A Hyperpolar Image of the Mandelbrot Set 139
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A HYPERPOLAR IMAGE OF THE MANDELBROT SET 139 A Hyperpolar Image of the Mandelbrot Set Frank Glaser Mathematics Andy J. Robles Mathematics Non-random complicated motions can exhibit a very rapid growth of errors and, despite perfect determinism, they inhibit the pragmatic ability to render accurate long term predictions. Technically, such non-random motions are termed “chaotic.” The geometry of chaotic behavior is known as “fractal geometry.” The term “fractal” was coined by Benoit Mandelbrot to mean “fractional dimension.” Fractals are structures which are elaborated upon at smaller and smaller scales differently at each point of an object. In 1979 a computer experiment made it possible to create the diagram of a certain object called the Mandelbrot set. It is a “limit fractal” that contains many other fractals. As such it is of great importance in fractal geometry. In this article we aim to map the Mandelbrot set from the normal complex plane into the hyperpolar complex plane and present a computer generated hyperpolar image of this “limit fractal.” In 1918 Gaston Julia (1893-1978) published a 199-page masterpiece entitled Memoire sur l’iteration des fonctiones rationelles. Pure et Appl. 8. (1918) 47-245. This paper deals with the iterations of complex polynomials like z2 + c, z3 + c, ..., zn+c, where c is a parameter such that for any fixed value of c, a sequence of complex numbers is generated when we choose some value for z and then start an iteration process. For example, if we have the polynomial z2+ c for fixed c, iterating means that we choose some value for z and obtain z2+ c. Next we substitute this resulting complex number into the polynomial obtaining (z2+ c)2 + c. Proceeding in this manner, we generate the sequence 2 2 2 2 2 2 z 0 = z, z1 = z + c, z2 = (z + c) + c, z3 = z + c + c + c, ... z 0 = z which can be written as 2 , (n = 0, 1, 2, ...). zn + 1 = z n + c This sequence of complex numbers is represented by a point-set in the complex plane. These types of point-sets are very hard to visualize and if we try to construct one by computing members of the sequence without a computer, we would get involved in endless, tedious computations. Indeed, the first visualization of such a set, now called a Julia set, was only partially achieved in 1925 by Hubert Cremer who had organized a seminar at the University of Berlin under the auspices of Erhard Schmidt and Ludwig Bieberbach. The participants were mathematicians of great renown such as Richard D. Brauer, Heinrich Hopf and Kurt Reidemeister. In a paper entitled über die Iteration rationaler Funktionen, Jahresberichte der Deutschen Mathematischen Vereinigung 33 (1925), Cremer produced a fractal consisting of thirteen triangles of different sizes and described a technique for generating more (Figure 1). 140 GLASER, ROBLES Fall 1998 Figure 1 First drawing by Cremer in 1925 visualizing a Julia set. First Visualization Due to the difficulty involved in visualizing Julia sets, the work of Julia was essentially forgotten until computer graphics became available at the end of the seventies, and Benoit Mandelbrot showed that Julia sets were a source of the some of the most beautiful fractals known today. (Peitgen, Jürgens & Saupe, 1992). Mandelbrot was born in Poland in 1924. His family emigrated to France in 1936 and his uncle, Szolem Mandelbrot, who was a professor of mathematics in Paris and the successor of Jacques Salomon Hadamard at the prestigious Collège de France, became responsible for his education. In 1945 Mandelbrot’s uncle recommended Julia’s paper to him as a master- piece and a source of good problems. However, Mandelbrot could not relate to Julia’s style and kind of mathematics and chose his own way which, nevertheless, brought him back to Julia’s work around 1977. It was in 1979 that Mandelbrot generated by computer graphics his famous “limit fractal,” the Mandelbrot set, which as a by-product, gave an ordering or “road map” for Julia sets. The Dichotomy of Julia Sets z = z A sequence of complex numbers such as 0 , (n = 0, 1, 2, ...) that defines 2 zn +1 = zn + c point-sets in the complex plane exhibits a basic dichotomy: either it becomes unbounded or it remains bounded. The collection of all points which lead to the first kind of behavior is called the escape set for the parameter c, while the collection of all points which lead to the second kind of behavior is called the prisoner set of c. Both sets cover some part of the complex plane and complement each other. The boundary of the prisoner set is simultaneously the boundary of the escape set, and that is the Julia set for c. Let us consider the example in which c= 0. Then we have the sequence z = z 0 , (n = 0, 1, 2, ...) 2 zn +1= z n A HYPERPOLAR IMAGE OF THE MANDELBROT SET 141 and we can construct choosing z0 = 0.8, z0 = 1.0, and z0 = 1.5, three sequences. The initial point z0 = 0.8, inside the unit circle, leads to a sequence which remains bounded forever. The point z0 = 1.0 leads to a sequence which remains forever on the unit circle. The point z0 = 1.5, which is outside the unit circle, leads to a sequence which escapes to infinity. This simple example shows an important dynamic dichotomy: the complex plane of initial values is subdivided into two subsets. The first subset collects points for which the iteration escapes; hence, it is called the escape set Ec of the parameter c. The iteration for all other initial values remains in a bounded region forever, and the resulting points are collected in a set called the prisoner set Pc of the parameter c. The parameter c = 0, P0 and E0 is the unit circle which is called the Julia set of the iteration. For initial values in the Julia set {z| |z| = 1}, the iteration generates only points which again lie on the unit cicle; i.e., the Julia set is invariant under iteration. The interior of P0 can be interpreted as a basin of attraction, the attractor being the origin z = 0. The iteration also has a second attractor: the point at infinity whose basin of attraction is the escape set E. The boundary between the two basins of attraction is the Julia set. For the parameter c = 0, the Julia set is the unit circle which is not a fractal. This is a very special case; in general most Julia sets are fractals. A basic example of a Julia set that is a fractal occurs if we choose c = -0.5 + 0.5i and compute the “orbits,” i.e. point sets, generated for the choices of z0 = 0 + 0i, z0 = 0.50 - 0.25i, and z0 = -0.25 + 0.50i and construct a table for their iterations . We construct also a second table for the iterations of the choices z0 = 1 + 0i, z0 = 0.50 + 0.25i, and z0 = 0.00 + 0.88i. In the first table, the iterated points do not escape but eventually settle down to a certain point whose value is approximately -0.408 + 0.275i, but in the second table all three orbits escape to infinity. There are again two basins of attraction, but zero is no longer one of the attracting points. The fixed point can be computed directly from the equation z2 - z + (-0.5 + 0.5i) = 0 The result gives two solutions z1 = 1.408 - 0.275i z2 = -0.408 + -.275i The first point is a repeller (i.e., points nearby are pushed away by the iteration) and z1 is in the Julia set. The second solution z2 is an attracting fixed point and, hence, not part of the Julia set. Many figures have been constructed that show the results of extended experiments which test all points of the complex plane. The prisoner set is pictured in black, and its complement, the escape set, is white. The boundary of the black region is the Julia set and it is a typical fractal. We can magnify any region near the Julia set and always find details that look similar. (Peitgen, H. O., Jürgens, H. & Saupe, D., 1992). The great variety of forms of Julia sets can be illustrated by producing the computer images of some basic examples. Figures showing the Julia sets for z2 + 0.25 + 0.52i, z2 + 0.377 - 0.248i, and z2 - 0.7382 + 0.0827i respectively have been generated by computer and are found below (Crownover, 1995). A fractal of this kind is usually called “Fatou” dust in honor of the mathematician Fatou, a contemporary of Julia. (Lauwerier, 1991) A HYPERPOLAR IMAGE OF THE MANDELBROT SET 143 The Mandelbrot Set And Its Hyperpolar Image 2 The Mandelbrot Set M for the quadratic polynomial fc(z) = z is defined as the collection of all complex numbers c for which the orbit of the point 0 is bounded. If C is the complex plane ε and c C , then the Mandelbrot set can be written as: z0 = c = {c ε C | the iteration remains bounded} M 2 zn + 1 = zn + c This was the definition used by Mandelbrot in his 1979 experiments. The above definition ofM is very similar to that of the prisoner set Pc. However, while the Julia set is part of the plane of initial values, z0 whose orbits reside in the complex plane, the Mandelbrot set is in the plane of parameter values c. The Mandelbrot set lies inside a circle of radius 2 because if | c | > 2, then the iteration escapes to infinity.