The core mathematical concept of MAT 104 is iteration, viewed from graphic, numeric, and symbolic perspectives. The construction of the black Mandelbrot set (above left) is based on the simple iteration
.To determine whether a selected fixed point (a, b) in the xy-plane representing the complex number c = a + b i should be colored black or white, this iteration is carried out repeatedly, starting with z = c. Thus
Now we ask the question whether the sequence of z-points thus obtained stays inside the circle of radius 2 centered at the origin in the complex plane. If the answer to this question is
To construct a computer image of the Mandelbrot set, the "viewing
window" in the xy-plane corresponding to our computer screen is
divided into a rectangular array of "pixels". Then the z-iteration
above is carried out separately for each c-point that is the center
of one of these pixels. If, after 50 (or perhaps 100) iterations, the
z-point is still within the circle of "escape radius" 2, then the
pixel represented by c is colored black. Otherwise, it is
colored white. For a more complete description see the slide
show that is provided.
Obviously a computer is required in order to construct an image
consisting of an array of (typically) tens of thousands of pixels on
a computer screen. But with a TI-83 calculator one can carry out the
z-iteration for a single c-point merely by pressing the Enter key
repeatedly. Hence MAT 104 students will be able to explore the
Mandelbrot using their graphing calculators, and computer programs
will be provided to those who want to go further.