For each of the following seeds,

(a) Find the first 5 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
   • Let x=-1.
   • x²+s=(-1)²+(-1)=1-1=0
   • Let x=result=0. s is still -1.
   • x²+s=0²+(-1)=0-1=-1
   • Let x=result=-1. s is still -1.

     At this point, we have exactly what we started with, so the result will be the same as well.

     x²+s=(-1)²+(-1)=1-1=0

   • Let x=result=0. s is still -1.

     Again, we've been here before. The result will be -1.

   The first several terms of the Mandelbrot sequence are -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

   {-1,0,-1,0,-1,0,...}

   The seed s=-1 is periodic.

   Thinking of -1 as a complex number, it is -1+0i, so 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 =-i
   • Let x=-i.
   • x²+s=(-i)²+(-i)=i²-i=-1-i.
   • Let x=result=-1-i. s is still -i.
   • x²+s=(-1-i)²+(-i)=(-1-i)(-1-i)-i=1+i+i+i²-i=1+2i-1-i=i.
   • Let x=result=i. s is still -i.
   • x²+s=i²+(-i)=-1-i.
   • Let x=result=-1-i. s is still -i. We've been here before, and we know that the result will be i.
   • Let x=result=i. s is still -i. We've also been here before, and we know that the result will be -1-i again.
   Thus the Mandelbrot sequence is
   {-i,-1-i,i,-1-i,i,-1-i, ...}

   The seed s=-i is periodic.

   -i can also be written 0-1i, and so 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
FAX (508) 285-8278