of IterationsPerimeter 0 (Original Figure)1+1+1 =3 112 * (1/3) =4 248 * (1/9) = (16/3) 3(16/3) * (4/3) = (64/9) This shows that the Koch Snowflake has perimeter that increases by 4/3 of the previous perimeter for each iteration.

What happens to the area of the Koch Snowflake? Area of original triangle= 1 unit 2 After the first iteration: area =1 + 3(1/3 2 )= 1 + 3(1/9)= 1 + (1/3) units 2 After the second iteration: area = 1 + 3(1/9) + 12(1/3 4 )= 1 + (1/3) + (4/27) units 2 After the third iteration: area = 1 + (1/3) + (4/27) + 48(1/3 6 )= 1 + (1/3) + (4/27) + (16/243) units 2 After infinitely many iterations: 1+ (1/3)[1+ (4/9) + (4/9) 2 + …+ (4/9) n + …] = 1 + (1/3)[1/ (1- (4/9))] = 1 + (1/3)[ 1 / (5/9)] = 1 + (1/3)[ (9/5) ] = 1 + (3/5) = 8/5 units 2, which is a finite area.

Sierpinski’s Triangle  First iteration: the original equilateral triangle is divided into four triangles, and the middle triangle is removed.  Second iteration: each of the three unshaded equilateral triangles are divided into four more equilateral triangles, and the middle triangle from each of these triangles is, once again, removed.  This process is repeated infinitely many times.  The video to the right shows the first nine iterations of the Sierpinski’s Triangle.

The Math Behind Sierpinski’s Triangle Using the steps on constructing Sierpinski’s Triangle from the previous slide, we can determine the formula for the number of triangles removed in Sierpinski’s Triangle, which is: a k = 3 k-1, in the k th iteration.  For iterations 0, 1, 2 and 3, the number of triangles removed (after each iteration) are 0, 1, 3 and 9, respectively. The determine the length of a side:  After every iteration, the length of a side is 1/2 the length of a side from the preceding stage. If we begin with an equilateral triangle with side length 1, then the length of a side in iteration ‘a’ is: length = 1*3 -a. For iterations 0 to 3, length = 1, 1/2, 1/4 and 1/8.

What is the perimeter of Sierpinski’s Triangle? Number of IterationsPerimeter 0 (Original Figure)1+1+1 = 3 19 sides* (length = 1/2) = 9/2 227 * (1/4) = 27/4 381 * (1/8) = 81/8 This shows that Sierpinski’s Triangle has perimeter that increases by 3/2 of the previous perimeter for each iteration.

What happens to the area of Sierpinski’s Triangle (unshaded section)? Area of original triangle= 1 unit 2 After the first iteration: area = 3(1/4) units 2 After the second iteration: area = 9(1/16) units 2 After the third iteration: area = 1 + 3(1/4) + 9(1/4 2 ) + 27(1/64)= 1 + 3(1/4) + 9(1/4 2 ) + 27(1/4 3 ) units 2 After infinitely many iterations: area= (3/4) n-1 units 2 (the area will eventually converge to 0).

Putting it All Together… As you can see from the information gathered, the curve around the boundary has an infinite perimeter, but encloses a finite area.

Other Examples of Fractals

Fractal Dimension Fractal dimension is a measure of how "complicated" a self- similar structure is.

Fractal Dimension Take the example of a self- similar figure such as a line segment (with a length of 1).  Split the line segment into 2 equal parts. This gives you 2 new line segments that are scaled down versions of the original line segment. Next, use the example of another self- similar figure, such as a square with dimensions 1 X 1.  Split each side by 2. This gives you 4 squares that are scaled down versions of the original square. Finally, Take a cube with dimensions 1 X 1 X 1.  Divide each side by 2. This gives you 8 smaller cubes, that again, are scaled down versions of the original cube.

Fractal Dimension Let N= number of self- similar pieces; r= scaling factor; D= dimension. For each self- similar figure from the previous slide (line segment, square, and cube), we have r=2 (since you split each side by 2) : Based on this information, we can define fractal dimension as: N= r D. FigureFractal Dimension (D) Number of self- similar pieces (N) Line Segment12 Square24 Cube38

Fractal Dimension Using the information from the previous slide, we can determine the fractal dimension for a Koch Snowflake: r=3, N=4, and N= r D : 4=3 D ln4 = ln3 D D= (ln4)/(ln3)= 1.26

Fun Facts In the movie Star Trek II- The Wrath of Kahn, the rebirth of a planet was created with great effect using fractals. Fractals can be found as part of the limited edition artwork from the alternative Outkast album “Stankonia”.

Potential Exam Question/ Answer What is the fractal dimension for Sierpinski’s Triangle when r=2, N=3? Answer: r=2, N=3, Find D: N= r D 3=2 D ln3= ln2 D D= (ln3)/(ln2)= 1.58

References Fractal. Retrieved November 25, 2014, from Lanius, C. (n.d.). Cynthia Lanius' Fractals Unit: Fractal Dimension. Retrieved November 25, 2014, from Sierpinski triangle. Retrieved November 25, 2014, from Su, Francis E., et al. "Koch Snowflake." Math Fun Facts. Taylor, T. (Professor) (2014, February 14). Fractals. Lecture conducted from Antigonish, Nova Scotia. Venit, S., & Bishop, W. (2008). Elementary linear algebra. Toronto: Nelson Education