ART MATRIX PO 880 Ithaca, NY 14851-0880 USA (607) 277-0959, Fax (607) 277-8913 'The Paths of Lovers Cross in the Line of Duty.' MANDELBROT SETS AND JULIA SETS. Copyright (C) 1985 by Homer Wilson Smith Consider the equation Y equals Z squared (Y = Z*Z) on the positive real number line from 0 to infinity. If Z starts off as a number bigger than 1, Y becomes a number even bigger. For example 2 squared is 4, 4 squared is 16, 16 squared is 256. This sequence soon approaches infinity. This process of using the OUTPUT as the next INPUT is called FORWARD ITERATION of the equation Y = Z*Z. For Y = Z*Z, infinity is ATTRACTIVE for all numbers greater than 1. Infinity is also a FIXED POINT because infinity squared is infinity (sort of). If Z starts off as less than 1, Y becomes smaller and smaller approaching 0. For example, .5 squared is .25, .25 squared is .0625 etc. For Y = Z*Z, 0 is ATTRACTIVE for all numbers less than 1. 0 is also a FIXED POINT because 0 squared is 0. The number 1 is a very important number because it forms a dividing line between those numbers that are REPELLED from it towards 0 and those that are REPELLED from it towards infinity. Yet 1 squared is 1, so 1 is also a FIXED POINT. 1 is a REPULSIVE FIXED POINT where 0 and infinity are ATTRACTIVE FIXED POINTS. 1 is UNSTABLE because any number even infinitesimally different from 1 will decay rapidly inward towards the center or outward towards infinity. Expanding this concept to include the entire COMPLEX PLANE, the number 1 becomes a CIRCLE of radius 1, called a JULIA SET, named after Gaston Julia, a French mathematician. This is shown in Fig. J1 on the other side. Using standard rules for COMPLEX MULTIPLICATION, any number chosen INSIDE the circle will FORWARD ITERATE inward towards 0 at the center, and any number chosen OUTSIDE the circle will FORWARD ITERATE outward towards infinity, and any number chosen ON the circle will FORWARD ITERATE to some other number ON the circle. This is shown by the green dots and red dots in Fig J1. The green dot outside the circle iterates to the red dot further outside the circle, etc. Thus there are three distinct regions. INSIDE the JULIA SET, ON the JULIA SET, and OUTSIDE the JULIA SET. Each region has its own FIXED POINT. 0 is the FIXED POINT for INSIDE the JULIA SET, 1 is the FIXED POINT for ON the JULIA SET and infinity is the FIXED POINT for OUTSIDE the JULIA SET. Notice that although 1 is a REPULSIVE FIXED POINT it does ATTRACT points ON the JULIA set just as it ATTRACTS itself. The Julia set is an extension of 1, and so acts similar to 1. Fig. J1 is called a Z-SPACE picture because it is in the space of all possible Z's. In this case Z ranges from -2 to 2 on both axes with 0 in the center. J1 is actually computed for the equation Y = Z*Z + C where C is 0. The question arises, what happens if C is NOT 0? The pretty color picture in the upper left corner is the MANDELBROT SET, named after Benoit Mandelbrot of IBM. It is a C-SPACE picture because it is the space of all possible C's. C = (0,0) is marked clearly by the + next to the letters J1. If C is chosen on the MANDELBROT SET from the point marked J2, then the JULIA SET of Fig. J2 becomes evident. Infinity is an ATTRACTIVE FIXED POINT for all JULIA SETS, but the two PAGE 2 yellow FIXED POINTS on J2 have moved off their original values of 1 and 0 to become some other numbers. Since both are now ON the JULIA SET, the are both REPULSIVE. Any point starting INSIDE this JULIA SET FORWARD ITERATES towards the two red points. The ITERATION ALTERNATES first near one, then near the other. Every TWO ITERATIONS the point is back near where it was. In Fig. J3 there is an ATTRACTIVE CYCLE of THREE POINTS. Notice how they surround the original yellow FIXED POINT that is ON the JULIA SET. J2 is taken from the TWO-BALL of the MANDELBROT SET, so called because all JULIA SETS taken from within this region have an ATTRACTIVE CYCLE OF TWO, and J3 is taken from the THREE- BALL. J32 is taken from the TWO-BALL off the THREE-BALL. This produces an ATTRACTIVE CYCLE of SIX POINTS. J0 is taken from the COLORED area of the MANDELBROT SET picture. Notice its JULIA set is not a CLOSED CURVE. Thus points INSIDE the JULIA SET escape to infinity as well as points OUTSIDE. This kind of JULIA SET is called a CANTOR SET and is composed of DUST. No point ON this JULIA SET is CONNECTED to any other point. Thus there is no CLOSED CURVE to fence in BASINS of ATTRACTION with ATTRACTIVE CYCLE POINTS in the middle. Which brings us to how the MANDELBROT SET was computed. For Y = Z*Z + C there is a theorem which says that if the JULIA SET is CLOSED and therefore there does exist a CYCLIC BASIN of ATTRACTION then 0 will fall into it. That is if you start with 0 and FORWARD ITERATE Y = Z*Z + C, then the points will be attracted to whatever ATTRACTIVE BASIN exists. If there is NO ATTRACTIVE BASIN because the JULIA SET is DUST, then 0 will go towards infinity very quickly as in Fig. J0. The MANDELBROT SET picture is colored according to how fast 0 escapes to infinity for all C's where the JULIA SET is DUST. If the JULIA SET is CLOSED and 0 finds an ATTRACTIVE BASIN, then the point will not have escaped even after 1000 iterations, and so there the MANDELBROT SET is colored BLACK. Each BALL on the MANDELBROT SET is surrounded by other BALLS which are surrounded by other BALLS ad infinitum. Each BALL relates to how many CYCLE POINTS the BASINS of ATTRACTION contain in the JULIA SET for that value of C. During FORWARD ITERATION the JULIA SET REPELS all points not ON it. In BACKWARD ITERATION the JULIA SET ATTRACTS all points not ON it. The equation for the BACKWARD ITERATION of Z1 = Z0*Z0 + C is Z0 = plus or minus the SQRT(Z1 - C). SQRT is the SQUARE ROOT function. This is shown in Fig. J3 where the red starting point BACKWARD ITERATES into two green BACKWARD IMAGES. Both green BACKWARD IMAGES would FORWARD ITERATE to the same red FORWARD IMAGE (starting point). Note that the BACKWARD IMAGES are closer to the JULIA set than the starting point. Repeated application of BACKWARD ITERATION to each of these points would create more points closer to the JULIA SET. This works from the INSIDE of the JULIA SET as well. INSIDE, points that would normally fall inward under FORWARD ITERATION will move outward towards the JULIA SET under BACKWARD ITERATION. The JULIA SETS created for this demonstration were generated by taking the repeated BACKWARD IMAGES of a point already ON the JULIA SET, namely the right hand yellow FIXED POINT. The BACKWARD IMAGES of this FIXED POINT are itself and another point. It is from this other point that the further BACKWARD IMAGES are taken. The application of all this is to systems which are a function of themselves and something else. That is, any system of Z's where Z = f(Z,C). Where Z at the next moment of time is equal to some function of Z at the previous moment of time plus all other influencing factors. PAGE 3 For example the number of moths in a forest is clearly a function of the number of moths in the forest just prior and all the other factors that affect how they breed, eat and get eaten, and grow and die. For a further discussion of this please read "The Cell and the Womb", obtainable from ART MATRIX. -(C) 1985 HWS PAGE 4 This page left blank for your comments.