A Doheny Page
We start out with a number x, and a function (formula) applied to the number. For example, suppose x is the number; then 2·x is a function. If we start out with x = 1, then 2·1 = 2 when the function is applied to 1. So applying the function to a given number means we merely double the number (when the function is 2x). Next, we can iterate a function. Using the above example, we start with the number 1; the result of applying the function is 2. Now we take 2 and apply the function; we get four. Iterating means we start with a number, get another number when we apply the function, then apply the function to the number we got. Do this over and over, and we get successive iterations of a function on a number. Notice in the above example that if we start with 1, and apply the function, we get 2. Applying it to 2 we get 4, and so on. Successive iterates lead away from 1, our original number. Successive iterates can lead away from our original number, and never get closer to any one number. Or, successive iterates can lead to a particular number, even the one we started with. For example, if our function is x·x, and we start with x = 0.9, successive iterates will lead to zero. A short sequence of the iterates are: 0.9, 0.81, 0.6551, 0.43046721, ... The Mandelbrot set is made using the function z·z + c, where z is a complex number, and c is a real number constant. We pick a point in the complex plane (think of the complex plane as the Cartesian coordinate system, where the y-axis is the complex part, and the x-axis is the real part), and iterate it in the function just given. If successive iterates lead away from the original point, we color the starting point a certain color. If successive iterates don't lead away, then we color the starting point a different color. Notice that there are many different color points in the set, not just two. If successive iterates lead away from the starting point, they may do so slowly or quickly. We color the points depending on how slowly or quickly the iterates move away from them. (My apologies to mathematicians who feel uncomfortable with my method of explanation here; I don't mean to give a rigorous mathematical expose'.) A BASIC program to generate a Mandelbrot set (written in QBASIC, the Microsoft BASIC with DOS). rem set up the screen domain and range rem in the complex plane SCREEN (12) WINDOW (-1.5, -1)-(.5, 1) rem go through the screen point by point to determine rem the iteration "rate" ss = .1 10 FOR x = -1.5 TO .5 STEP ss FOR y = -1 TO 1 STEP ss t = x s = y FOR i = 0 TO 16 imf = 2 * t * s ref = t ^ 2 - s ^ 2 s = imf + y t = ref + x IF t ^ 2 + s ^ 2 > 4 THEN 40 30 NEXT i 40 LINE (x, y)-(x + ss, y + ss), i, BF LINE (x, -y)-(x + ss, (-y) + ss), i, BF 60 NEXT y NEXT x rem pick points closer togeher, and do the same as above rem to increase detail ss = .5 * ss GOTO 10 end
Kevin R. Doheny (kdoheny@carpet.daltonstate.edu)