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A HYPERPOLAR IMAGE OF THE MANDELBROT 139

A Hyperpolar Image of the Frank Glaser 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 of chaotic behavior is known as “ geometry.” The term “fractal” was coined by to mean “fractional dimension.” are structures which are elaborated upon at smaller and smaller scales differently at each point of an object. In 1979 a 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 into the hyperpolar complex plane and present a computer generated hyperpolar image of this “limit fractal.”

In 1918 (1893-1978) published a 199-page masterpiece entitled Memoire sur l’ des fonctiones rationelles. Pure et Appl. 8. (1918) 47-245. This paper deals with the of complex like z2 + c, z3 + c, ..., zn+c, where c is a parameter such that for any fixed value of c, a of complex numbers is generated when we choose some value for z and then start an iteration process. For example, if we have the z2+ c for fixed c, iterating means that we choose some value for z and obtain z2+ c. Next we substitute this resulting 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 , 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 each other. The 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 . 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 .

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 under iteration.

The interior of P0 can be interpreted as a basin of attraction, the 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 of the point 0 is bounded. If C is the complex plane and cεC , 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. This is easy to show; just assume that | z | ≥ | c | and | z | > 2. Then there exists someε > 0 such that | z | = 2 + ε . Apply the triangle inequality for complex numbers obtaining | z2 | = | z2 + c - c | ≥ | z2 + c | + | c |.

Then | z2 + c | ≥ | z2 | - | c | ≥ | z2 | - | z |

| z2 + c | ≥ (| z | - 1) | z | = (1 + e) | z |

The first iteration will increase in by the factor 1 + ε , and the K-th iteration will increase the absolute value by the factor (1 + ε )k which tends to infinity as K→ ∞. In the previous section we saw that for 0 < | c | < 1 , Julia sets are simple, closed, connected 4 fractal curves. It has also been shown that for real parameters c with Ð 2 ≤ | c | ≤ 1 , the 4 iteration is bounded and the Julia set is connected. Therefore, the interval Ð 2, 1 on the real 4 axis belongs to M and M itself is contained in a of radius 2 centered at the origin. The point c = -2 is the only point of the Mandelbrot set that has an absolute value of 2.

The fact that the Mandelbrot set is contained inside a circle of radius 2, with its center at the origin, allows us to transform its image into the hyperpolar plane by directly applying the hyperpolar transformation. (Glaser, 1992) The parameter c is written in terms of its real and imaginary parts as c = ℜ(c ) + i ℑm (c) and the iteration:

Im (c) ℜ ℑ ℜ u0 = e cos (c) = exp m (c) cos (c)

Im (c) ℜ ℑ ℜ v0 = e sin (c) = exp m(c) sin (c) ℜ ℑ 2 ℜ 2 un + 1= exp l m(zn + 1) cos (c) = exp m(zn+c) cos (zn+c)

ℜ ℑ 2 ℜ 2 vn + 1= exp l m (zn + 1) sin (c) = exp m (zn+c) sin (zn+c)

is performed and a computer generated hyperpolar image is constructed. The result is shown in Figure 9 where the hyperpolar image of M is compared with M itself. 144 GLASER, ROBLES Fall 1998

Figure 9 The Mandelbrot set and its hypolar image

Implementation of the Computer Program for the Hyperpolar Image of the Mandelbrot Set

For the Hyperpolar image of the Mandelbrot set, a modified version of the Mapping Method or Escape Time was used. This algorithm was chosen because it is simple to use and easy to modify to include the hyperpolar transformation. The disadvantage of this method is that the algorithm is inherently slow. In a 400x300 pixel screen with maximum iterations of 100, there are 120,000 points to check at up to 100 calculations per point. The program fragment is given below:

procedure Mapping begin deltaxPerPixel := (Right - Left) / (PackingFactor * Xscreen); deltayPerPixel := (Top - Bottom) / (PackingFactor * Yscreen); y := Bottom; for yRange := 0 to PackingFactor * Yscreen do begin if MandelbrotComputeAndTest(x, y) then begin u := exp(y) * cos(x); v := exp(y) * sin(x); SetUniversalPoint(u, v); end; x := x + deltaxPerPixel; end; y := y + deltayPerPixel; end; A HYPERPOLAR IMAGE OF THE MANDELBROT SET 145

The program was modified so that points of the Mandelbrot set were first determined by the procedure MandelbrotComputeAndTest. Each point was then transformed into the hyperpolar plane by u := exp(y)*cos(x) and v := exp(y)*sin(x) and plotted. The procedure SetUniversalPoint was used to plot the transformed u,v points onto the screen by first scaling the point to the screen size and then finding the associated pixel to color. The program also worked well in transforming contour lines but does not have zooming capabilities yet.

One initial disadvantage was the resolution of the image. Due to the exp(y) term in the transformation, it created a contraction for y < 0 and an expansion for y > 0. Thus for y > 0, two pixels next to each other in the complex Cartesian plane were mapped to two pixels not next to each other in their image in the hyperpolar plane. To remedy this problem, a method of “point packing” was created where the screen size (Xscreen, Yscreen) was not increased but the pixel lattice was increased by a user-defined factor called PackingFactor. While this greatly slowed the programs running time, it did fill in the gaps in the image and improve the resolution.

All programs were written using Symantec’s THINK Pascal 4.0.2 for Macintosh.

Conclusion

The results obtained in this article, as well as those obtained in two previous articles, (Glaser, 1996, 1997), indicate that side by side with the normal fractal geometry, there is also a hyperpolar form of fractal geometry which gives the hyperpolar images of fractals. The mapping of hyperpolar transformation will be one-to-one provided the fractal mapped is contained in the Cartesian or in the complex plane inside a circle of radius less than p; otherwise there will be overlapping as in the case of hyperpolar circles of radii larger than or equal to p. (Glaser, 1992). The hyperpolar image of the Mandelbrot set is one-to-one with that of M because M is contained inside a circle of radius 2. 146 GLASER, ROBLES Fall 1998

References Becker, Karl-Heinz and Dorfler, Michael. (1991). Dynamical systems and fractals: Computer graphics experiments in Pascal. Cambridge University Press. Crownover, . M. (1995). Introduction to fractals and chaos, Jones and Bartlett Publishers. Glaser, F. (1992). "Images of Cartesian Lines and Circles in the Hyperpolar Plane". The Cal Poly Scholar, Volume 5 (Fall). Glaser, F. (1996).The Sierpinski Gasket and a in the Hyperpolar Plane. The Journal of Interdisciplinary Studies, Vol. 9 (Fall). Glaser, F. (1997). "The Universality of the Hyperpolar Images of the and the ". The Journal of Interdisciplinary Studies, Vol. 10 (Fall). Lauwerier, H. (1991). Fractals. Princeton University Press. Peitgen, H. O., Jürgens, H., and Saupe, D. (1992). "Chaos and Fractals". New Frontiers of Science. Springer-Verlag.