Mandelbrot Fractals

1. Mandelbrot Fractals Betsey Davis MathScience Innovation Center

2. 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. Mandelbrot Fractals B. Davis MathScience Innovation Center

3. Benoit Mandelbrot Sterling Professor of Mathematical SciencesMathematics DepartmentYale UniversityIBM Fellow Emeritus Mandelbrot Fractals B. Davis MathScience Innovation Center

4. 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) = x2 – 0.75. http://aleph0.clarku.edu/~djoyce/julia/julia.html Mandelbrot Fractals B. Davis MathScience Innovation Center

5. Julia Sets • We will iterate this function when initially applied to an initial value of x, say x = a0. Let a1 denote the first iterate f(a0), let a2 denote the second iterate f(a1), which equals f(f(a0)), and so forth. Then we'll consider the infinite sequence of iterates • a0, a1 = f(a0), a2 = f(a1), a3 = f(a2), ... http://aleph0.clarku.edu/~djoyce/julia/julia.html Mandelbrot Fractals B. Davis MathScience Innovation Center

6. http://aleph0.clarku.edu/~djoyce/julia/julia.html Julia Sets • It may happen that these values stay small or perhaps they don't, depending on the initial value a0. For instance, if we iterate our sample function f(x) = x2 – 0.75 starting with the initial value a0 = 1.0, we'll get the following sequence of iterates (easily computed with a handheld calculator) • a0 =  1.0, • a1 = f(1.0) = 1.02 – 0.75 = 0.25 • a2 = f(0.25) = 0.252 – 0.75 = –0.6875 • a3 = f(–0.6875) = (–0.6875)2 – 0.75 = –0.2773 • a4 = f(–0.2773) = (–0.2773)2 – 0.75 = –0.6731 • a5 = f(–0.6731) = (–0.6731)2 – 0.75 = –0.2970 Mandelbrot Fractals B. Davis MathScience Innovation Center

7. 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 a0 = 1.0, the iterates stay small, and, in particular, they approach –0.5. Mandelbrot Fractals B. Davis MathScience Innovation Center http://aleph0.clarku.edu/~djoyce/julia/julia.html

8. 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 Mandelbrot Fractals B. Davis MathScience Innovation Center http://aleph0.clarku.edu/~djoyce/julia/julia.html

9. 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) Mandelbrot Fractals B. Davis MathScience Innovation Center http://aleph0.clarku.edu/~djoyce/julia/julia.html

10. 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) = x2 - 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 Mandelbrot Fractals B. Davis MathScience Innovation Center

11. 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 Mandelbrot Fractals B. Davis MathScience Innovation Center

12. 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. • A cardioid is a heart-shaped figure. http://aleph0.clarku.edu/~djoyce/julia/julia.html Mandelbrot Fractals B. Davis MathScience Innovation Center

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

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

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

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

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