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 i 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 Z^2 + C (0,0) (-1,0) C = 0 + 0i or 0 C = i or -1 Z^2 + 0 Z^2 + (-1) 0^2 + 0 = 0 0^2 + (-1) = -1 0^2 + 0 = 0 -1^2 + (-1) = 0 0^2 + 0 = 0 0^2 + (-1) = -1 0^2 + 0 = 0 -1^2 + (-1) = 0 0^2 + 0 = 0 0^2 + (-1) = -1

More Examples Z^2 + C Z^2 + C (2,0) (0,1) C = 2 + 0i or 2 C = 0 +1i or i Z^2 + 2 Z^2 + i 0^2 + 2 = 2 0^2 + i = i 2^2 + 2 = 6 i^2 + i = -1 + i 6^2 + 2 = 38 (-1 + i)^2 + i = -i 38^2 + 2 = 1444 (-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 Goldsmith, Jeffrey. (1994) The Geometric Dreams of Benoit Mandlebrot. Wired. Niall,Ryan. Penrose, Roger. The Emperor’s New Mind: Concerning computers, Minds, and the laws of Physics. Mathematics and Reality.Ch3.pgs Oxford University Press, New York:1989