1. Mandelbrot Fractals Betsey Davis MathScience Innovation Center

2. Mandelbrot Fractals B. Davis MathScience Innovation Center Benoit Mandelbrot largely responsible for the present interest in fractal geometry. He showed how fractals can occur in many different places in both mathematics and elsewhere in nature. Mandelbrot was born in Poland in 1924 into a family with a very academic tradition.

3. Mandelbrot Fractals B. Davis MathScience Innovation Center Benoit Mandelbrot Sterling Professor of Mathematical Sciences Mathematics Department Yale University IBM Fellow Emeritus

4. Mandelbrot Fractals B. Davis MathScience Innovation Center Let’s start with Julia Sets Gaston Julia studied the iteration of polynomials and rational functions in the early twentieth century. If f(x) is a function, various behaviors can arise when f is iterated. Let's take, for example, the function f(x) = x 2 – 0.75. http://aleph0.clarku.edu/~djoyce/julia/julia.html

5. Mandelbrot Fractals B. Davis MathScience Innovation Center Julia Sets We will iterate this function when initially applied to an initial value of x, say x = a 0. Let a 1 denote the first iterate f(a 0 ), let a 2 denote the second iterate f(a 1 ), which equals f(f(a 0 )), and so forth. Then we'll consider the infinite sequence of iterates a 0, a 1 = f(a 0 ), a 2 = f(a 1 ), a 3 = f(a 2 ),... http://aleph0.clarku.edu/~djoyce/julia/julia.html

6. Mandelbrot Fractals B. Davis MathScience Innovation Center Julia Sets It may happen that these values stay small or perhaps they don't, depending on the initial value a 0. For instance, if we iterate our sample function f(x) = x 2 – 0.75 starting with the initial value a 0 = 1.0, we'll get the following sequence of iterates (easily computed with a handheld calculator) a 0 = 1.0, a 1 = f(1.0) = 1.0 2 – 0.75 = 0.25 a 2 = f(0.25) = 0.25 2 – 0.75 = –0.6875 a 3 = f(–0.6875) = (–0.6875) 2 – 0.75 = –0.2773 a 4 = f(–0.2773) = (–0.2773) 2 – 0.75 = –0.6731 a 5 = f(–0.6731) = (–0.6731) 2 – 0.75 = –0.2970 http://aleph0.clarku.edu/~djoyce/julia/julia.html

7. Mandelbrot Fractals B. Davis MathScience Innovation Center Julia Sets If you extend this table far enough, you'll see the iterates slowly approach the number –0.5. The iterates are above or below –0.5, but they get closer and closer to –0.5. In summary, when the initial value is a 0 = 1.0, the iterates stay small, and, in particular, they approach –0.5. http://aleph0.clarku.edu/~djoyce/julia/julia.html

8. Mandelbrot Fractals B. Davis MathScience Innovation Center Two things can happen In our example, they approach –0.5. So, one thing that can happen is that the value of f(x) approaches a limit but never exceeds it Another is that it can grow without bound http://aleph0.clarku.edu/~djoyce/julia/julia.html

9. Mandelbrot Fractals B. Davis MathScience Innovation Center Two things can happen: If value of f(x) approaches a limit but never exceeds it, it stays black –oscillation back and forth creates “bulbs” If it grows without bound, and it is assigned a different color depending on when it “breaks out” (escapes) http://aleph0.clarku.edu/~djoyce/julia/julia.html

10. Mandelbrot Fractals B. Davis MathScience Innovation Center Mandelbrot Sets Consider a whole family of functions parameterized by a variable. Although any family of functions can be studied, we'll look at the most studied family, that being the family of quadratic polynomials f(x) = x 2 - µ, where µ is a complex parameter. As µ varies, the Julia set will vary on the complex plane. Some of these Julia sets will be connected, and some will be disconnected, and so this character of the Julia sets will partition the µ- parameter plane into two parts. http://aleph0.clarku.edu/~djoyce/julia/julia.html

11. Mandelbrot Fractals B. Davis MathScience Innovation Center Mandelbrot Sets Those values of µ for which the Julia set is connected is called the Mandelbrot set in the parameter plane. The boundary between the Mandelbrot set and its complement is often called the Mandelbrot separator curve. The Mandelbrot set is the black shape in the picture. This is the portion of the plane where x varies from -1 to 2 and y varies between -1.5 and 1.5. http://aleph0.clarku.edu/~djoyce/julia/julia.html

12. Mandelbrot Fractals B. Davis MathScience Innovation Center Mandelbrot Sets There are some surprising details in this image, and it's well worth exploring. The bulk of the Mandelbrot set is the black cardioid. exploring A cardioid is a heart- shaped figure. http://aleph0.clarku.edu/~djoyce/julia/julia.html

13. Mandelbrot Fractals B. Davis MathScience Innovation Center The period of this bulb is 5 we include the spoke holding to the bulb numbers in this region repeat cycle in 5 steps

14. Mandelbrot Fractals B. Davis MathScience Innovation Center Guess the period of this bulb 3

15. Mandelbrot Fractals B. Davis MathScience Innovation Center Guess the period of this bulb 5

16. Mandelbrot Fractals B. Davis MathScience Innovation Center Here’s another zoom

17. Mandelbrot Fractals B. Davis MathScience Innovation Center To Create your own… Mandelbrot Explorer –http://www.softlab.ece.ntua.gr/miscellaneous/m andel/mandel.htmlhttp://www.softlab.ece.ntua.gr/miscellaneous/m andel/mandel.html Julia and Mandelbrot Set Explorer –http://aleph0.clarku.edu/~djoyce/julia/explorer. htmlhttp://aleph0.clarku.edu/~djoyce/julia/explorer. html