Fractals! Bullock Math Academy March 22, 2014 Brian Shelburne Dept. of Math & CS Wittenberg Univ. Website http://www4.wittenberg.edu/academics/mathcomp/shelburne/Fractals/Fractalindex.html

Bullock Math Academy – Saturday March 22, 2014 Kahn Academy Video - Doodling in Math: Dragons https://www.khanacademy.org/math/recreational-math/vi-hart/doodling-in-math/v/dragons Iteration Generating Fractals Measuring Areas and Lengths of Some Common Fractals Self-Similarity and Fractal Dimension The Mandelbrot Set Fractal

Prerequisites Pythagorean Theorem for Right Triangles FOIL Exponents: 2×2×2 = 23 and (1/3)×(1/3) = (1/3)2. Square Roots: 3 = 3 1 2 (and fractional exponents) Formulas for Area and Perimeter Ability to see and describe patterns

Geometric Iteration A Seed and a Rule Orbits and Eventual Orbits Seed: A square with side 1 Rule: Shrink each side by half

More Geometric Iteration Seed Rule: Rotate 90° clockwise Seed Orbit: Rule?

Numeric Iteration Seed: 12 Numeric Iteration Seed: 12.3 → x Rule: (1/3)∙x → x Seed: anything → x Rule: 𝑥 → x Seed: 0.67 → x 0.67 [sto→] x Rule: 3.5∙x∙(1-x) → x 3.5∙x∙(1-x) [sto→] x Seed: 0.45 → x 0.45 [sto→] x Rule: 3.2∙x∙(1-x) → x 3.2∙x∙(1-x) [sto→] x

You Try It Seed: Rule: Shrink each linear dimension by ½ and rotation 180° What is the Orbit? What is the Eventual Orbit?

The Fractal Plus Seed: Rule: Tic-Tac-Toe and Remove the 4 Corners

The Sierpinski Carpet Fractal Seed: Rule: Tic Tac Toe & Remove Middle Square Orbit:

Fractals by Removals Seed: An equilateral triangle Rule: Connect the midpoints of the 3 sides and remove the interior of the middle triangle Assume the area of the Seed is 1. What is the area of the 2nd and 3rd iterations?

Self-Similar Copies - Another Method for Generating Fractals Seed: Generator: Rule:

Dragon Fractal

The Dragon Fractal Seed: Rule: i.e. “jag” right then left Orbit:

The Koch Fractal Seed: Line of length 1: Rule: Divide by 3 – replace middle 3rd with a tent The Thunderbolt Fractal Seed: A Line of length 1: Rule: Divide by 4 and replace two middle pieces with opposite tents

Area and Perimeter

Sierpinski Carpet – Area and Perimeter Iteration 0 1 2 3 … n Number of Squares 1 8 64 512 8n Length of Side 1 1/3 1/9 1/27 (1/3)n Area 1 8/9 Perimeter 4 4+4/3

Sierpinski Gasket – Area and Perimeter Iteration 0 1 2 3 … n Number of Triangles 1 3 9 27 3n Length of Side 1 1/2 1/4 1/8 (1/2)n Area 1 3/4 Perimeter 3 9/2

Koch Curve – Perimeter Iteration 0 1 2 3 … n Number of Segments 1 4 16 64 4n Length of Segment 1 1/3 1/9 1/27 (1/3)n Length 1 4/3 Area under curve

Self-Similarity & Fractal Dimension

Dimension Line Number of Copies 1 2 3 4 … Scaling Factor 1 2 3 4 … Square Number of Copies 1 4 9 16 … Scaling Factor 1 2 3 4 ... Cube Number of Copies 1 8 27 64 …

Fractal Dimension Sierpinski Gasket Number of Copies 1 3 9 … Scaling Factor 1 2 4 … Sierpinski Carpet Number of Copies 1 8 64 … Scaling Factor 1 3 9 ... Koch Curve Number of Copies 1 4 16 … Scaling Factor 1 3 9 …

Logarithms: Solving 3 = 2n. What is n. Examples: Solve 9 = 3n Logarithms: Solving 3 = 2n. What is n? Examples: Solve 9 = 3n. Solve 16 = 4n. Solve 2 = 4n. Solve 32 = 4n. Solve 10 = 2m. Since 23 = 8 and 24 = 16 it follows that 3 < m < 4. If n is the unique number such that 3 = 2n , the we say n = log2 3. Using your calculator log2 3 = log3/log2

The Mandelbrot Set Complex Numbers (a brief introduction) Let 𝑖= −1 The Mandelbrot Set Complex Numbers (a brief introduction) Let 𝑖= −1 . So 𝑖 2 =−1. The i sometimes stands for imaginary although it’s not a very good name. A Complex Number has a real part and an imaginary part For example: (2 + 3i) All the usual rules for arithmetic work for complex numbers (Law of Least Astonishment) To add or subtract complex numbers add or subtract real parts and the imaginary parts separately For example (2 + 3i) + (4 – i) = (6 + 2i)

To multiply complex numbers use First Outside Inside Last For Example: (2 + 3i) × (4 – i) = 8 - 2i + 12i - 6i2 except i2 = -1 so 8 - 2i + 12i - 6i2 = 8 – 2i +12i + 6 = 14 -10i You Try It: Finally – In the same way you can plot real numbers on the number line, you can plot complex numbers in the complex plane (using the vertical y-axis to mark off the imaginary distance) .

Complex Iteration Seed: 1 → z Rule: Multiply by each term by 1 2 + 3 2 𝑖 : 1 2 + 3 2 𝑖 ×𝑧→𝑧 Orbit: 1, 1 2 + 3 2 𝑖 , −1 2 + 3 2 𝑖 , -1, −1 2 + − 3 2 𝑖 , 1 2 + − 3 2 𝑖, 1 … Geometric Picture -0.5 + 0.866i 0.5 + 0.866i + + + + -1 1 + + -0.5 - 0.866i 0.5 - 0.866i

The Mandelbrot Iteration Seed: Let z = 0 and let c be any complex number Rule: Square z and add c ( z = z2 + c) Try c = i Try c = -1.76 + 0.01i The Mandelbrot Set is the set of all complex numbers c such that the iteration formula 0 →z; z2 + c → z does not escape to infinity!

Coloring the Mandelbrot Set Black: Elements that do not escape – these points are in the Mandelbrot Set Colors: speed of escape – from red (fast) to magenta (slow)