Practice with the Mandelbrot Set
    (Last modified: Wednesday, November 14, 2012, 10:59 AM )

    For each of the following seeds,

    (a) Find the first 6 terms of the Mandelbrot sequence with seed s.
    (b) Decide whether that Mandelbrot sequence is escaping, periodic, or attracted.
    (c) Will the point in the plane corresponding to the seed s be a black point or a non-black point?

    1. s =-1

      Work with: zn=zn-12-1

      • Let z0=0.
      • z1=z02-1=-1.
      • z2=z12-1=(-1)2-1=1-1=0
      • z3=z22-1=02-1=0-1=-1
      • z4= z32-1=(-1)2-1=1-1=0
      • z5=z42-1=0^2-1= -1.
      Thus the first six terms of the Mandelbrot sequence are 0,-1,0,-1,0,-1.

      We can see that as we repeat this process ad infinitum, we always bounce back and forth between -1 and 0. Thus we can say that the Mandelbrot sequence is

      {0,-1,0,-1,0,-1,0,...}
      (In other words, we can state the pattern continues indefinitely)

      The seed s=-1 is periodic, because the sequence bounces back and forth.

      Thinking of -1 as the complex number -1+0i, we can see it corresponds to the point (-1,0).

      Since it is not escaping, the point (-1,0) is in the Mandelbrot set.

      Color the point (-1,0) black.

    2. s =-2

      Work with: zn=zn-12-2

      • Let z0=0
      • z1=z02-2=02-2=-2.
      • z2=z12-2=(-2)2-2=4-2=2
      • z3=z22-2=22-2=4-2=2
      • z4= z32-2=22-2=4-2=2
      • z5= 2.
      Thus the first six terms of the Mandelbrot sequence are 0,-2, 2, 2, 2, 2.

      We can see that as we repeat this process ad infinitum, from here on out we will always get 2. Thus we can say that the Mandelbrot sequence is

      {0,-2, 2, 2, 2, 2, ...}
      (In other words, we can state the pattern continues indefinitely)

      The seed s=-2 is attracted, because it approaches a specific number. (In this case, it reaches that number, and reaches it quickly, but that doesn't have to happen.

      Thinking of -2 as the complex number -2+0i, we can see it corresponds to the point (-2,0).

      Since it is not escaping, the point (-2,0) is in the Mandelbrot set.

      Color the point (-2,0) black.

    3. s =-i

      Work with: zn=zn-12-i

      • Let z0=0
      • z1=z02-i=02-i=-i.
      • z2=z12-i=(-i)2-i=-1-i
      • z3=z22-i=(-1-i)2-i=(1+2i-1)-i=2i-i=i
      • z4= z32-i=i2-i=-1-i
      • z5=z42-i=(-1-i)^2-i= i.
      Thus the first five terms of the Mandelbrot sequence are 0,-i, -1-i, i, -1-i, i.

      We can see that as we repeat this process ad infinitum, from here on out we will continue to bounce between -1-i and i. Thus we can say that the Mandelbrot sequence is

      {0,-i, -1-i, i, -1-i, i, ...}
      (In other words, we can state the pattern continues indefinitely)

      The seed s=-i is periodic, because it bounces between two numbers.

      Thinking of -i as the complex number 0-1i, we can see it corresponds to the point (0,-1).

      Since it is not escaping, the point (0,-1) is in the Mandelbrot set.

      Color the point (0,-1) black.

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    Janice Sklensky
    Wheaton College
    Department of Mathematics and Computer Science
    Science Center, Room 109
    Norton, Massachusetts 02766-0930
    TEL (508) 286-3973
    FAz (508) 285-8278