Each of the fractals we discuss here is generated by a single equation and may be classified into one of the three categories, which I call Fatou, Julia, and Newton patterns, using the names of giants in the history of mathematics contributed to fractal geometry. I should warn the audience at this stage, however, that the terms such as Fatou pattern are used for convenience in this site and not universally adopted in the field. The prerequisite for the subjects is to know (1) fundamental algebra and geometry of complex numbers, (2) beginning calculus, and (3) basic computer programming in such language as Java in a graphics mode. I use a modernized “object-oriented” version of Pascal called Delphi marketed by Borland. (1) includes the practice of writing a complex number z as a point (x, y) in the xy-plane as well as the standard algebraic expression z = x + yi. At the outset of each plotting process, we choose a rectangular canvas, part or whole of which will appear in the computer’s display screen and which will be filled with a fractal at the end. The canvas comprises finitely many small rectangular picture elements, a.k.a. pixels. One of the basic routines in (3) (usually given in a programming manual) is a simple change of coordinates based on the “point-slope” equations of lines, which allows us to convert the canvas to a rectangular region R in the xy-plane consisting of points, er complex numbers (x, y). As in a basic graphics calculator routine, four real numbers called xmin, xmax, ymin, and ymax bound R.
For the sake of simplicity in our argument, we will identify each pixel in the canvas and the corresponding complex number in R, and consequently, regard the canvas and R to be the same. In particular, the number of complex numbers we consider is finite so a computer can deal with “all” of them in R. Also, at the beginning of the process, we choose an equation such as zn+1= zn2 + p involving two indexed variables and the third variable p called a parameter. After following the program, the computer will light up each pixel (complex number) in the canvas with a color determined by a simple interaction between the complex number (the pixel) and the equation and eventually display a fractal image on the canvas. Here comes an example:
(A) The Divergence Scheme
We begin our discussion with the rectangular region R in the xy-plane
bounded by say xmin = -2.1, xmax = 2.1, ymin = -2.1, and ymax = 2.1 and the equation
(*)
zn+1 = zn2 + p ,
where the parameter p is a complex number in R (which is our canvas). (*) is called the Mandelbrot equation or Fatou equation (sometimes) as Pierre Fatou came up with the following idea nearly a century ago.
The best-known figure in fractal geometry is given by iterating the Fatou equation using the initial value z0= (0, 0) for n = 0, 1, 2, ... , M, and for “every” parameter p in R. Here M is a prescribed maximum number of iterations that rescues the computer from getting trapped in an infinite loop, and M = 500 is used in this example. M is almost always a number between 50 and 500,000 but it occasionally gets as large as 2,000,000 in my program.
Assuming that our object is to plot a picture in black and red with a background or canvas color say white, our painting scheme is basically given by the IF statement:
If
|z1| > 2 then color the pixel p black,
else if |z2| > 2 then color the pixel p red,
else if |z3| > 2 then color the pixel p black,
else if |z4| > 2 then color the pixel p red,
etc.,
where |zn| stands for the modulus or absolute value of zn. It can be shown that |zn| > 2
for some n if and only if the sequence |zn| tends
to infinity as n increases; this implies that the above scheme assigns a color to each parameter p (which is a pixel, remember?) according to how fast the complex number zn escapes from the circle of radius 2 before taking a long journey toward infinity. This basic scheme can be streamlined in the actual program by using:
just two variables zold and znew instead of the large array z0 , z1 , z2 , ...;
|zn|2 > 4 in place of |zn|
> 2 to avoid the hidden square root operation; and
MOD operation to replace the awkward IF statement by a few good looking lines.
Also, the use of any radius greater than 2 (say 100) in place of 2 in the above scheme works just as well and in some cases even better. Computer programming has two faces, art and science, and requires artful wits as well as rigidly logical thinking. Writing, streamlining and debugging computer programs, therefore, provide us with an ideal way of exercising our brain and maintaining normal connections among the neurons. I would recommend it highly to people with PCs to take up computer programming.
As we noted, the color of a pixel is determined according to the "escape speed" of the corresponding sequence zn = z0 , z1 , z2 , ... from a prescribed circle about the origin. Since all of the escaping sequences diverge to infinity, I call the coloring scheme (without limiting a number of colors) the divergence scheme. Figure 2 is given by the above basic red-black divergence scheme, and the area that retains the white canvas color depicts the famous Mandelbrot set (or M set, for short). The set consists of the parameters (pixels) p attached to the sequences zn that stay within the circle forever regardless of the value of n. If we alter the values of xmin, etc. and run the same program to zoom in on a small area of the plot near the boundary of the Mandelbrot set, we will discover the self-similarity property of fractals along with a hairy characteristic of the "snowman". Many microscopic shapes resembling the snowman are connected by a network through curious and extremely intricate patterns, which we can find by zooming intently and patiently; for example, see the
Mandelbrot Set or Elephant M Set in the gallery. Not surprisingly, it has been verified that the Mandelbrot set is one of the most complex figures ever plotted on a piece of paper (in terms of the so-called Hausdorff-Besicovitch dimension and other advanced properties).
Here is the very essence of our fractal plotting:
The closer the zoomed area is to the boundary, the finer the fractal pattern.
The reason, which leads us to the idea of chaos, is that parameters p near the boundary often come from sequences
zn with diverse properties interlaced in a complex fashion. Thus, a very slight change in the "starting" value p = z02 + p in this area may result in an utterly different "future behavior" of zn which dictates its escape speed and the pixel color. Yes, these sequences zn are just like people, most of whom would behave unpredictably when they are placed near an infernal border between life and death and cause chaos for the whole population in the area. What is interesting in mathematics is that chaos near the border between life (staying within the circle) and death (diverging to infinity) stems from an equation such as (*) which looks totally well behaving and instead of ugly consequences it produces beautiful fractal patterns. You might remember that Steven Spielberg's Jurassic Park premiered in 1993 has a mathematician who introduced himself as a “chaotician.” A chaotician obviously deals with chaotic behaviors of various dynamic mathematical systems, a.k.a. dynamical systems, including our generating equation (*). It was in fact around the 1970s and the early 80s when several significant events coincidentally took place in mathematics: People saw the computer-generated bizarre-looking and epoch-making Mandelbrot set for the first time; the new field called "chaos" was born from a computer experiment and created a great sensation among younger mathematicians; powerful computers found their way to become main tools among theoretical mathematicians who used to pride themselves on using only pencil and paper to discover recondite theories. Like in chemistry and physics, experimental investigations are now important part of mathematics.
(C) Fatou Patterns
Since it was published in 1980, the Mandelbrot set became so popular that zillions of digital artists, mathematicians, and computer programmers and hackers have explored around it and shown their fractal images on a variety of objects including web pages, posters, book covers, T-shirts, and coffee mugs. Although the complexity of the M set is boundless and the hidden beauty inexhaustible, it has become quite a challenging task to unearth new patterns from the Fatou (or Mandelbrot) equation (*) using available computers and software. Consequently, creative work calls for modification of this formula, and there are infinitely many formulas available for this purpose. I call a fractal given by the divergence/convergence scheme applied on a dynamical system of the form
(**)
zn+1 = fp(zn)
a Fatou Pattern, where fp is a function of complex numbers that contains a parameter p. For example, we may find
Fatou Patterns using the logistic equation
zn+1 = fp(zn) = p(1 - zn) zn .
An initial value z0 is usually chosen from the so-called critical points of the function fp at which the derivative vanishes, but this is not necessarily mandatory from an artistic viewpoint. As you can see in the picture (D) Julia Patterns
A fractal, which I call a Julia pattern,
is given by fixing the parameter value p in the dynamical system (**) and
iterating it for every choice of the initial value z0 (instead of p in plotting Fatou Patterns) in the rectangular region R (which is our canvas). Thus, z0 plays the role of a pixel in the canvas. As in the case of a Fatou Pattern, a Julia pattern appears more interesting when the parameter value
p is chosen in an area where the sequences zn
behave unpredictably. A Julia pattern with an appropriate p value contains what appears to be the boundary between two sets, one consisting of every z0 which keeps the sequence zn within a fixed circle and the second set consisting of every other z0. The boundary is called a Julia set, which is an important object in fractal geometry. Gaston Julia, Pierre Fatou, and several other mathematicians, amazingly envisioned fractal images generated by certain dynamical systems about a hundred years ago using sheer imaginations.
Gatekeepers in Figure 1 above, for example, is a Julia pattern given by
zn+1= zn3 + zn + p,
p = (0.185, 0.00007666),
on the rectangular region with xmin = -1.1, xmax = 1.1, ymin = -0.7, and ymax = 0.7. The gatekeepers are painted by the convergence scheme with various shades of gold and the red background by the divergence scheme; thus, the Julia set in this pattern is depicted by the outlines (or the boundary) of the gatekeepers. Once you find a value of p yielding a nice Julia pattern, change the value slightly to see if you can improve the picture while retaining the same motif. The corresponding change in the picture sometimes appears gradual and sometimes quite dramatic as you can see in
