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The Mandlebrot Set
Derek Ball
University of Kentucky
Math 341- College Geometry
The Mandlebrot Set
• It is a type of fractal and was discovered in 1980 by
Benoit Mandlebrot.
• It uses Z^2 + C to test complex numbers on the
Argand plane to see if they are contained within the
boundaries of the set. This set being the region on
the Argand plane for which upon repeating this
sequence it remains bounded and does not approach
infinity.
• Points that are in the set are colored black in the
picture and the ones colored white are not. The
pictures are drawn with the aid of a computer.
Complex Numbers
• The square root of -1 is denoted as “i”
• “i” and all its multiples, such as 5i, 7.9i,
423i,etc., are referred to as imaginary
numbers.
• A complex number is any number that
results from a combination of a real number
with an imaginary number.
• It has the form of a + bi, thus 5 + 32i is a
complex number.
The Argand Plane
• The Argand plane is a way of organizing
complex numbers into a useful geometric
interpretation.
• It is an ordinary Euclidean Plane using
standard Cartesian coordinates, x and y
• x, the horizontal axis, represents the real
numbers and y, the vertical axis, represents
the imaginary numbers
• So the point 7 + 4i, this is the a + bi form,
and represents the point (7,4) on the Argand
plane.
Testing points using Z^2 + C
• C is always the complex number you are testing
• The first value for Z is always zero. After that the
resulting value is taken and substituted in the
formula for z, and the process is repeated.
• This is repeated N number of times to test each
point on the Argand plane.
• After the the process is repeated if the obtained
values do not approach infinity, and stay less than
two, they are believed to be contained in the
Mandelbrot Set.
Examples using Z^2 + C
Z^2 + C
(0,0)
C = 0 + 0i or 0
Z^2 + 0
0^2 + 0 = 0
0^2 + 0 = 0
0^2 + 0 = 0
0^2 + 0 = 0
0^2 + 0 = 0
Z^2 + C
(-1,0)
C = -1 + 0i or -1
Z^2 + (-1)
0^2 + (-1) = -1
-1^2 + (-1) = 0
0^2 + (-1) = -1
-1^2 + (-1) = 0
0^2 + (-1) = -1
More Examples
Z^2 + C
(2,0)
C = 2 + 0i or 2
Z^2 + 2
0^2 + 2 = 2
2^2 + 2 = 6
6^2 + 2 = 38
38^2 + 2 = 1444
Z^2 + C
(0,1)
C = 0 +1i or i
Z^2 + i
0^2 + i = i
i^2 + i = -1 + i
(-1 + i)^2 + i = -i
(-i)^2 + i = -1 + i
(-1 + i)^2 + i = -i
More Examples
Z^2 + C
(2,-3)
C = 2 - 3i
Z^2 + (2 - 3i)
(0)^2 + (2 - 3i) = 2 - 3i
(2 - 3i)^2 + (2 - 3i) = -9i - 3
(-9i - 3)^2 + (2 - 3i) = -45i + 75
Conclusion
The Mandlebrot Set was an important
discovery in a branch of mathematics that
opened up doors to areas that had been
virtually unexplored. It allowed for
computers to be used to help visualize
things that could not be seen and in many
cases would not have been proven
otherwise.
Sources
http://cs.gettysburg.edu/~jfink/vanworkshop/intro.html
Goldsmith, Jeffrey. (1994) The Geometric Dreams of Benoit
Mandlebrot. Wired. http://www.wired.com/wired/archive
Niall,Ryan.
http://www.maths.tcd.ie/~nryan/mandlebrot/how.html
Penrose, Roger. The Emperor’s New Mind: Concerning
computers, Minds, and the laws of Physics. Mathematics
and Reality.Ch3.pgs. 75-95. Oxford University Press,
New York:1989