FRACTALS The Geometry of Nature

1. π Π ϕ ε γ FRACTALSThe Geometry of Nature α ξ β λ θ Ω μ ρ Ξ ζ ψ τ ω Σ 𝜕 𝜎 By Michael Duong η 𝜈

2. What does this look like to you?

3. Georg Cantor: Infinite Insanity -Georg Cantor was born on March 3, 1845 in St. Petersburg, Russia. He studied number theory in the University of Berlin. He later became a mathematics professor of the University of Halle, a position he would remain in for the rest of his miserable life.

4. Georg Cantor: Infinite Insanity -He began to investigate the infinity. After developing parts of set theory, Cantor wondered what would happen if he took a line, split it in thirds, took away the middle, and repeated. Would this continue to infinity. YES! This was the birth of the first “mathematical monster,” the earliest form of a fractal.

5. Georg Cantor: Infinite Insanity -After numerous failed attempts to solve his famous continuum hypothesis, and the deaths of his loved ones, Cantor’s trips to the insane asylum became more and more frequent. His work had turned him insane. -He died in 1918.

6. What does this look like to you? Usual answers: -Beetle -Sting ray -Heart shaped thingy It is actually a… FRACTAL! But, what is a FRACTAL, and who made this one?

7. Benoit Mandelbrot: Father of Fractals -Benoit Mandelbrot was born on November 20, 1924 in Lithuania. His family moved to Paris, France when he was a child, where his uncle, Szolem, a studious mathematician, worked. -Unlike his uncle, Benoit Mandelbrot did not inherit a love for numbers. He was a misfit. DUCK SITING?

8. Benoit Mandelbrot: Father of Fractals -Mandelbrot was practically illiterate, and never learned multiplication past the 5 times tables. -One day, in geometry class, the professor asked the students to graph an equation. -Mandelbrot usually struggled at this because he lacked algebra skills, but as Mandelbrot pondered at the equation, he could suddenly visualize the graph. He had found his gift.

9. IBM and the FRACTALS -In the 1950’s a computer company called IBM was looking for new mathematicians and computer programmers -Mandelbrot was hired in 1958. His first task was to find out why there was static in the computer telephone lines. -When graphed for 1 day, 1 hour, and 1 second, he realized that all the static always remained the same. He called this “self-similarity.”

10. Self Similarity -characteristic in which the smaller and smaller details of a shape have the same original form. -Notice how as you zoom into this self similar shape, it looks as if you always end up in the place you originated from. Mandelbrot’s static was self similar, and so are those hands. FRACTALS are also self similar.

11. Recursive Formulas -While at IBM, Mandelbrot tried to solve a problem presented by a young mathematician named Gaston Julia. (recursive formula) So substitute a number for the x on the right. Get an answer for x on the left, and substitute that value on the right again, etc. Using the IBM computers, Mandelbrot graphed this equation. He got… Right x Left x 1 2 2 5 5 26 26 677 677 458,330

12. The Mandelbrot Set- -the graph of the equation x=x2+c. Notice that as you zoom into it, it is self similar. Mandelbrot called this kind of shape a fractal. Before Mandelbrot, other mathematicians called fractals monsters, because they iterated infinitely. (Cantor, Weierstrass, Koch, Sierpinski, Hausdorff, Julia.)

13. Fractals -a geometric figure that is created using iteration. -a process of repeating the same procedure over and over again. So, when we were creating the table for the Mandelbrot Set, we were iterating the equation x=x2+1. We continued the same procedure of x2 +1 and substitute in the x 5 times. So, we went through 5 iterations, or STAGES of the fractal. Iteration

14. Examples: -This is the Koch snowflake by Helge von Koch. It starts of as a triangle, and smaller triangles are added in the middle third of every straight line. How many stages does the animation go through? This is Sierpinski’s Triangle by Waclaw Sierpinski. It begins as a triangle and continues to add black triangles inside itself. What stage does the animation begin at? How many stages does the animation go through?

15. Real World Fractals- Many people believe fractals are just pretty pictures. Can you think of any fractals in the real world? Mandelbrot died on October 14, 2010, knowing that without his contributions, we would still be living in the Dark Ages.

16. π Π ϕ ε γ THE END? α ξ β λ θ Ω μ ρ Ξ ζ ψ τ ω Σ 𝜕 𝜎 η 𝜈