Mandelbrot Set Project
I.
Name ________________________
Date ____________ Pd _________
Draw real and imaginary axes on a sheet of
graph paper, numbering maybe -5 to 5, counting
by quarters or halves. Pick a complex number,
any complex number: c = a + bi. Plot it on your
graph in pencil.
Use this flow chart to find a sequence
of numbers: z0, z1, z2.
That is, take your z0, square it by
FOILing the multiplication of it times
itself, then add the original c to get
your next number, z1. Now take z1,
square it and add c to get z2.
z0 = c =
z1 =
z2 =
Show your work to get z1 and z2 here:
How big is z2? Is it big (numbers bigger than 100)? Is it medium
(numbers from 5 to 100)? Or is it small (small numbers and
fractions/decimals)?
If it is small, color it red, since after many times through the feedback loop
it will probably still be small (it stays in the same area of the graph, close
to zero). If it is big, color it green, since it goes far away from its original
location: every time through the loop it will get bigger and bigger.
II.
Try the following points: compute z1 and z2, showing your FOIL
work! Then color the point red or green, depending if it stays small or
goes off large toward infinity. If the number is medium, leave it
uncolored (in pencil): we might be able to color it in later.
1) c = 5 + 3i
2) c = 2 + 4i
3) c = 1 + 1i
4) c = -0.5 +0.5i
III.
Open the Mandelbrot Excel spreadsheet from Blackboard
(www.bcpss.org). Please don’t change anything in columns A through
D; C and D especially are essential for calculating the operation
zn = zn-12 + c.
Notice columns E and F, which is just repeated copies of the starting
point c = a + bi. When you first get the spreadsheet from Blackboard,
a=1 and b=1, so the point c we are beginning with is 1+1i, which you
examined in II.3.
Does z2 = z(2) agree with what you found above? If it doesn’t get a
classmate or Mr. Yates to check over your work!
Notice that, even though you probably called this point “medium”, the
numbers do get large pretty soon, with z4 = -9407 – 193i, and z6 getting
into scientific notation (7.811 x 1015), and z11 exceeding the computer’s
number size limit!
Try changing columns E and F to represent the point -0.5+0.5i. The
best way to do this is to change cell E2 to -0.5, change F2 to 0.5, then
highlight all of cells E2-32, and go to Edit  Fill  Down, then do the
same for F2-32. Notice this time the numbers stay very small, even
past the z2 you calculated in II.4. They never exceed 1 or -1.
IV.
Try each of the following seeds as your c value, using the method
described above to alter the entire E and F columns. Then plot these
points and color them in appropriately.
0+0i
0+1i
0+2i
0+-1i
0+-2i
1+0i
2+0i
V.
-1+0i
-2+0i
1+2i
2+1i
2+2i
1+-1i
-1+1i
-1+-1i
0.5+0.5i
0.5+-0.5i
-0.5+-0.5i
0.25+0.25i
0.25+-0.25i
-0.25+0.25i
-0.25+-0.25i
What do you notice about your red points and green points? Find an
online picture of the Mandelbrot Set, print it out to attach to this page
and your graph paper. Explain in about five sentences what the
Mandelbrot Set is and how it relates to what we just did.