John D. Melville and Dr. Robert P. Burton, Computer Science
Considered the most complex object in mathematics, the Mandelbrot set has dazzled and mystified mathematicians and the public for more than twenty years. Unfortunately little progress has been made in understanding the phenomena that generate the Mandelbrot set. Recent discoveries by the Hyperspace Research Group (HRG) at BYU suggested a novel and plausible explanation for the complexity of the Mandelbrot Set as an infinite set of geometric objects. The purpose of the proposed project has been to investigate and develop the mathematics required to validate this theory.
We successfully demonstrated that the Mandelbrot set is an infinite set of geometric objects. Although we did not prove the circles and cardoids we saw are the same geometric regions discussed in our proof, empirical evidence suggests that it is likely. We proposed to find a test to determine in a given point is in the set, as opposed to not having diverged yet, as an intermediate goal. This test was not needed to accomplish our objective, and thus it was not pursued. I will first give the proof outline for our result, and then I will discuss the implications of our result on our understanding of fractals.
The Geometric Regions: To prove that the Mandelbrot set is an infinite set of geometric objects, I reference an appropriate definition of the Mandelbrot set. Peitgen, Jurgens, and Saupe report that the portion of the Mandelbrot set represented by any given orbit length can be expressed as the solution of two equations. The union of these solutions for all orbit lengths is the Mandelbrot set. Using theorems from algebraic geometry I prove that the solution set of the two equations is not fractal.
Let Mn(x,c) be the orbit polynomial of length n evaluated at x and c, as discussed in the proposal. Let M’n(x,c) be the first derivative with respect to x of Mn(x,c). Since Mn(x,c) is a polynomial, M’n(x,c) is also a polynomial.
According to Peitgen, Jurgens, and Saupe, the values of c that form a solution to Mn(x,c)=x and M’n(x,c) <= 1 for any x are the portion of the Mandelbrot set represented by orbit length n. If the <= is replaced with =, the solution becomes the border of the region.
I expand these equations into three real-valued polynomials in four real variables. Mn(x,c) is a polynomial and therefore can be expanded into its real and imaginary parts MnR(x,c) and MnI(x,c), respectively, such that (MnR(x,c) + i MnI+(x,c)) = Mn(x,c). I can do the same for the length of the derivative as well; however it is done differently, using the Pythagorean theorem to eliminate the length of the derivative. It is then easy to split the arguments of each polynomial into real and imaginary parts.
Now we have a system of three polynomials on four variables. Algebraic geometry tells us that the solution to these three equations is algebraic in R4. An algebraic figure cannot be the border of a fractal because fractals possess infinite detail, and algebraic figures do not. It is easy to show that the figures do not become fractal when we convert back to the complex plane. Hence, each bud on the Mandelbrot set is not a fractal.
The Noncycling Points: The previous discussion assumes that the points in the set enter a cycle. Jacobsen’s theorem guarantees that there exists a positive Lebesgue measure set of points that do not enter a cycle, and yet are still in the set. Thus it appears that the regions previously discussed do not cover the Mandelbrot set. In this section I demonstrate that the previous discussion completely accounts for the Mandelbrot set.
The central theme of this section is the equality portion of the second equation used above: Mn(x,c) <= 1. As mentioned earlier the area where the length of the derivative is one is the border of the region. This border region is where we find the noncycling points. This is convenient because we have already included the border region in the set. Thus we do not need an alternate explanation for the noncycling regions of the set.
It can be shown that any bud of the Mandelbrot set is surrounded by a region where the derivative has length one. The points on the border cannot enter a cycle. Because the length of the derivative is not less than one, these points do not converge to a fixed point. Thus they cannot enter cycles as all orbits in cycles converge to a fixed point under the orbit polynomial. However, the orbit also does not diverge because the orbit is not greater than one. Therefore the point is a chaotic point, one that neither enters a cycle nor diverges to infinity. The region surrounding any single bud is at least a curve; therefore it represents a set of positive Lebesgue measure, as required by Jacobsen’s theorem.
Having satisfied Jacobsen’s theorem, the proposed description covers the Mandelbrot set. The above definition satisfies all reported descriptions of the set. Algorithms that detect cycles directly produce graphics indistinguishable from the Mandelbrot set. Other authors also have reported that the buds of the Mandelbrot set appear to be correlated with orbit length. Thus, on a macroscopic level the orbit polynomial definition of the Mandelbrot set covers the set. As a future project, we hope to verify that this last point generalizes to a scales.
This research has raised many new questions about fractals. It is possible that the Mandelbrot set could be generated by plotting some subset of the geometric regions. However, I think the most significant aspect of this research is that it establishes a link between fractals and more traditional mathematics. This gives us a bridge from conduct further studies of this interesting field.